A triangle is the simplest polygon, made of three sides and three angles, and it is also rigid: once its three side lengths are fixed, its shape cannot change.
In an isosceles triangle, whose two sides are equal, the two base angles are always equal, and in every triangle the three interior angles add up to 180°.
The exterior angle made by extending a side at a vertex equals the sum of the two interior angles not next to it, which lets many angle problems be finished in a single line.
Changing the angles in the two pictures shows how these properties fit together in numbers.
The two base angles of an isosceles triangle
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An isosceles triangle ABC with adjustable base angles, its apex angle and the bisector of the apex angle
65°
👀 Properties of an isosceles triangle
①In an isosceles triangle with AB = AC, the base angles ∠B and ∠C are equal.
②The bisector of the apex angle cuts the base in half at right angles; it is the dashed line in the picture.
③Conversely, a triangle with two equal angles has equal sides opposite them, so it is isosceles.
④As the base angles grow, the apex angle shrinks to 180° − 2 × (base angle).
Interior and exterior angles
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Triangle ABC with two chosen angles, and the exterior angle at C made by extending BC
50°
60°
Angle sum of a triangle
∠A + ∠B + ∠C = 180°
Drawing the line through A parallel to BC, the alternate angles gather all three angles on one straight line, making 180°.
Exterior angle property
(exterior angle at C) = ∠A + ∠B
The exterior angle and ∠C add up to 180°, and so do ∠A + ∠B + ∠C, so the exterior angle equals ∠A + ∠B.
Equilateral and right triangles
💡 Special triangles
①An equilateral triangle has three equal sides, and all three angles are 60°.
②A right triangle has one 90° angle, so its two acute angles add up to 90°.
③A right isosceles triangle has angles of 90°, 45° and 45°, and its hypotenuse is √2 times as long as either of the other sides.
Conditions for congruent triangles
Congruence conditions
SSS · SAS · ASA
SSS: three sides; SAS: two sides and the angle between them; ASA: one side and the two angles at its ends.
📝 The three congruence conditions
①SSS: if all three pairs of corresponding sides are equal, the triangles are congruent.
②SAS: if two pairs of corresponding sides are equal and the angles between them are equal, the triangles are congruent.
③ASA: if one pair of corresponding sides is equal and the angles at both ends are equal, the triangles are congruent.
④Right triangles have an extra condition, hypotenuse and one other side (RHS), which cannot be used for triangles in general.
Worked examples
Example 1
In an isosceles triangle the apex angle is 40°. Find one base angle.
1
The base angles of an isosceles triangle are equal, and the three angles add up to 180°.
sum of base angles = 180° − 40° = 140°
2
The base angles are equal, so divide by 2.
one base angle = 140° ÷ 2 = 70°
▸ 70°
The two base angles of an isosceles triangle are equal.
Example 2
Two interior angles of a triangle are 50° and 60°. Find the exterior angle at the third vertex.
1
The exterior angle at the third vertex equals the sum of the non-adjacent interior angles 50° and 60°.
exterior angle = 50° + 60°
2
Add the two angles.
= 110°
▸ 110°
Finding the third interior angle, 70°, first and subtracting it from 180° gives the same answer.
Exam checklist
Key facts about triangles
angle sum 180° · exterior angle = sum of the two non-adjacent interior angles · congruence by SSS, SAS, ASA
The equal base angles of an isosceles triangle are tested most often.
Grade-8 school exam type
In an isosceles triangle a base angle is 65°. What is the apex angle?
①40°
②45°
③50°
④55°
⑤65°
▸ ③ 50°
1
The base angles are equal, so together they make 65° × 2 = 130°.
sum of base angles = 65° × 2 = 130°
2
Subtract from the angle sum of 180°.
apex angle = 180° − 130° = 50°
🎯 Exam points
①In an isosceles triangle, "two equal sides" and "two equal base angles" each guarantee the other.
②An exterior angle is the sum of the two non-adjacent interior angles, so it can be found without working out each interior angle.
③To prove congruence, find which of SSS, SAS and ASA the given information matches.
④Each interior angle of an equilateral triangle is 60°.
⑤In SAS the angle must be the one between the two sides; equal angles that are not between them may not give congruence.