☰ Year plan · Mathematics

Congruence

Week 8 · Congruence: congruent polygons and the five cases of congruent triangles · 3 lessons
1

Congruent figures: equal sides and angles in corresponding order

Mathematics Prep 2 Term 1, Unit 3, Lesson 1
GoalSay when two figures are congruent, name corresponding vertices in order, and find unknown sides and angles.
New words
congruent (≅) · exactly the same shape and size: one fits on the othercorresponding parts · the sides or angles that match in position in the two figuresvertex order · in Δ ABC ≅ Δ DEF, A matches D, B matches E and C matches Fsame shape · not enough: a small and a large square have the same shape but are not congruent
Explanation

Two figures are congruent when one can be moved (slid, turned or flipped) to fit exactly on the other. Then all the corresponding sides are equal and all the corresponding angles are equal. For polygons both are needed: a square of side 4 cm and a square of side 5 cm have equal angles but not equal sides, so they are not congruent; a 3 × 6 rectangle and a 2 × 9 rectangle have the same area, 18, but are not congruent. We write the vertices in corresponding order: Δ ABC ≅ Δ DEF means A → D, B → E, C → F, so AB = DE, BC = EF, CA = FD and m(∠A) = m(∠D), and so on. Find corresponding parts by their position (opposite the same angle, between the same sides), not by their names or by the way the picture is turned.

ABCDEF
Worked examples
Are a 3 × 6 rectangle and a 2 × 9 rectangle congruent? Their areas are both 18 cm², but the sides are different (3 and 6, against 2 and 9). The sides are not equal, so they are not congruent. Equal area is not enough.
Δ ABC ≅ Δ DEF with m(∠A) = 50°, m(∠B) = 60°, AB = 7 cm and BC = 5 cm. Then DE = 7 cm and EF = 5 cm. Angle C is 180° − 50° − 60° = 70°, and F matches C, so m(∠F) = 70°.
Congruent quadrilaterals ABCD ≅ EFGH have AB = 5x − 2 and EF = 3x + 4. Equal sides: 5x − 2 = 3x + 4, so 2x = 6 and x = 3. Check: 5 × 3 − 2 = 13 and 3 × 3 + 4 = 13.
Class activity

Cut and match. Draw a scalene triangle on card and cut out two copies. Label the first A, B, C and the second D, E, F so that they match. Lay them on top of each other in different positions. Which sides coincide?

Practice
1
If Δ ABC ≅ Δ DEF, which side matches AC and which angle matches ∠B?
2
Δ PQR ≅ Δ XYZ with PQ = 8 cm and m(∠R) = 40°. Which side equals PQ and which angle equals ∠R?
3
Two congruent triangles have corresponding sides 3x − 5 and x + 7. Find x and the side length.
4
In Δ ABC, ∠A = 90°. In Δ DEF, ∠E = 90°. If AB = EF and AC = ED, complete: Δ ABC ≅ Δ E__ __.
5
A square of side 6 cm and a rhombus of side 6 cm: congruent? Explain.