☰ Year plan · Mathematics

Congruence

Week 8 · Congruence: congruent polygons and the five cases of congruent triangles · 3 lessons
2

Congruent triangles: three sides (SSS) and two sides with the included angle (SAS)

Mathematics Prep 2 Term 1, Unit 3, Lesson 1
GoalUse SSS and SAS to show that two triangles are congruent, and write a short proof with a reason for every step.
New words
SSS · three sides of one triangle equal three sides of the otherSAS · two sides and the angle between them are equalincluded angle · the angle that lies between the two given sidescommon side · a side that belongs to both triangles, so it equals itself
Explanation

You do not need to compare all six parts. Three equal sides are enough (SSS): a triangle made from three sticks has only one possible shape, which is why triangular frames are rigid, while a four-bar frame can lean over. Two equal sides and the angle BETWEEN them are enough (SAS): the angle fixes how far apart the free ends are. The angle must be the included one: if the angle is not between the sides, the triangles can differ. When you write a proof, give a reason for each step: ‘AB = DE (given)’, ‘AD is a common side’, ‘M is the midpoint, so BM = MC’. Then say which case you use, write the congruence with the vertices in order, and state what follows, such as ∠B = ∠E.

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Worked examples
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Are the triangles congruent? In the picture the sides are 5, 6, 7 in both. Each side of Δ ABC equals the matching side of Δ DEF, so Δ ABC ≅ Δ DEF by SSS. Then ∠A = ∠D, ∠B = ∠E and ∠C = ∠F.
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Two triangles have AB = DE = 4 cm, AC = DF = 6 cm and m(∠A) = m(∠D) = 50°. The angle is between the equal sides, so Δ ABC ≅ Δ DEF by SAS, and therefore BC = EF.
ABCD
Prove that the diagonals of a rectangle ABCD are equal. In Δ ABC and Δ DCB: AB = DC (opposite sides of a rectangle), ∠ABC = ∠DCB = 90°, and BC is a common side. So Δ ABC ≅ Δ DCB by SAS, and AC = DB.
Class activity

Straws. Cut straws of 5, 6 and 7 cm and join them with thread or tape into a triangle. Ask a friend to make another with the same three straws. Do the triangles match? Now try four straws of 5 cm: do the shapes all look the same?

Practice
1
Δ ABC has AB = 7, BC = 8, CA = 9 and Δ XYZ has XY = 7, YZ = 8, ZX = 9. Are they congruent? Which case?
2
PQ = LM = 5 cm, PR = LN = 8 cm and ∠P = ∠L = 70°. Congruent? What else is equal?
3
In Δ ABC and Δ DEF, AB = DE = 6, BC = EF = 7 and ∠A = ∠D = 50°. Can we say they are congruent?
4
In Δ ABC, AB = AC and D is the midpoint of BC. Prove Δ ABD ≅ Δ ACD.
5
Why is a triangular shelf bracket rigid but a four-sided frame can lean over?