Triangle Properties

Properties of Triangles

A triangle is the simplest polygon, made of three sides and three angles. In an isosceles triangle the two base angles are always equal, and the three interior angles always sum to 180°. An exterior angle equals the two non-adjacent interior angles added together. Here you can drag the base angle and interior angles to watch these rules hold.

Intuition — Isosceles Triangle

Base Angle65°

👀 Key Idea: Isosceles Triangle

①Two sides equal → their base angles are equal

②The bisector of the apex angle ⊥ bisects the base

③Conversely, two equal angles ⇒ opposite sides equal

Interior & Exterior Angles

∠A50°

∠B60°

Interior Angle Sum

∠A + ∠B + ∠C = 180°

The three interior angles always add to 180°

Exterior Angle Theorem

Exterior angle = sum of the two non-adjacent interior angles

An exterior angle equals the two interior angles not next to it

Equilateral & Right Triangles

💡 Special Triangles

①Equilateral: all sides equal, all angles 60°

②Right triangle: one 90°, the other two sum to 90°

③Right isosceles: 90°, 45°, 45° → hypotenuse = leg × √2

Triangle Congruence

Three Congruence Tests

SSS, SAS, ASA

SSS: three sides; SAS: two sides + included angle; ASA: one side + two adjacent angles

📝 Tests Explained

①SSS: three sides equal → congruent

②SAS: two sides and the included angle equal → congruent

③ASA: one side and the two adjacent angles equal → congruent

Work It Out

Example 1

In an isosceles triangle the apex angle is 40°. Find one base angle.

The two base angles are equal, and all three angles sum to 180°.

sum of base angles = 180° − 40° = 140°

The two base angles are equal, so divide by 2.

one base angle = 140° ÷ 2 = 70°

▸ 70°

An isosceles triangle has two equal base angles.

Example 2

Two interior angles of a triangle are 50° and 60°. Find the exterior angle at the third vertex.

An exterior angle equals the sum of the two non-adjacent interior angles.

exterior angle = 50° + 60°

Compute.

= 110°

▸ 110°

An exterior angle of a triangle equals the sum of the two remote interior angles.

Exam Wrap-up

Core Properties

Interior sum 180° + Exterior thm + Congruence SSS/SAS/ASA

The "equal base angles" of an isosceles triangle is the most-tested fact

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°

The two base angles are equal, so they sum to 65° × 2 = 130°.

sum of base angles = 65° × 2 = 130°

Subtract from the 180° angle sum.

apex angle = 180° − 130° = 50°

🎯 Exam Points

①Isosceles: "two equal sides ↔ two equal angles" (and the converse)

②Exterior angle = sum of two non-adjacent interior angles (fast for finding angles)

③Congruence proof: spot SSS/SAS/ASA from the givens

④An equilateral triangle has 60° angles — never forget

⑤Know the meaning of "included angle" in SAS

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