DWG NO. 04.3 — Lesson 3 of 6

Proving Congruence: SSS and SAS

Unit 4: Congruent Triangles · ~15–30 min

Objective

Use the Side-Side-Side and Side-Angle-Side postulates to prove that two triangles are congruent.

What "congruent triangles" means

Two triangles are congruent when all three pairs of corresponding sides and all three pairs of corresponding angles are congruent — one triangle is just a copy of the other, possibly moved, rotated, or flipped. Writing "triangle ABC ≅ triangle DEF" is a promise about every part matching up, in that specific letter order: A with D, B with E, C with F.

Checking all six pairs every time would be slow. Instead, geometry gives us shortcuts — sets of just three matching parts that guarantee the rest must match too.

Side-Side-Side (SSS)

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

Once you fix three side lengths, there's only one possible triangle shape that can be built from them — the angles are locked in automatically.

Side-Angle-Side (SAS)

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 triangles are congruent.

The angle has to be included — meaning it's the angle formed by the two sides you're using, not some other angle in the triangle. An angle that isn't between the two known sides doesn't lock the triangle's shape the same way.

Triangle ABC ≅ triangle DEF by SSS — three pairs of matching side tick marks are enough to fix the whole shape.

Worked Example 1 · Identify the shortcut

ProblemIn triangles PQR and STU, PQ = ST, QR = TU, and angle Q ≅ angle T. Angle Q is between sides PQ and QR; angle T is between sides ST and TU. Which postulate applies?

1.Two pairs of sides are given congruent (PQ = ST, QR = TU), plus one pair of angles. Check whether the angle is included between those two sides.

2.Angle Q sits between PQ and QR, and angle T sits between ST and TU — both are the included angles, matching the same position in each triangle.

Triangle PQR ≅ triangle STU by SAS

Worked Example 2 · Use SSS with shared side

ProblemTriangles ABD and CBD share side BD. Given AB = CB and AD = CD, prove the triangles are congruent.

1.AB ≅ CB is given, and AD ≅ CD is given.

2.BD ≅ BD by the Reflexive Property — any segment is congruent to itself, and both triangles use the exact same segment.

3.All three sides of triangle ABD match all three sides of triangle CBD.

Triangle ABD ≅ triangle CBD by SSS

Guided practice

1.Two triangles have all three pairs of corresponding sides congruent. What postulate proves them congruent?
Show answer
SSS — Side-Side-Side.
2.Triangle XYZ has XY = 6, angle Y = 50°, YZ = 9. Triangle LMN has LM = 6, angle M = 50°, MN = 9, with angle Y and angle M included between the matching sides. Are the triangles congruent, and by what postulate?
Show answer
Yes, by SAS — two sides and the included angle match in both triangles.
3.Why does SSS work — why can't you build two different-shaped triangles from the same three side lengths?
Show answer
Once three side lengths are fixed, the angles between them are forced — there's no way to hinge the sides to a different angle without changing at least one of the lengths. Only one triangle shape fits three given side lengths.
4.Two triangles share two congruent sides and a congruent angle, but the angle is not between the two given sides. Can you conclude the triangles are congruent by SAS?
Show answer
No. SAS requires the angle to be included between the two known sides. An angle that isn't included doesn't guarantee congruence by this postulate.
5.Triangles GHI and JKL share side HI = KL. Given GH = JK and GI = JL, name the postulate that proves congruence, and state which parts matched.
Show answer
SSS — all three sides match: GH ≅ JK, GI ≅ JL, and HI ≅ KL.

← Lesson 4.2 Unit 4 Overview Lesson 4.4 →