☰ Year plan · Mathematics

The isosceles triangle

Week 11 · The isosceles triangle: base angles, the equilateral triangle, axis of symmetry and the 30°–60°–90° triangle · 3 lessons
1

The isosceles triangle: equal base angles and its converse

Mathematics Prep 2 Term 1, Unit 3, Lesson 5
GoalUse the equal base angles of an isosceles triangle, and the converse, to find angles and sides.
New words
isosceles triangle · a triangle with two equal sides, called the legsbase · the third side, which joins the ends of the two legsvertex angle · the angle between the two equal legsbase angles · the two angles at the ends of the base; they are equal
Explanation

In an isosceles triangle the two equal sides AB and AC are the legs and BC is the base. The angle at A, between the legs, is the vertex angle. The two angles at B and C, at the ends of the base, are the base angles. The first rule: the base angles of an isosceles triangle are equal. Why? Fold the triangle along the line from A to the midpoint of BC: the two halves fit exactly, so the angle at B lands on the angle at C. The converse is also true: if two angles of a triangle are equal, the sides opposite them are equal, so the triangle is isosceles. The angle sum is 180°, so each base angle is (180° − vertex angle) ÷ 2. In the picture the vertex angle is 72° and each base angle is (180° − 72°) ÷ 2 = 54°. Do not mix up the vertex angle and the base angles: the base angles are never the angle between the equal sides.

72°54°54°ABC
Worked examples
In Δ ABC, AB = AC and the vertex angle A = 72°. The base angles are equal, so each is (180° − 72°) ÷ 2 = 108° ÷ 2 = 54°. If instead the base angle B = 65°, then C = 65° and A = 180° − 130° = 50°.
In an isosceles triangle with AB = AC the base angles are B = 3x + 10 and C = 5x − 20. They are equal, so 3x + 10 = 5x − 20, 2x = 30 and x = 15. Each base angle is 55° and the vertex angle is 180° − 110° = 70°.
The converse. In Δ ABC the angles are B = C = 40° (so A = 100°). The sides opposite equal angles are equal, so AB = AC. If AB = 2x + 3 and AC = 4x − 5, then 2x + 3 = 4x − 5, so x = 4 and AB = AC = 11 cm.
Class activity

Fold and check. Cut a paper triangle with two equal sides (use a ruler). Fold it so the two equal sides lie on each other. Which angles lie on each other? Measure them with a protractor.

Practice
1
In an isosceles triangle the vertex angle is 40°. Find each base angle.
2
A base angle of an isosceles triangle is 80°. Find the vertex angle.
3
In an isosceles triangle with AB = AC, B = 2x + 6 and C = 4x − 18. Find x and the vertex angle.
4
A triangle has angles 50°, 50° and 80°. Which two sides are equal?
5
Omar says the vertex angle of an isosceles triangle is 100°, so the base angles are 100° too. What is wrong?