Proving Triangles Congruent: SSS and SAS
State and apply the SSS and SAS congruence postulates. Write formal two-column proofs using these postulates and distinguish between what is given and what must be proved.
A lesson of Geometry, in Triangles and Congruence, about 10 minutes.
By the end of this lesson, you will be able to apply the SSS and SAS congruence postulates, identify sufficient conditions for triangle congruence, and write two-column proofs using SSS and SAS.
The Big Idea
You don't need all six pairs of congruent parts to establish triangle congruence. Three carefully chosen pairs are sufficient. This is why we have congruence shortcuts — SSS and SAS — that let us prove congruence efficiently without measuring all six parts.
The key insight: once certain parts are fixed, the triangle's shape is determined. There is no "wiggle room."
SSS — Side-Side-Side Congruence Postulate
If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.
$$\overline{AB} \cong \overline{DE}, \quad \overline{BC} \cong \overline{EF}, \quad \overline{AC} \cong \overline{DF} \implies \triangle ABC \cong \triangle DEF$$
Why it works: Given three side lengths, there is only one possible triangle shape (up to reflection). The angles are forced.
SAS — Side-Angle-Side Congruence Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent.
The included angle is the angle formed between the two sides — it must be the angle between them, not a random angle.
$$\overline{AB} \cong \overline{DE}, \quad \angle B \cong \angle E, \quad \overline{BC} \cong \overline{EF} \implies \triangle ABC \cong \triangle DEF$$
Why SSA fails: Side-Side-Angle (SSA) does NOT guarantee congruence. Two triangles can share two sides and a non-included angle yet be different. This is the "ambiguous case."
Example 1 — Identifying SSS
From a marked diagram: $\overline{AB} \cong \overline{XY}$, $\overline{BC} \cong \overline{YZ}$, $\overline{AC} \cong \overline{XZ}$.
Three pairs of congruent sides → SSS. $\triangle ABC \cong \triangle XYZ$.
Example 2 — Identifying SAS
From a marked diagram: $\overline{PQ} \cong \overline{TU}$, $\angle Q \cong \angle U$, $\overline{QR} \cong \overline{UV}$.
Two sides and the included angle → SAS. $\triangle PQR \cong \triangle TUV$.
Example 3 — Two-Column Proof (SSS)
Given: $\overline{AB} \cong \overline{CD}$, $\overline{BC} \cong \overline{DA}$
Prove: $\triangle ABC \cong \triangle CDA$
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