Congruence and proofs

Congruence tests SSS, SAS, AAS and RHS, writing a congruence proof with reasons, and using congruent triangles to prove properties of shapes.

Writing a proof

Example 1●●●

$ABCD$ is a parallelogram with diagonal $AC$. Prove $\triangle ABC \equiv \triangle CDA$.

$AB = CD$ (opposite sides of a parallelogram)

$BC = DA$ (opposite sides of a parallelogram)

$AC$ is common.

$\therefore \triangle ABC \equiv \triangle CDA$ (SSS)

Example 2●●●

$M$ is the midpoint of $AB$, and $PM \perp AB$. Prove $PA = PB$.

$AM = BM$ ($M$ is the midpoint); $\angle PMA = \angle PMB = 90^\circ$; $PM$ is common.

$\triangle PMA \equiv \triangle PMB$ (SAS), so $PA = PB$ (matching sides of congruent triangles).

Flashcards

Common side

A side shared by both triangles

Proof layout

Statement, then reason in brackets.

After proving congruence

Matching sides and angles are equal.

Practice

In Example 1, which test was used?
In Example 2, which test was used?
Two right-angled triangles share their hypotenuse and have one other pair of sides equal. Test?
The reason "$AC$ is common" means:
Which is NOT a congruence test?
Two angles and the side between them are equal. Which test applies?
Vertically opposite angles are equal. This is useful in proofs because it gives:
After proving $\triangle PQR \equiv \triangle XYZ$, which must be true?