Congruence Conditions for Triangles

Grade 6

The four conditions

Two triangles are congruent if they are exactly the same shape and size. There are four valid conditions — proving any one of them is enough.

SSS: all three sides equal. SAS: two sides and the included angle equal. ASA (or AAS): two angles and a corresponding side equal. RHS: right angle, hypotenuse, and one side equal.

SSA (two sides and a non-included angle) is not a valid congruence condition — two different triangles can share those measurements. Never use SSA as a proof.

Proving congruence in a question

State each pair of equal sides or angles, give the reason for each one, then name the congruence condition used. That structure earns all three marks.

Worked example

EXAMPLE

Triangles ABC and PQR have AB = PQ, BC = QR, and angle ABC = angle PQR. Are the triangles congruent? If so, state the condition.

AB = PQ (given). BC = QR (given). Angle ABC = angle PQR (given). These are two sides and the included angle. Therefore the triangles are congruent by SAS (side-angle-side).

Try this one

Each interior angle of a regular polygon is \( 156^\circ \).

Work out how many sides the polygon has. [3]

Nova's hint:

This uses the link between interior and exterior angles, then the exterior-angle total.

Ask yourself: 'if I know the interior angle, how do I find the exterior angle, and how does that give the number of sides?' The exterior angles share \( 360^\circ \) equally.

Try to Solve it with Nova?