Calculating Angles
In this lesson, you will learn how to calculate angles using triangles, polygons, and parallel lines.
Basic Properties of Angles
- Sum of the interior angles of a triangle: 180°
- Sum of the interior angles of a quadrilateral: 360°
- Sum of the interior angles of an n-sided polygon: 180×(n－2)°
- Vertical angles are equal.
- Alternate interior angles and corresponding angles formed by parallel lines are equal.
Example 1
Two angles of a triangle are 55° and 75°. What is the measure of the remaining angle?
Solution:
180－55－75＝50°
Two angles of a triangle are 55° and 75°. What is the measure of the remaining angle?
Solution:
180－55－75＝50°
Example 2 (Exterior Angle Theorem)
An exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.
When two angles of a triangle are 40° and 65°, what is the measure of the exterior angle of one of the angles?
Solution:
Exterior angle＝40＋65＝105°
An exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.
When two angles of a triangle are 40° and 65°, what is the measure of the exterior angle of one of the angles?
Solution:
Exterior angle＝40＋65＝105°
Example 3 (Angles of a Star)
What is the sum of the angles at the tips of a regular pentagram (five-pointed star)?
Solution:
The sum of the five angles at the tips of the star is 180° (the equivalent of one triangle)
Measure of each tip angle: 180÷5＝36°
What is the sum of the angles at the tips of a regular pentagram (five-pointed star)?
Solution:
The sum of the five angles at the tips of the star is 180° (the equivalent of one triangle)
Measure of each tip angle: 180÷5＝36°
Key Points
- The sum of the interior angles of a triangle, 180°, is the most important formula.
- Exterior Angle Theorem: exterior angle＝sum of the two non-adjacent interior angles
- When you find parallel lines, use alternate interior angles and corresponding angles.
Practice Problems
- What is the sum of the interior angles of a pentagon?
- When the vertex angle of an isosceles triangle is 40°, what is the measure of each base angle?
Answers and Explanations
- Answer: 540°
Explanation: Sum of the interior angles of an n-sided polygon＝180×(n-2)°. For a pentagon (n=5): 180×(5-2)＝180×3＝540°. - Answer: 70°
Explanation: In an isosceles triangle, the two base angles are equal. Vertex angle＋2×base angle＝180°. 40＋2×base angle＝180. Base angle＝（180-40）÷2＝70°.