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## 2.2: The SAS Theorem

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• Page ID 34125

• Henry Africk
• CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

We have said that two triangles are congruent if all their correspond­ ing sides and angles are equal, However in some cases, it is possible to conclude that two triangles are congruent, with only partial information about their sides and angles.

Suppose we are told that $$\triangle ABC$$ has $$\angle A = 53^{\circ}$$, $$AB = 5$$ inches, and $$AC = 3$$ inches. Let us attenpt to sketch $$\triangle ABC$$. We first draw an angle of $$53^{\circ}$$ with a protractor and label it $$\angle A$$. Using a ruler, we find the point 5 inches from the vertex on one side of the angle and label it $$B$$, On the other side of the angle, we find the point 3 inches from the vertex and label it $$C$$, See Figure $$\PageIndex{1}$$, There is now only one way for us to complete our sketch of $$\triangle ABC$$, and that is to connect points $$B$$ and $$C$$ with a line segment, We could now measure $$BC$$, $$\angle B$$, and $$\angle C$$ to find the remaining parts of the triangle.

Suppose now $$\triangle DEF$$ were another triangle, with $$\angle D = 53^{\circ}$$, $$DE = 5$$ inches, and $$DF = 3$$ inches. We could sketch $$\triangle DEF$$ just as we did $$\triangle ABC$$, and then measure $$EF$$, $$\angle E$$, and $$\angle F$$ (Figure $$\PageIndex{2}$$). It is clear that we must have $$BC = EF$$, $$\angle B = \angle E$$, and $$\angle C = \angle F$$ because both triangles were drawn in exactly the same way. Therefore $$\triangle ABC \cong \triangle DEF$$.

• In $$\triangle ABC$$, we say that $$\angle A$$ is the angle included between sides $$AB$$ and $$AC$$.
• In $$\triangle DEF$$, we say that $$\angle D$$ is the angle included between sides $$DE$$ and $$DF$$.

Our discussion suggests the following theorem:

## Theorem $$\PageIndex{1}$$ (SAS or Side-Angle-Side Theorem)

Two triangles are congruent if two sides and the included angle of one are equal respectively to two sides and the included angle of the other,

In Figure $$\PageIndex{1}$$ and $$\PageIndex{2}$$, $$\triangle ABC \cong \triangle DEF$$ because $$AB, AC$$, and $$\angle A$$ are equal respectively to $$DE, DF$$ and $$\angle D$$.

We sometimes abbreviate Theorem $$\PageIndex{1}$$ by simply writing $$SAS = SAS$$.

## Example $$\PageIndex{1}$$

In $$\triangle PQR$$ name the angle included between sides

• $$PQ$$ and $$QR$$,
• $$PQ$$ and $$PR$$,
• $$PR$$ and $$QR$$,

Note that the included angle is named by the letter that is common to both sides, For (1), the letter "$$Q$$" is common to $$PQ$$ and $$QR$$ and so $$\angle Q$$ is included between sides $$PQ$$ and $$QR$$. Similarly for (2) and (3).

Answer : (1) $$\angle Q$$, (2) $$\angle P$$, (3) $$\angle R$$.

## Example $$\PageIndex{2}$$

For the two triangles in the diagram

• list two sides and an included angle of each triangle that are respectively equal, using the infonnation given in the diagram,
• write the congruence statement,

and (3) find $$x$$ by identifying a pair of corresponding sides of the congruent triangles.

(1) The angles and sides that are marked the same way in the diagram are assumed to be equal, So $$\angle B$$ in $$\triangle ABD$$ is equal to $$\angle D$$ in $$\triangle BCD$$. Therefore, "$$B$$" corresponds to "$$D$$." We also have $$AB = CD$$. Therefore "$$A$$" must corresponds to "$$C$$". Thus, if the triangles are congruent, the correspondence must be

Finally, $$BD$$ (the same as $$DB$$) is a side common to both triangles, Summaryzing,

$$\begin{array} {ccrclcl} {} & \ & {\underline{\triangle ABD}} & \ & {\underline{\triangle CDB}} & \ & {} \\ {\text{Side}} & \ & {AB} & = & {CD} & \ & {\text{(marked = in diagram)}} \\ {\text{Included Angle}} & \ & {\angle B} & = & {\angle D} & \ & {\text{(marked = in diagram)}} \\ {\text{Side}} & \ & {BD} & = & {DB} & \ & {\text{(common side)}} \end{array}$$

(2) $$\triangle ABD \cong \triangle CDB$$ because of the SAS Theorem ($$SAS = SAS$$).

(3) $$x = AD = CB = 10$$ because $$AD$$ and $$CB$$ are corresponding sides (first and third letters in the congruence statement) a.~d corresponding sides of congruent triangles are equal.

(1) $$AB$$, $$\angle B$$, $$BD$$ of $$\triangle ABD = CD$$, $$\angle D$$, $$DB$$ of $$\triangle CDB$$.

(2) $$\triangle ABD \cong \triangle CDB$$.

(3) $$x = AD = CB = 10$$.

## Example $$\PageIndex{3}$$

• list two sides and an included angle of each triangle that are respectively equal, using the information given in the diagram.
• write the congruence statement, and
• find $$x$$ and $$y$$ by identifying a pair of corresponding sides of the congruent triangles.

(1) $$AC = CE$$ and $$BC = CD$$ because they are mar!,;:ed the same way. We also know that $$\angle ACB = \angle ECD = 50^{\circ}$$ because vertical angles are equal. Therefore "$$C$$" in $$\triangle ABC$$ corresponds to "$$C$$" in $$\triangle CDE$$. Since $$AC = CE$$, we must have that "$$A$$" in $$\triangle ABC$$ corresponds to "$$E$$" in $$\triangle CDE$$. Thus, if the triangles are congruent, the correspondence must be

We summarize:

$$\begin{array} {ccrclcl} {} & \ & {\underline{\triangle ABC}} & \ & {\underline{\triangle EDC}} & \ & {} \\ {\text{Side}} & \ & {AC} & = & {EC} & \ & {\text{(marked = in diagram)}} \\ {\text{Included Angle}} & \ & {\angle ACB} & = & {\angle ECD} & \ & {\text{(vertical angles are =)}} \\ {\text{Side}} & \ & {BC} & = & {DC} & \ & {\text{(marked = in diagram)}} \end{array}$$

(2) $$\triangle ABC \cong \triangle EDC$$ because of the SAS theorem. ($$SAS = SAS$$)

(3) $$\angle A = \angle E$$ and $$\angle B = \angle D$$ because they are corres:9onding angles of the congruent triangles. $$\angle D = 85^{\circ}$$ because the sum of the angles of $$\triangle EDC$$ must be $$180^{\circ}$$. ($$\angle D = 180^{\circ} - (50^{\circ} + 45^{\circ}) = 180^{\circ} - 95^{\circ} = 85^{\circ}$$). We obtain a system of two equations in the two unknowns $$x$$ and $$y$$:

Substituting for $$x$$ in the first original equation,

$\begin{array} {rcl} {2x + y} & = & {45} \\ {2(20) + y} & = & {45} \\ {40 + y} & = & {45} \\ {y} & = & {45- 40} \\ {y} & = & {5} \end{array}$

• $$AC$$, $$\angle ACB$$, $$BC$$ of $$\triangle ABC$$ = $$EC, \angle ECD, DC$$ of $$\triangle EDC$$.
• $$\triangle ABC \cong \triangle EDC$$.
• $$x = 20, y = 5$$.

## Example $$\PageIndex{4}$$

The following procedure was used to measure the d.istance AB across a pond: From a point $$C$$, $$AC$$ and $$BC$$ were measured and found to be 80 and 100 feet respectively. Then $$AC$$ was extended to $$E$$ so that $$AC = CE$$ and $$BC$$ was extended to $$D$$ so that $$BC = CD$$. Finally, $$DE$$ we found to be 110 feet.

• Write the congruence statement.
• Give a reason for (1).
• Find $$AB$$.

(1) $$\angle ACB = \angle ECD$$ because vertical angles are equal. Therefore the "$$C$$'s" correspond, $$AC = EC$$ so $$A$$ must correspond to $$E$$. We have

(2) $$SAS = SAS$$. Sides $$AC$$, $$BC$$, and included angle $$C$$ of $$ABC$$ are equal respectively to $$EC, DC$$, and included angle $$C$$ of $$\angle EDC$$.

(3) $$AB = ED$$ ecause they are corresponding sides of congruent triangles, Since $$ED = 110$$, $$AB = 110$$.

(1) $$\triangle ABC \cong \triangle EDC$$.

(2) $$SAS = SAS$$: $$AC$$, $$\angle C$$, $$BC$$ of $$\triangle ABC = EC$$, $$\angle C$$, $$DC$$ of $$\triangle EDC$$.

(3) $$AB = 110 feet$$.

## Historical Note

The SAS Theorem is Proposition 4 in Euclid's Elements , Both our discussion and Suclit's proof of the SAS Theoremimplicitly use the following principle: If a geometric construction is repeated in a different location (or what amounts to the same thing is "moved" to a different location) then the size and shape of the figure remain the same, There is evidence that Euclid used this principle reluctantly, and many mathematicians have since questioned its use in formal proofs, They feel that it makes too strong an assumption about the nature of physical space and is an inferior form of geometric reasoning. Bertrand Russell (1872 - 1970), for example, has suggested that we would be better off assuming the SAS Theorem as a postulate , This is in fact done in a system of axioms for Euclidean geometry devised by David Hilbert (1862 - 1943), a system that has gained much favor with modern mathematicians. Hilbert was the leading exponent of the "formalist school," which sought to discover exactly what assumptions underlie each branch of mathematics and to remove all logical ambiguities, Hilbert's system, however, is too formal for an introductory course in geometry,

1 - 4. For each of the following (1) draw the triangle with the two sides and the included angle and (2) measure the remaining side and angles:

1. $$AB = 2$$ inches, $$AC = 1$$ inch, $$\angle A = 60^{\circ}$$.

2. $$DE = 2$$ inches, $$DF = 1$$ inch, $$\angle D = 60^{\circ}$$.

3. $$AB = 2$$ inches, $$AC = 3$$ inches, $$\angle A = 40^{\circ}$$.

4. $$DE = 2$$ inches, $$DF = 3$$ inches, $$\angle D = 40^{\circ}$$.

5 - 8. Name the angle included between sides

5. $$AB$$ and $$BC$$ in $$\triangle ABC$$.

6. $$XY$$ and $$YZ$$ in $$\triangle XYZ$$.

7. $$DE$$ and $$DF$$ in $$\triangle DEF$$.

8. $$RS$$ and $$TS$$ in $$\triangle RST$$.

9 - 22. For each of the following.

(1) list two sides and an included angle of each triangle that are respectively equal, using the information given in the diagram,

(2) write the congruence statement,

(3) find $$x$$, or $$x$$ and $$y$$.

Assume that angles or sides marked in the same way are eQual.

## Triangle Congruence – SSS and SAS

We have learned that triangles are congruent if their corresponding sides and angles are congruent. However, there are excessive requirements that need to be met in order for this claim to hold. In this section, we will learn two postulates that prove triangles congruent with less information required. These postulates are useful because they only require three corresponding parts of triangles to be congruent (rather than six corresponding parts like with CPCTC). Let’s take a look at the first postulate.

## SSS Postulate (Side-Side-Side)

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

As you can see, the SSS Postulate does not concern itself with angles at all. Rather, it only focuses only on corresponding, congruent sides of triangles in order to determine that two triangles are congruent. An illustration of this postulate is shown below.

We conclude that ?ABC??DEF because all three corresponding sides of the triangles are congruent.

Let’s work through an exercise that requires the use of the SSS Postulate.

The only information that we are given that requires no extensive work is that segment JK is congruent to segment NK . We are given the fact that A is a midpoint, but we will have to analyze this information to derive facts that will be useful to us.

In the two triangles shown above, we only have one pair of corresponding sides that are equal. However, we can say that AK is equal to itself by the Reflexive Property to give two more corresponding sides of the triangles that are congruent.

Finally, we must make something of the fact A is the midpoint of JN . By definition, the midpoint of a line segment lies in the exact middle of a segment, so we can conclude that JA?NA .

After doing some work on our original diagram, we should have a figure that looks like this:

Now, we have three sides of a triangle that are congruent to three sides of another triangle, so by the SSS Postulate , we conclude that ?JAK??NAK . Our two column proof is shown below.

We involved no angles in the SSS Postulate , but there are postulates that do include angles. Let’s take a look at one of these postulates now.

## SAS Postulate (Side-Angle-Side)

If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.

A key component of this postulate (that is easy to get mistaken) is that the angle must be formed by the two pairs of congruent, corresponding sides of the triangles. If the angles are not formed by the two sides that are congruent and corresponding to the other triangle’s parts, then we cannot use the SAS Postulate . We show a correct and incorrect use of this postulate below.

The diagram above uses the SAS Postulate incorrectly because the angles that are congruent are not formed by the congruent sides of the triangle.

The diagram above uses the SAS Postulate correctly. Notice that the angles that are congruent are formed by the corresponding sides of the triangle that are congruent.

Let’s use the SAS Postulate to prove our claim in this next exercise.

For this solution, we will try to prove that the triangles are congruent by the SAS Postulate . We are initially given that segments AC and EC are congruent, and that segment BC is congruent to DC .

If we can find a way to prove that ?ACB and ?ECD are congruent, we will be able to prove that the triangles are congruent because we will have two corresponding sides that are congruent, as well as congruent included angles. Trying to prove congruence between any other angles would not allow us to apply the SAS Postulate .

The way in which we can prove that ?ACB and ?ECD are congruent is by applying the Vertical Angles Theorem . This theorem states that vertical angles are congruent, so we know that ?ACB and ?ECD have the same measure. Our figure show look like this:

Now we have two pairs of corresponding, congruent sides, as well as congruent included angles. Applying the SAS Postulate proves that ?ABC??EDC . The two-column geometric proof for our argument is shown below.

• High School Advanced Geometry Help - Tangents and Circles inscribed in a Triangle
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## SAS- Side Angle Side Congruence and Similarity

The word "congruent" for figures means equal in every aspect, majorly in terms of shape and size. The relation of two congruent figures is described by congruence. Congruence is the term used to describe the relation of two figures that are congruent. Now let's discuss the SAS congruence of triangles. In order to be sure that the two triangles are similar, we do not necessarily need to inquire about all sides and all angles . Congruence of triangles means:

• All corresponding angle pairs are equal.
• All corresponding sides are equal.

In this mini-lesson, we will learn about the SAS similarity theorem in the concept of the SAS rule of congruence, using similar illustrative examples.

## What do you mean by Side Angle Side?

SAS congruence is the term which is also known as Side Angle Side congruence, which is used to describe the relation of two figures that are congruent. Let's discuss the SAS congruence of triangles in detail to understand the meaning of SAS. Look at ΔABC and ΔPQR:

These two triangles are of the same size and shape. Thus, we can say that these are congruent. They can be considered as congruent triangle examples. We can represent this in a mathematical form using the congruent triangles symbol (≅). ( ΔDEF≅ΔPQR). This means D falls on P, E falls on Q, and F falls on R. ED falls on PQ, EF falls on QR, and DF falls on PR. Thus, we can conclude that the corresponding parts of the congruent triangles are equal.

## SAS Criterion

Under this criterion, if the two sides of a triangle are equal to the two sides of another triangle, and the angle formed by these sides in the two triangles are equal, then these two triangles are congruent. The SAS Criterion stands for the 'Side-Angle-Side' triangle congruence theorem.

## The SAS Congruence Rule

The Side-Angle-Side theorem of congruency states that, if two sides and the angle formed by these two sides are equal to two sides and the included angle of another triangle, then these triangles are said to be congruent.

Verification:

Let's perform an activity to show the proof of SAS. Given: AB=PQ, BC=QR, and ∠B=∠Q. To prove: ΔABC ≅ ΔPQR

Place the triangle ΔABC over the triangle ΔPQR such that B falls on Q and side AB falls along the side PQ.

• Since AB=PQ, so point A falls on point P.
• Since ∠B=∠Q, so the side BC will fall along the side QR.
• BC=QR, so point C falls on point R. Thus, BC coincides with QR and AC coincides with PR.

So, ΔABC will coincide with ΔPQR. Therefore, ΔABC≅ΔPQR. This demonstrates SAS criterion of congruence.

## The SAS Similarity Rule

The SAS similarity criterion states that If two sides of one triangle are respectively proportional to two corresponding sides of another, and if the included angles are equal, then the two triangles are similar.

Given: DE/AB=DF/AC and ∠D=∠A. To prove: ΔDEF is similar to ΔABC The SAS criterion tells us that ΔABC ~ ΔDEF. Let us see the justification of this.

Construction:

• Take a point X on AB such that AX = DE.
• Through X, draw segment XY∥ BC, intersecting AC at Y.

Since XY II BC, we can note that ΔAXY ~ ΔABC, and thus: AX/AB = AY/AC....(1)

Now, we will show that ΔAXY and ΔDEF are congruent. It is given that DE/AB=DF/AC....(2)

Since AX=DE (By construction) and from (1) and (2), we have: DE/AB = AX/AB = AY/AC = DF/AC. Thus, AY=DF

Now, by the SAS congruency criterion, ΔAXY≅ΔDEF⇒ΔAXY∼ΔDEF

While we already have, ΔAXY ~ ΔABC. This means ΔDEF and ΔABC are similar. Hence Proved.

## How do you Solve a SAS Triangle?

Observe the triangle ABC where

AB= 2 units, BC= 4 units and ∠ABC=50°

Now, to find the value of side AC, we will use the law of cosine .

The low of cosine gives the formula b 2 = a 2 + c 2 − 2ac cos B, where AB = c; BC = a; and AC = b.

= b 2 = a 2 + c 2 − 2ac cosB = x 2 = 4 2 + 2 2 − 2 (2) (4) cos 50° = 16 + 4 − 16 (0.643) = 20 − 10.288 = x = √9.712 = 3.116

Now apply the sine rule in the triangle ABC and calculate the value of ∠C.

sin C/2 = sin B/x sin C/2 = sin 50°/3.116 sin C = 2 × 0.766/3.116 = 0.492 ∠C = sin −1 0.492 = 29.47°

Then to calculate the value of ∠A use the sum of interior angles of a triangle is 180°

∠A + ∠B + ∠C = 180° ∠A + 50° + 29.47° = 180° ∠A = 180° − 79.47° =100.53°

Importat Notes:

• Two triangles are said to be congruent if their shape and size are exactly the same.
• We represent the congruence of triangles using the symbol (≅).
• If the two sides and the angle formed at their vertex of one triangle are equal to the two corresponding sides and the angle formed at their vertex of another triangle then the triangles are congruent by SAS Criterion for Congruence.

## Related Articles on SAS-Side Angle Side Congruence and Similarity

• Properties of Triangle
• Congruent Angles
• Congruent Triangles
• SSS Formula
• SSS Criterion in Triangles
• SSS-Congruence
• SAS Triangle Formula

## Examples on SAS

Example.1: James wanted to know which congruency rule says that these triangles are congruent. Let's help him.

Here, EF = MO = 3in, FG = NO = 4.5in, ∠EFG = ∠MON = 110°. Thus, △EFG ≅ △MNO ( By SAS rule ). ∴ These triangles are congruent by the SAS rule.

Example 2: Triangle ABC is an isosceles triangle and the line segment AD is the angle bisector of the angle A. Can you prove that ΔADB is congruent to the ΔADC by using SAS rule?

The triangle, ABC is an isosceles triangle where it is given that AB=AC. Now the side AD is common in both the triangles ΔADB and ΔADC. As the line segment AD is the angle bisector of the angle A then it divides the ∠A into two equal parts. Therefore, ∠BAD=∠CAD. Now according to the SAS rule, the two triangles are congruent. Hence, ΔADB≅ΔADC.

Example 3: Jolly was doing geometrical construction assignments in her notebook. She drew an isosceles triangle PQR on a page. She marked L, M as the midpoints of the equal sides (PQ and QR) of the triangle and N as the midpoint of the third side. She states that LN=MN. Is she right?

We will prove that ΔLPN ≅ ΔMRN. We know that ΔPQR is an isosceles triangle and PQ=QR. Angles opposite to equal sides are equal. Thus, ∠QPR=∠QRP. Since L and M are the midpoints of PQ and QR respectively, PL = LQ = QM = MR = QR/2. N is the midpoint of PR, hence, PN = NR. In ΔLPN and ΔMRN:

• ∠LPN = ∠MRN
• PN = NR Thus, by SAS Criterion of Congruence, ΔLPN ≅ ΔMRN. Since congruent parts of congruent triangles are equal, LN=MN.

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## Practice Questions on SAS

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## FAQs on SAS

How do you prove the sas congruence rule.

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are said to be congruent by the SAS congruence rule.

## What is a SAS Triangle?

A triangle whose two sides and the angle formed by them is known as a SAS triangle.

## What is the SAS Axiom?

SAS axiom is the rule which says that if the two sides of a triangle are equal to the two sides of another triangle, and the angle formed by these sides in the two triangles are equal, then these two triangles are congruent by the SAS criterion.

## How do you Prove the SAS Similarity Theorem?

SAS similarly can be proved by showing that one pair of side lengths of one triangle is proportional to one pair of side lengths of the other triangle and included angles are equal.

## What is the Difference Between SAS and SSS?

Both SAS and SSS rules are the triangle congruence rules. The full form of SAS is "Side-Angle-Side" and SSS stands for "Side-Side-Side."

• In the SAS postulate, two sides and the angle between them in a triangle are equal to the corresponding two sides and the angle between them in another triangle.
• In the SSS postulate, all three sides of one triangle are equal to the three corresponding sides of another triangle.

## What are the Four Triangle Similarity Theorems?

The Four triangle similarity theorems are:

• Angle-Angle (AA)
• Side-Angle-Side (SAS)
• Side-Side-Side (SSS)
• Right-Hand-Side (RHS)

## What does SAS Mean in Math?

SAS stands for the Side-Angle-Side theorem in the congruency of triangles. The SAS congruency is used when two triangles have one angle common and two sides equal, so as to prove that such triangles are congruent.

## What is SAS and ASA Congruence Rule?

The SAS congruence rule states that if two sides of a triangle along with an angle in between is equal to two sides and included the angle of another triangle, then the two triangles are said to be congruent. Whereas the ASA congruence rule states that when two angles with an included side are equal to the two angles along with the included side of another triangle, then these two triangles are said to be congruent.

## What is SAS, SSS, ASA, and AAS?

The four congruence rules of a triangle are:

• Angle-Angle-Side (AAS): When two angles of a triangle along with a side i.e excluded are equal to two angles and a side of another triangle, the triangles are considered to be congruent.
• Side-Angle-Side (SAS): When two sides with an angle included are equal to two sides with an angle of another triangle, the two triangles are considered to be congruent.
• Side-Side-Side (SSS): When all three sides of a triangle are equal to all three sides of another triangle, the two triangles are considered to be congruent.
• Angle-Side-Angle (ASA): When two angles including the side are equal to two angles with a side of another triangle, the two triangles are said to be congruent.

## Triangle congruence theorems (SSS, SAS, & ASA Postulates)

Triangles can be similar or congruent. Similar triangles will have congruent angles but sides of different lengths. Congruent triangles will have completely matching angles and sides. Their interior angles and sides will be congruent. Testing to see if triangles are congruent involves three postulates, abbreviated SAS , ASA , and SSS .

## Congruence definition

Two triangles are  congruent  if their corresponding sides are equal in length and their corresponding interior angles are equal in measure.

We use the symbol ≅ to show congruence.

Corresponding sides and angles mean that the side on one triangle and the side on the other triangle, in the same position, match. You may have to rotate one triangle, to make a careful comparison and find corresponding parts.

## How can you tell if triangles are congruent?

You could cut up your textbook with scissors to check two triangles. That is not very helpful, and it ruins your textbook. If you are working with an online textbook, you cannot even do  that .

Geometricians prefer more elegant ways to prove congruence. Comparing one triangle with another for congruence, they use three postulates.

## Postulate definition

A  postulate  is a statement presented mathematically that is assumed to be true. All three triangle congruence statements are generally regarded in the mathematics world as postulates, but some authorities identify them as  theorems  (able to be proved).

Do not worry if some texts call them postulates and some mathematicians call the theorems. More important than those two words are the  concepts  about congruence.

## Triangle congruence theorems

Testing to see if triangles are congruent involves three postulates. Let's take a look at the three postulates abbreviated  ASA ,  SAS , and  SSS.

Angle Side Angle (ASA)

Side Angle Side (SAS)

Side Side Side (SSS)

## ASA theorem (Angle-Side-Angle)

The  Angle Side Angle Postulate (ASA)  says triangles are congruent if any two angles and their included side are equal in the triangles. An included side is the side between two angles.

In the sketch below, we have  △CAT  and  △BUG . Notice that  ∠C  on  △CAT  is congruent to  ∠B  on  △BUG , and  ∠A  on  △CAT  is congruent to  ∠U  on  △BUG .

See the included side between  ∠C  and  ∠A  on  △CAT ? It is equal in length to the included side between  ∠B  and  ∠U  on  △BUG .

The two triangles have two angles congruent (equal) and the included side between those angles congruent. This forces the remaining angle on our  △CAT to be:

This is because interior angles of triangles add to  180° . You can only make one triangle (or its reflection) with given sides and angles.

You may think we rigged this, because we forced you to look at particular angles. The postulate says you can pick  any  two angles and their included side. So go ahead; look at either  ∠C  and  ∠T  or  ∠A  and  ∠T  on  △CAT .

Compare them to the corresponding angles on  △BUG . You will see that all the angles and all the sides are congruent in the two triangles, no matter which ones you pick to compare.

## SAS theorem (Side-Angle-Side)

By applying the  Side Angle Side Postulate (SAS) , you can also be sure your two triangles are congruent. Here, instead of picking two angles, we pick a side and its corresponding side on two triangles.

The  SAS Postulate  says that triangles are congruent if any pair of corresponding sides and their included angle are congruent.

Pick any side of  △JOB  below. Notice we are not forcing you to pick a particular side, because we know this works no matter where you start. Move to the next side (in whichever direction you want to move), which will sweep up an included angle.

or the two triangles to be congruent, those three parts – a side, included angle, and adjacent side – must be congruent to the same three parts – the corresponding side, angle and side – on the other triangle,  △YAK .

## SSS theorem (Side-Side-Side)

Perhaps the easiest of the three postulates,  Side Side Side Postulate (SSS)  says triangles are congruent if three sides of one triangle are congruent to the corresponding sides of the other triangle.

This is the only postulate that does not deal with angles. You can replicate the  SSS Postulate  using two straight objects – uncooked spaghetti or plastic stirrers work great.

Cut a tiny bit off one, so it is not quite as long as it started out. Cut the other length into two distinctly unequal parts. Now you have three sides of a triangle. Put them together. You have one triangle. Now shuffle the sides around and try to put them together in a different way, to make a different triangle.

Guess what? You can't do it. You can only assemble your triangle in one way, no matter what you do. You can think you are clever and switch two sides around, but then all you have is a reflection (a mirror image) of the original.

So once you realize that three lengths can only make one triangle, you can see that two triangles with their three sides corresponding to each other are identical, or congruent.

## Checking congruence in polygons

You can check polygons like parallelograms, squares and rectangles using these postulates.

Introducing a diagonal into any of those shapes creates two triangles. Using any postulate, you will find that the two created triangles are  always  congruent.

Suppose you have parallelogram  SWAN  and add diagonal SA . You now have two triangles,  △SAN  and  △SWA . Are they congruent?

You already know line SA , used in both triangles, is congruent to itself. What about ∠SAN ? It is congruent to ∠WSA because they are alternate interior angles of the parallel line segments SW and NA (because of the Alternate Interior Angles Theorem).

You also know that line segments SW and NA are congruent, because they were part of the parallelogram (opposite sides are parallel and congruent).

So now you have a side SA , an included angle ∠WSA , and a side SW of  △SWA . You can compare those three triangle parts to the corresponding parts of  △SAN:

Side SA ≅ Side SA (sure hope so!)

Included angle ∠WSA ≅ ∠NAS

Side SW ≅ Side NA

## Lesson summary

After working your way through this lesson and giving it some thought, you now are able to recall and apply three triangle congruence postulates, the Side Angle Side Congruence Postulate, Angle Side Angle Congruence Postulate, and the Side Side Side Congruence Postulate. You can now determine if any two triangles are congruent!

## How do we prove triangles congruent?

Theorems and Postulates: ASA, SAS, SSS & Hypotenuse Leg

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## Triangle Congruence by SSS and SAS

Before you can ever start with proofs your students need to have a clear understanding of what makes sides and angles of triangles congruent. This lesson on Triangle Congruence by SSS and SAS is one of the more memorization based lessons to teach. With that said the only way to memorize something and master is it to see it multiple times in multiple ways. Here are a couple of things to help you out!

## Congruent Figures

Two figures are said to be congruent if they have same  shape and the same size . The corresponding angles of the two congruent figures are equal . The corresponding sides are congruent . The two figures shown below are congruent .

The sides of two rectangles have a ratio of 1: 1. The rectangles are congruent .

## Congruent Triangles

Triangle congruence by side side side.

The congruence of triangles can be proved using its three sides and three  angles .

If all the three sides of one triangle are congruent to all the three corresponding sides of another triangle, the two triangles are said to be congruent.

The two triangles shown below are congruent by SSS Postulate .

## Triangle Congruence by Side Angle Side

If the two sides and their included angle of one triangle is congruent to the two sides and their included angle of another triangle, then the two triangles are said to be congruent.

The included angle is the angle made at the point where two sides of a triangle meet. The two triangles shown below are congruent by SAS Postulate.

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## 7.9: SAS Similarity

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Triangles are similar if two pairs of sides are proportional and the included angles are congruent.

## SAS Similarity Theorem

By definition, two triangles are similar if all their corresponding angles are congruent and their corresponding sides are proportional. It is not necessary to check all angles and sides in order to tell if two triangles are similar. In fact, if you know only that two pairs of sides are proportional and their included angles are congruent, that is enough information to know that the triangles are similar. This is called the SAS Similarity Theorem.

SAS Similarity Theorem: If two sides in one triangle are proportional to two sides in another triangle and the included angle in both are congruent, then the two triangles are similar.

If $$\dfrac{AB}{XY}=\dfrac{AC}{XZ}$$ and $$\angle A\cong \angle X$$, then $$\Delta ABC\sim \Delta XYZ$$.

What if you were given a pair of triangles, the lengths of two of their sides, and the measure of the angle between those two sides? How could you use this information to determine if the two triangles are similar?

Example $$\PageIndex{1}$$

Determine if the following triangles are similar. If so, write the similarity theorem and statement.

We can see that $$\angle B\cong \angle F$$ and these are both included angles. We just have to check that the sides around the angles are proportional.

\begin{aligned} \dfrac{AB}{DF} &=\dfrac{12}{8}=\dfrac{3}{2} \\ \dfrac{BC}{FE}&=\dfrac{24}{16}=\dfrac{3}{2} \end{aligned}

Since the ratios are the same $$\Delta ABC\sim \Delta DFE$$ by the SAS Similarity Theorem.

Example $$\PageIndex{2}$$

The triangles are not similar because the angle is not the included angle for both triangles.

Example $$\PageIndex{3}$$

Are the two triangles similar? How do you know?

We know that $$\angle B\cong \angle Z$$ because they are both right angles and $$\dfrac{10}{15}=\dfrac{24}{36}$$. So, $$\dfrac{AB}{XZ}=\dfrac{BC}{ZY}$$ and $$\Delta ABC\sim \Delta XZY$$ by SAS.

Example $$\PageIndex{4}$$

Are there any similar triangles in the figure? How do you know?

$$\angle A$$ is shared by $$\Delta EAB$$ and $$\Delta DAC$$, so it is congruent to itself. Let’s see if $$\dfrac{AE}{AD}=\dfrac{AB}{AC}$$.

\begin{aligned} \dfrac{9}{9+3}&=\dfrac{12}{12+5} \\ \dfrac{9}{12}&=\dfrac{3}{4}\neq \dfrac{12}{17}\qquad \text{ The two triangles are not similar. }\end{aligned}

Example $$\PageIndex{5}$$

From Example 4, what should $$BC$$ equal for $$\Delta EAB\sim \Delta DAC$$?

The proportion we ended up with was $$\dfrac{9}{12}=\dfrac{3}{4}\neq \dfrac{12}{17}$$. AC needs to equal 16, so that $$\dfrac{12}{16}=dfrac{3}{4}$$. $$AC=AB+BC$$ and $$16=12+BC$$. $$BC$$ should equal 4.

Fill in the blanks.

• If two sides in one triangle are _________________ to two sides in another and the ________________ angles are _________________, then the triangles are ______________.

Find the value of the missing variable(s) that makes the two triangles similar.

Determine if the triangles are similar. If so, write the similarity theorem and statement.

• $$\Delta ABC$$ is a right triangle with legs that measure 3 and 4. $$\Delta DEF$$ is a right triangle with legs that measure 6 and 8.
• $$\Delta GHI$$ is a right triangle with a leg that measures 12 and a hypotenuse that measures 13. $$\Delta JKL$$ is a right triangle with legs that measure 1 and 2.

• $$\overline{AC}=3$$

$$\overline{DF}=6$$

Interactive Element

Video: Congruent and Similar Triangles

Activities: SAS Similarity Discussion Questions

Study Aids: Polygon Similarity Study Guide

Practice: SAS Similarity

Real World: Triangle Similarity

## SAS (Side Angle Side) Theorem | Definition, Congruence, Examples

What is sas theorem (side-angle-side theorem) in geometry, sas (side-angle-side) congruence theorem, sas (side-angle-side) similarity rule, solved examples on  sas theorem, practice problems on sas theorem, frequently asked questions on  sas theorem.

The SAS theorem, which stands for Side-Angle-Side theorem, is a criterion used to prove triangle congruence and also triangle similarity. However, the terms or the conditions of the SAS theorem for triangle congruence and triangle similarity are slightly different.

Congruent Triangles: Two triangles are said to be congruent if they have the same shape and the same size. Corresponding angles and corresponding sides of two congruent triangles are also congruent. The triangle congruence is denoted using the symbol ≅.

Similar Triangles: Two triangles are said to be similar if they have the same shape but different sizes. Corresponding angles of similar triangles are congruent. Corresponding sides of two similar triangles are proportional. We denote the triangle similarity using the symbol .

The SAS theorem helps to prove the triangle congruence (or similarity) based on the lengths of any two sides and the measurement of the included angle between these sides.

## SAS Definition in Geometry

SAS Congruence Theorem: When the two sides of a triangle are equal to the two sides of another triangle, and the angles formed by these sides (the included angles) are also equal, then the two triangles are congruent.

SAS Similarity Criterion: If two sides of a triangle are proportional with two sides of another triangle, and if the corresponding included angles formed by these sides are congruent, then the two triangles are similar.

SAS congruence theorem states that, if two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.

SSS Congruence Theorem Example:

Are the given two triangles congruent?

AB= EF = 7 units

BC = DE = 5 units

∠B= ∠E = $50^{\circ}$

Thus,  ΔABC ≅ ΔFED by SAS congruence theorem

SAS congruence theorem states that, if two sides of one triangle are proportional to the corresponding sides of another triangle, and if the corresponding included angles are also equal, then the triangles are similar.

SAS Similarity Theorem Example:

Are the two triangles similar?

Here,

$\frac{ML}{RQ} = \frac{LN}{QS} = \frac{1}{2}$

m∠ L = m∠ Q $= 80^{\circ}$

By SAS similarity rule, ΔMLN ≅ ΔQRS

• SAS theorem is applicable to all types of triangles as long as the conditions of the SAS rule are satisfied.
• The positions of letters in SAS are significant. It refers to the combination of sides and angles used to prove the congruence or similarity. Since the angle A lies between two sides (S), we use the term “included angle.”

In this article, we learned about the SAS theorem for both congruence and similarity rule examples. Now let us practice solving problems on the SAS Theorem.

1. Are triangles ABC and BCD similar?

In ΔABC and ΔBCD,

$\frac{AB}{BD} = \frac{27}{18} = \frac{3}{2}$

$\frac{EB}{BC} = \frac{36}{24} = \frac{3}{2}$

Thus, $\frac{AB}{BD} = \frac{EB}{BC}$

Also, ∠ABE = ∠CBD (vertical angles)

Thus, ΔABC~ ΔBCD by SAS similarity rule

2. Prove that ΔYXZ ≅ ΔSRT.

XY = RS = 5 in

XZ = RT = 8.5 in

∠YXZ is the included angle between XY and XZ.

∠SRT is the included angle between SR and RT.

∠YXZ = ∠SRT = 60°

Thus, by SAS congruence rule, ΔYXZ ≅ ΔSRT.

3. Prove that ΔABC ≅ ΔCDA.

Solution:

AC = AC (common side)

∠DAC = ∠BCA (Alternate angles formed by a transversal cutting two parallel lines)

∠DAC is included angle between AC and AD.

∠BCA is included angle between BC and AC.

Thus,   ΔABC ≅ ΔCDA By SAS congruence rule.

Attend this quiz & Test your knowledge.

## What is the full form of the SAS theorem?

In sas theorem, the angle a refers to, which of the following measurements of sides and angles are required to prove congruence using the sas theorem.

What is the difference between SAS congruence rule and ASA congruence rule?

We apply the SAS rule when two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle. The ASA congruence rule is used when two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle.

What is the difference between SAS and SSS rules?

SAS stands for “Side-Angle-Side” and SSS stands for “Side-Side-Side.” Both rules are the triangle congruence rules.

Can the SAS theorem be used to prove similarity of triangles?

Yes, SAS is the criterion in both triangle congruence and triangle similarity theorems.

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Sas triangle, what are sas triangles.

SAS means ‘Angle-Side-Angle’. SAS triangles are triangles where two sides and the angle between them are known. Shown below is a SAS triangle, △ABC, where b and c are the two known sides, while ∠B is the known, common angle present between the two sides.

## How to Solve SAS Triangle

It involves three steps:

Step 1 : Use the Law of Cosines to calculate the missing side

Step 2 : Use The Law of Sines to find the smaller of the two unknown angles

Step 3 : Use the angle sum rule of a triangle to find the last angle

Let us take some examples to understand the concept better.

## Solved Examples

In the triangle, the given angles and side is: Side b = 4 Side c = 6 ∠A = 45° Step 1: Using theLaw of Cosines, we will calculate the missing side, side a a 2 = b 2 + c 2 − 2bc cos A, here b = 4, c = 6, ∠A = 45° a 2 = 4 2 + 6 2 – 2 x 4 x 6 x cos 45° a 2 = 16 + 36 – 48 x cos 45° a 2 = 52 – 48 x 0.70 a 2 = 52 – 33.6 a 2 = 18.4 a = 4.28 Step 2: Now, we use The Law of Sines to find the smaller of the two unknown angles Why the Smaller Angle to be Determined First? The smaller angle is determined first because the inverse sine function gives answers less than 90° even for angles greater than 90°. By choosing the smaller angle a triangle cannot have two angles greater than 90°. a/sin A = b/sin B, here ∠A = 45°, a = 4.28, b = 4 4.28/sin 45° = 4/sin B sin B = 4 x sin 45°/4.28 sin B = 4 x 0.70/4.28 sin B = 0.65 B = sin -1 (0.65) B = 40.54° Step 3: Finally, we will use the angle sum rule of a triangle to find the last undetermined angle, ∠C ∠A + ∠B + ∠C = 180°, here ∠A = 45°, ∠B = 40.54° 45° + 40.54° +∠C = 180° ∠C = 180° – (45° + 40.54°) ∠C = 94.46°

The given triangle is an SAS triangle. Here, Side a = 6.8 Side b = 4.2 ∠C = 110° Step 1: Using theLaw of Cosines, we will calculate the missing side, side c c 2 = a 2 + b 2 − 2ab cos C, here a = 6.8, b = 4.2, ∠C = 110° c 2 = 6.8 2 + 4.2 2 – 2 x 6.8 x 4.2 x cos 110° c 2 = 46.24 + 17.64 – 57.12 x (-0.34) c 2 = 63.88 + 19.42 c 2 = 83.3 c = 9.12 Step 2: Now, for calculating the Law of Sines, we don’t need to look for the smaller angle as ∠C is greater than 90°, so ∠A and ∠B must be less than 90°. a/sin A = c/sin C, here, a = 6.8, c = 9.12, ∠C = 110° 6.8/sin A = 9.12/ sin 110° sin A = 6.8 x sin 110°/9.12 sin A = 6.8 x 0.93/9.12 sin A = 0.69 A = sin -1 (0.69) ∠A = 43.63° Step 3: Finally, we will use the angle sum rule of a triangle to find the last undetermined angle, ∠B ∠A + ∠B + ∠C = 180°, here ∠A = 43.63°, ∠C = 110° 43.63° + ∠B + 110° = 180° ∠B = 180° – (43.63° + 110°) ∠B = 26.37°

## SAS Triangle Congruence Theorem

Let us take an example to understand the concept.

## Proving Triangles Congruent using the Side-Angle-Side Theorem

To prove: ΔABO ≅ ΔCDO

Option (b). Here the triangles ΔHIJ and ΔKLM are congruent by the SAS relationship.

Given that BC ≅ DC and C is the midpoint of AE Now, according to the ‘ Vertical Angles Theorem ‘ ∠ACB ≅∠ECD Again, AC ≅ EC as the midpoint C bisects AE into two congruent parts AC and EC Hence ΔABC ≅ ΔDEC by SAS postulate (Proved)

## Area of a SAS Triangle

If ΔABC is an SAS triangle with side lengths a, b and c, then we can find its area with one of the three formulas given below:

## How It Works

The method discussed above is just an extension of the general formula for determining the area of a triangle.

As we know, the general formula of area of a triangle is given by,

Area (A) = ½ x a x h, here a = base, h = height

But, if we are not given the height but we are given side b and angle C in the right triangle, ΔCDA

Using trigonometry, we can write

sin C = h/b

=> h = b sin C

Substituting the value of h in the general formula of area of a triangle, we get,

Area (A) = ½ x a x b sin C = ½ ab sin C

Let us solve an example using the above formula.

As the given triangle is an SAS triangle, we will use the formula Area (A) = ½ ab sin C, here a = 10, b = 16, ∠C = 55°  = ½ 10 x 16 sin 55° = 64.8 square units

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• Trigonometry

## How To Prove Triangles Congruent - SSS, SAS, ASA, AAS Rules

In these lessons, we will learn

• the SSS, SAS, ASA and AAS rules,
• how to use two-column proofs to prove triangles congruent.

Related Pages Congruent Triangles More Geometry Lessons

## Congruent Triangles

Rules for triangle congruency.

Congruent triangles are triangles that have the same size and shape. This means that the corresponding sides are equal and the corresponding angles are equal.

We can tell whether two triangles are congruent without testing all the sides and all the angles of the two triangles. In this lesson, we will consider the four rules to prove triangle congruence. They are called the SSS rule, SAS rule, ASA rule and AAS rule. In another lesson, we will consider a proof used for right triangles called the Hypotenuse Leg rule . As long as one of the rules is true, it is sufficient to prove that the two triangles are congruent.

The following diagrams show the Rules for Triangle Congruency: SSS, SAS, ASA, AAS and RHS. Take note that SSA is not sufficient for Triangle Congruency. Scroll down the page for more examples, solutions and proofs.

## Side-Side-Side (SSS) Rule

Side-Side-Side is a rule used to prove whether a given set of triangles are congruent.

The SSS rule states that: If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.

In the diagrams below, if AB = RP , BC = PQ and CA = QR , then triangle ABC is congruent to triangle RPQ .

## Side-Angle-Side (SAS) Rule

Side-Angle-Side is a rule used to prove whether a given set of triangles are congruent.

The SAS rule states that: If two sides and the included angle of one triangle are equal to two sides and included angle of another triangle, then the triangles are congruent.

An included angle is an angle formed by two given sides.

Included Angle           Non-included angle

For the two triangles below, if AC = PQ , BC = PR and angle C< = angle P , then by the SAS rule, triangle ABC is congruent to triangle QRP .

## Angle-Side-Angle (ASA) Rule

Angle-side-angle is a rule used to prove whether a given set of triangles are congruent.

The ASA rule states that: If two angles and the included side of one triangle are equal to two angles and included side of another triangle, then the triangles are congruent.

## Angle-Angle-Side (AAS) Rule

The AAS rule states that: If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then the triangles are congruent.

In the diagrams below, if AC = QP , angle A = angle Q , and angle B = angle R , then triangle ABC is congruent to triangle QRP .

## Three Ways To Prove Triangles Congruent

A video lesson on SAS, ASA and SSS.

• SSS Postulate: If there exists a correspondence between the vertices of two triangles such that three sides of one triangle are congruent to the corresponding sides of the other triangle, the two triangles are congruent.
• SAS Postulate: If there exists a correspondence between the vertices of two triangles such that the two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.
• ASA Postulate: If there exits a correspondence between the vertices of two triangles such that two angles and the included side of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.

## Using Two Column Proofs To Prove Triangles Congruent

Triangle Congruence by SSS How to Prove Triangles Congruent using the Side Side Side Postulate? If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

Triangle Congruence by SAS How to Prove Triangles Congruent using the SAS Postulate? If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Prove Triangle Congruence with ASA Postulate How to Prove Triangles Congruent using the Angle Side Angle Postulate? If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Prove Triangle Congruence by AAS Postulate How to Prove Triangles Congruent using the Angle Angle Side Postulate? If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the two triangles are congruent.

## Free Printable congruent triangles sss sas and asa worksheets

Explore a variety of free printable math worksheets focusing on congruent triangles using SSS, SAS, and ASA postulates. Discover comprehensive resources to enhance your students' learning experience and strengthen their geometry skills.

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## Explore printable congruent triangles sss sas and asa worksheets

Congruent triangles SSS SAS and ASA worksheets are essential tools for teachers who want to help their students master the concepts of congruence in Math and Geometry. These worksheets provide a variety of exercises and problems that focus on the three primary methods for proving triangles congruent: Side-Side-Side (SSS), Side-Angle-Side (SAS), and Angle-Side-Angle (ASA). By incorporating these worksheets into their lesson plans, teachers can ensure that their students develop a strong foundation in understanding congruent triangles and their properties. Moreover, these worksheets are designed to cater to different grade levels, making it easy for teachers to select the most appropriate materials for their students. With congruent triangles SSS SAS and ASA worksheets, teachers can effectively engage their students in the fascinating world of Math and Geometry.

Quizizz is an excellent platform that offers a wide range of resources, including congruent triangles SSS SAS and ASA worksheets, to make learning Math and Geometry more interactive and enjoyable for students. Teachers can easily create and share quizzes, polls, and other activities that are tailored to their students' grade levels and learning objectives. In addition to worksheets, Quizizz also provides teachers with valuable insights into their students' progress, allowing them to identify areas where students may need additional support or reinforcement. Furthermore, the platform offers various game-based learning activities that can help students develop a deeper understanding of congruent triangles and other essential Math and Geometry concepts. By incorporating Quizizz into their teaching strategies, teachers can create a more engaging and effective learning environment for their students, ensuring that they excel in their studies.

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## Sas Theorem

Sas Theorem - Displaying top 8 worksheets found for this concept.

Some of the worksheets for this concept are 4 s and sas congruence, Hypotenuse leg theorem work and activity, Unit 3 syllabus congruent triangles, 4 s sas asa and aas congruence, Geometry, Congruent triangles proof work, Congruent triangles work 1, Sas similarity theorem.

Found worksheet you are looking for? To download/print, click on pop-out icon or print icon to worksheet to print or download. Worksheet will open in a new window. You can & download or print using the browser document reader options.

## 1. 4-SSS and SAS Congruence

2. hypotenuse leg theorem worksheet and activity, 3. unit 3 syllabus: congruent triangles, 4. 4-sss, sas, asa, and aas congruence, 5. geometry, 6. congruent triangles proof worksheet, 7. congruent triangles worksheet #1, 8. sas similarity theorem -.

#### IMAGES

1. Fillable Online Congruent Triangles SSS and SAS Theorems Matching

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3. Triangle Congruence Sss And Sas Worksheet Answer Key

4. Eighteen SSS or SAS Triangle Congruence Problems Worksheet for 9th

5. Congruent Triangles Sss And Sas Worksheet Answers

6. Triangle Congruence Sss And Sas Worksheet Answer Key

#### VIDEO

1. Homotopy Type Theory in LeanProver

2. Statistics in SAS

3. SAS FINAL REVISION PART-1

4. Geometry Section 5.8 Hinge Theorem (SAS Inequality)

5. NCERT CLASS 10, MATHEMATICS, CHAPTER

6. MTH 22 SAS Theorem Intro

1. PDF 4-SSS, SAS, ASA, and AAS Congruence

©g j2z0 01S1 S MK6uwt Paq iS Oo 1f 5t4woanr geL CLtLACT.r M CAQlql0 Sr1isg3h 8tUsC VrIe7skevrVvPeadx. i w VMDaDdyeR ewGiXtrh u WIkn AfBiPndi Vt0e M YGge HoZm0eUt4roy A.l Worksheet by Kuta Software LLC Kuta Software - Infinite Geometry Name_____ SSS, SAS, ASA, and AAS Congruence Date_____ Period____

2. PDF 4-SSS and SAS Congruence

C Worksheet by Kuta Software LLC State what additional information is required in order to know that the triangles are congruent for the reason given. 11) SAS J H I E G IJ ≅ IE 12) SAS L M K G I H ∠L ≅ ∠H 13) SSS Z Y D X YZ ≅ DX 14) SSS R S T Y X Z TR ≅ ZX 15) SAS V U W X Z Y WU ≅ ZX 16) SSS E G F Y W X GE ≅ WY 17) SAS E F G Q ...

3. 2.2: The SAS Theorem

The SAS Theorem is Proposition 4 in Euclid's Elements, Both our discussion and Suclit's proof of the SAS Theoremimplicitly use the following principle: If a geometric construction is repeated in a different location (or what amounts to the same thing is "moved" to a different location) then the size and shape of the figure remain the same ...

4. Proving Congruence with SSS and SAS

first postulate. SSS Postulate (Side-Side-Side) If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. As you can see, the SSS Postulate does not concern itself with angles at all. Rather, it only focuses only on corresponding, congruent sides of triangles in order to

5. Side Angle Side Congruence and Similarity

The Side-Angle-Side theorem of congruency states that, if two sides and the angle formed by these two sides are equal to two sides and the included angle of another triangle, then these triangles are said to be congruent. Verification: Let's perform an activity to show the proof of SAS. Given: AB=PQ, BC=QR, and ∠B=∠Q. To prove: ΔABC ≅ ΔPQR

6. PDF Infinite Geometry

Worksheet by Kuta Software LLC Geometry HW: SSS, SAS and AA similarity Name_____ ©_ w2G0G1u7i RKBuptTat OSkokfytdwmaZrieZ aLnL[CG.t B RAKl_lH HrYiLgYhTtqsZ Nr\easSeYrhvyevd_.-1-State if the triangles in each pair are similar. If so, state how you know they are similar and complete the similarity statement. 1) 66 GF 1818 VW U

7. PDF 7.4 Showing Triangles are Similar: SSS and SAS

EXAMPLE 1 Use the SSS Similarity Theorem Determine whether the triangles are P S similar. If they are similar, write a 6 4 similarity statement and find the scale 8 factor of Triangle B to Triangle A. 12 A 5 T B U Solution P 10 R Find the ratios of the corresponding sides.

8. Triangle Congruence Theorems

January 11, 2023 Fact-checked by Paul Mazzola Congruence definition Postulate definition Triangle congruence theorems Triangle congruence theorems (SSS, SAS, & ASA Postulates) Triangles can be similar or congruent. Similar triangles will have congruent angles but sides of different lengths.

9. Proving triangles congruent with SSS, ASA, SAS, Hypotenuse Leg and

Theorems and Postulates: ASA, SAS, SSS & Hypotenuse Leg Preparing for Proof Proof Theorems Quiz Corresponding Sides and Angles Properties, properties, properties! Triangle Congruence Side Side Side (SSS) Angle Side Angle (ASA) Side Angle Side (SAS) Angle Angle Side (AAS) Hypotenuse Leg (HL) CPCTC Worksheets on Triangle Congruence

10. Triangle Congruence by SSS and SAS

If the two sides and their included angle of one triangle is congruent to the two sides and their included angle of another triangle, then the two triangles are said to be congruent. The included angle is the angle made at the point where two sides of a triangle meet. The two triangles shown below are congruent by SAS Postulate.

11. IXL

IXL's SmartScore is a dynamic measure of progress towards mastery, rather than a percentage grade. It tracks your skill level as you tackle progressively more difficult questions. Consistently answer questions correctly to reach excellence (90), or conquer the Challenge Zone to achieve mastery (100)! Learn more. 0.

12. 7.9: SAS Similarity

7: Similarity 7.9: SAS Similarity Expand/collapse global location 7.9: SAS Similarity Page ID Triangles are similar if two pairs of sides are proportional and the included angles are congruent. SAS Similarity Theorem

13. SSS and SAS Congruence Worksheets

These worksheets and lessons will help you master the use and application of both of these valuable theorems. The focus pf the series is on proving triangle congruence through the use of side-side-side and side-angle-side theorems. Aligned Standard: High School Geometry - HSG-SRT.B.5

14. SAS

The SAS theorem, which stands for Side-Angle-Side theorem, is a criterion used to prove triangle congruence and also triangle similarity. However, the terms or the conditions of the SAS theorem for triangle congruence and triangle similarity are slightly different.

15. PDF 8.3 Proving Triangle Similarity by SSS and SAS

Section 8.3 Proving Triangle Similarity by SSS and SAS 439 Proving Slope Criteria Using Similar Triangles You can use similar triangles to prove the Slopes of Parallel Lines Theorem (Theorem 3.13). Because the theorem is biconditional, you must prove both parts. 1. If two nonvertical lines are parallel, then they have the same slope. 2.

16. PDF 5.2 Proving Triangles are Congruent: SSS and SAS

POSTULATE 12 Side-Side-Side Congruence Postulate (SSS) Words If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. Symbols

17. PDF Similar Triangles Date Period

State if the triangles in each pair are similar. If so, state how you know they are similar and complete the similarity statement. 7) similar; SSS similarity; ∆QRS8) not similar. Find the missing length. The triangles in each pair are similar. Free trial available at KutaSoftware.com.

18. SAS Triangle

Solution: In the triangle, the given angles and side is: Side b = 4 Side c = 6 ∠A = 45° Step 1: Using theLaw of Cosines, we will calculate the missing side, side a a 2 = b 2 + c 2 − 2bc cos A, here b = 4, c = 6, ∠A = 45° a 2 = 4 2 + 6 2 - 2 x 4 x 6 x cos 45° a 2 = 16 + 36 - 48 x cos 45° a 2 = 52 - 48 x 0.70 a 2 = 52 - 33.6 a 2 = 18.4 a = 4.28

19. How To Prove Triangles Congruent

Angle-side-angle is a rule used to prove whether a given set of triangles are congruent. The AAS rule states that: If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then the triangles are congruent. In the diagrams below, if AC = QP, angle A = angle Q, and angle B = angle ...

20. PDF Congruent Triangles: SSS and SAS Theorems Guided ...

Congruent Triangles: SSS and SAS Theorems - Guided Lesson Explanation Explanation#1 SAS (side-angle-side) Two sides and the angle between them are congruent The SAS Theorem states that two triangles are congruent if and only if two ... Tons of Free Math Worksheets at: ...

21. PDF Geometry: Proofs and Postulates Worksheet

Side-Angle-Side (SAS) CPCTC (Coresponding Parts of Congruent Triangles are Congruent) Reasons 1. Given Sides-Angles Theorem 2. (Isosceles Triangle) Given 3. Definition of median 4. Definition of midpoint 5. Division property (like division of 6. congruent segments) Reflexive property 8. Side-Angle-Side (SAS) 9. CPCTC 3) Given: AT and CS are medians

22. Free Printable congruent triangles sss sas and asa worksheets

These worksheets provide a variety of exercises and problems that focus on the three primary methods for proving triangles congruent: Side-Side-Side (SSS), Side-Angle-Side (SAS), and Angle-Side-Angle (ASA). By incorporating these worksheets into their lesson plans, teachers can ensure that their students develop a strong foundation in ...

23. Sas Theorem Worksheets

Sas Theorem - Displaying top 8 worksheets found for this concept. Some of the worksheets for this concept are 4 s and sas congruence, Hypotenuse leg theorem work and activity, Unit 3 syllabus congruent triangles, 4 s sas asa and aas congruence, Geometry, Congruent triangles proof work, Congruent triangles work 1, Sas similarity theorem. ...