Writing congruence proofs

Warm-up

Three quick questions from earlier lessons.

Discover

To use SAS in a proof, what must you have shown first?

Two pairs of equal sides and the equal angles between them
That the triangles are congruent

Key idea

A proof is a chain of statements, each with a reason.

Typical chain: state the given facts → derive new facts (vertical angles, midpoints, parallel lines, shared sides) → apply a criterion (SSS, SAS, ASA, AAS) → conclude, then use corresponding parts of the congruent triangles.

Worked example

$M$ is the midpoint of $AD$ and of $BC$ .

AM = MD

,

BM = MC

Definition of midpoint.

\angle AMB = \angle DMC

(vertical angles)

They are opposite at M.

\triangle AMB \cong \triangle DMC

, so

AB = DC

SAS.

Your turn, guided

Your turn: how many statements (facts) does SAS need before the conclusion?

3

Side, angle, side.

Practice ★

In a proof, two segments of length 3 cross at $M$ . Which reason justifies $\angle AMB = \angle DMC$ ?Not to scale
1. Vertical angles are equal
2. Corresponding angles of parallel lines
3. Corresponding parts of congruent triangles

Segments $AD$ and $BC$ cross at their common midpoint $M$ (so $AM = MD$ and $BM = MC$ , each pair 6 cm). Put the steps proving $\triangle AMB \cong \triangle DMC$ in order.Not to scale
- $AM = MD$ and $BM = MC$ (M is the midpoint)
- $\angle AMB = \angle DMC$ (vertical angles)
- Two sides and the included angle match: SAS
- So $\triangle AMB \cong \triangle DMC$

Watch out!

Practice ★★

Two triangles have two equal angles ( $54^\circ$ and $70^\circ$ ) and an equal side that is not between them. Which criterion applies?
1. AAS (equivalent to ASA, since the third angles are equal too)
2. SSA
3. None: they need not be congruent

Exit ticket

Three questions to check you have got it.

In a proof, two segments of length 3 cross at $M$ . Which reason justifies $\angle AMB = \angle DMC$ ?Not to scale
1. Vertical angles are equal
2. Corresponding angles of parallel lines
3. Corresponding parts of congruent triangles
In a proof, two segments of length 6 cross at $M$ . Which reason justifies $\angle AMB = \angle DMC$ ?Not to scale
1. Vertical angles are equal
2. Corresponding angles of parallel lines
3. Corresponding parts of congruent triangles
In a proof, two segments of length 9 cross at $M$ . Which reason justifies $\angle AMB = \angle DMC$ ?Not to scale
1. Vertical angles are equal
2. Corresponding angles of parallel lines
3. Corresponding parts of congruent triangles

Summary card

Congruence proofs

Facts with reasons → criterion → conclusion → corresponding parts.

Watch out: never use what you are proving as a reason.

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