Equilateral
Three equal sides and three 60° angles.
Learning Objectives
- classify triangles by sides and angles;
- use the triangle angle sum;
- use properties of isosceles and equilateral triangles;
- find exterior angles of triangles;
- solve algebraic triangle problems.
Introduction
A triangle is a three-sided polygon. Its side markings and angle sizes tell us which properties we can use.
Begin by noticing what is equal or special; the calculation comes after the geometry.
Classifying by sides
Isosceles
Two equal sides and two equal base angles.
Scalene
All sides have different lengths.
Key Notes
Acute triangle
All three angles are less than 90°.
Right triangle
One angle is exactly 90°.
Obtuse triangle
One angle is greater than 90°.
Triangle Angle Sum
The interior angles of every triangle add up to 180°.
Exterior Angle Rule
An exterior angle of a triangle equals the sum of the two opposite interior angles.
Two opposite interior angles are 48° and 67°.
Exterior angle = 48°+67°=115°.
Worked Example: Isosceles Triangle
The vertex angle is 44°. Find each base angle.
Remaining total=180°−44°=136°.
Base angles are equal, so each is 136°÷2=68°.
Try It: Complete the Triangle
50° + 60° + 70° = 180°
| A | B | C | Triangle type |
|---|---|---|---|
| 50° | 60° | 70° | acute |
🧠 Let’s Think Together
An isosceles triangle has a vertex angle of 44°. The other two angles are equal. First remove the vertex angle from 180°: 180° − 44° = 136°. Then share 136° equally, giving two base angles of 68°.
Worked Examples
Example 1: Find a missing interior angle
A triangle has angles 52°, 73° and x. Since the angles total 180°, x = 180° − 52° − 73° = 55°.
Example 2: Use the exterior-angle rule
Two opposite interior angles are 48° and 67°. The exterior angle is 48° + 67° = 115°.
Example 3: Use algebra
The angles are x, 2x and 3x. Then 6x = 180°, so x = 30°. The angles are 30°, 60° and 90°.
Practice Questions
Foundation- Find the third angle of a triangle with angles 52° and 73°.
- An equilateral triangle has side length 8 cm. State all its angles.
- An isosceles triangle has equal base angles of 71°. Find the vertex angle.
- The angles of a triangle are x, 2x and 3x. Find all angles.
- An exterior angle is 128° and one opposite interior angle is 49°. Find the other opposite angle.
- The vertex angle of an isosceles triangle is 20° less than each base angle. Find all three angles.
✅ Check Your Work
Show practice answers
- 55°.
- 60°, 60°, 60°.
- 38°.
- 6x=180, x=30. Angles: 30°, 60°, 90°.
- 79°.
- Let each base angle=x, vertex=x−20. Then 3x−20=180, x=200/3=66⅔°. Vertex=46⅔°.
Common Mistakes
- Assuming a triangle is isosceles without equal-side markings or a statement.
- Forgetting that all three interior angles total 180°.
- Adding the exterior angle to the wrong pair of interior angles.
Exam Focus
Mark equal angles before calculating and state the property used. For an exterior angle, identify the two non-adjacent interior angles clearly.
Summary
- Triangle angles total 180°.
- Base angles in an isosceles triangle are equal.
- Each angle of an equilateral triangle is 60°.
- An exterior angle equals the two opposite interior angles combined.