Congruence · Grade 9
In Grade 9 you learn: proving triangles congruent.
Congruent shapes are identical: the same shape and the same size, so all their matching sides and angles are equal. Two triangles are congruent when they match by one of four conditions: SSS, SAS, ASA (or AAS) or RHS.
Key facts
- Congruent shapes have equal matching sides and equal matching angles.
- Triangles are congruent if they match by SSS, SAS, ASA (or AAS) or RHS. AAA is not enough.
- To prove two triangles congruent, give three matching facts and name the condition they meet.
Worked examples
Example 1
Two triangles have two pairs of equal sides and the angles between those sides are equal. Are they congruent?
Answer: Yes (SAS)
Example 2
Two right triangles both have a hypotenuse of 13 cm and another side of 5 cm. Are they congruent? Give the reason.
Answer: Yes (RHS)
Example 3
Two triangles both have angles of 50° and 70°, and the side between those two angles is 8 cm in both. Are they congruent?
Answer: Yes (ASA)
Example 4
Two triangles both have angles of 40°, 60° and 80°. The shortest side of one is 3 cm; the shortest side of the other is 6 cm. Are they congruent?
Answer: No: they are similar, not congruent
Example 5
Triangles PQR and XYZ are congruent, with P matching X, Q matching Y and R matching Z. PQ = 7 cm and angle Q = 52°. Find XY and angle Y.
Answer: XY = 7 cm, angle Y = 52°
Example 6
ABCD is a parallelogram. Prove that triangles ABC and CDA are congruent.
Answer: SSS: AB = CD, BC = DA and AC is shared