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## 4.2 Triangle Congruence by SSS and SAS – Part 2

This post covers both triangle congruence using side-angle-side and side-side-side. If you have not checked out the first post on Triangle Congruence by SSS and SAS, follow the preceding link.

## Side Side Side Postulate

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

## 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.

Example 1 (Additional Example 2 from the video below)

Given that segment AD is congruent to segment BC, what other information is needed to prove triangle ACD congruent to triangle BCD by SAS?

It can be established that segment DC is congruent to itself by the reflexive property of congruence.  So, there are two pairs of corresponding sides congruent in each triangle. You will need to prove that angle ADC is congruent to angle BCD to prove triangle ACD congruent to triangle BCD by SAS. Remember, the angle is the included side in the SAS postulate and angle ACD is the included angle of sides AD and DC, while angle BCD is the included of sides BC and DC.

Example 2 (Quick Check #1 from the video)

Given: segment HF is congruent to segment JH, segment FG is congruent to segment JK and H is the midpoint of segment GK

Prove: triangle FGH congruent to triangle JKH

Completing proofs in geometry requires some planning. It may not be planning that you write out as will in this post, but you should at least think about what you are given and how to prove the triangles congruent. In this problem, you are given two sets of corresponding sides as congruent. Also, you are given information in regards to the third side of each triangle. The previous sentence should indicate the use of the side-side-side postulate of triangle congruence.

Paragraph Proof

It is given that segment HF is congruent to segment JH, segment FG is congruent to segment JK and H is the midpoint of segment GK. Since H is the midpoint of segment GK, it can be concluded that segment GH is congruent to segment KH by the definition of midpoint. Now that all pairs of corresponding sides are congruent, it can be stated that triangle FGH congruent to triangle JKH by the Side-Side-Side Postulate. QED

Example 3 (Additional Example 3 from the video)

This example is a Side-Angle-Side proof.

Given: angle RSG is congruent to angle RSH and segment SG is congruent to SH

Prove: triangle RSG is congruent to triangle RSH

Planning this proof involves realizing that SAS is going to be used. Once you mark the given information on the diagram, it should be clear that you need prove side SR is congruent to itself to establish the SAS postulate in both triangles.

Paragraph Proof

It is given that angle RSG is congruent to angle RSH and segment SG is congruent to SH and triangle RSG is congruent to triangle RSH needs to be proved . Both triangles share the side SR, thus segment SR is congruent to segment SR by the reflexive property of congruence. From the given statement and the previous sentence, Side-Angle-Side has been established in each triangle so, triangle RSG is congruent to triangle RSH. QED

## Side Side Side Postulate

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

Example 1

The first example is proving two triangles congruent using the SSS postulate. You are given two pairs of corresponding sides congruent. Thus, you need to only prove one pair of corresponding sides is congruent. As you can see in the diagram, the two triangles share a side segment BD. You will be able to establish this side is congruent to itself with the reflexive property of congruent segments. Now that all three sides are congruent, the triangles are congruent by the SSS postulate.

Example 2

Given that segment AX is congruent to segment AY and M is the midpoint of segment XY, Prove that triangle AMX is congruent to triangle AMY. First one can establish that segment AM is congruent to itself by the reflective property of congruence. That gives two pairs of corresponding sides of each triangle congruent. The third pair of corresponding sides, MX and MY, is congruent, because M is the midpoint of segment XY which is made up the segments MX and MY. Since all three pairs of corresponding sides are congruent, the triangles, AMX and AMY are congruent by SSS.

## 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.

Example 3 Using SAS

This is actually example 2 in the video, but it is example 3 in the blog! Given that segment RS is congruent to segment TK.

What other information is needed to prove triangle RSK is congruent to triangle TKS By Side-Angle-Side?

Since the triangles share side KS, it can be stated that segment KS is congruent to itself by the reflexive property of congruence. You now have two pairs of corresponding sides congruent. To prove the triangles congruent by SAS, angle RSK must be proved congruent to angle TKS.﻿