1.3b Computing Exterior Angle and its Adjacent Interior Angle
LEARNING OBJECTIVES
- Compute for the measure of the exterior angle given the adjacent interior angle and vice versa.
Review
From the previous reading, we defined exterior angle as the angle formed when a side of a polygon is extended outward. Meanwhile, an interior angle is the angle formed inside the polygon between two adjacent sides. The exterior angle and the adjacent interior angle (the interior angle next to the exterior angle) are supplementary angles.
Exterior Angle Theorem
Exterior Angle Theorem
The exterior angle and its adjacent interior angle are supplementary or adds up to 180°.
We already mentioned that exterior angle and its adjacent interior angle are supplementary angles. This means that when you add the measure of an exterior angle to the measure of its adjacent interior angle, you will always get a sum of 180°. This is also known as the Exterior Angle Theorem (EAT).
Drag any vertex of the triangle and observe what happens to the measure of the exterior angle and its adjacent interior angle.
Following the Exterior Angle Theorem, we can solve for the measure of the exterior angle or its adjacent interior angle when either one is given. For example, if the exterior angle of a polygon measures 100°, then the adjacent interior angle will measure 80° because 100° + 80° = 180°, making them supplementary angles. Or if the adjacent interior angle is 20, then the exterior angle will be 160° because 20° + 160° = 180°.
Sample Problems
Example 1
If ∠1 measures 120° (or m∠1 = 120°), what is the measure of its adjacent interior angle?
Solution: Let’s name the adjacent interior angle of ∠1 as ∠3
| Step-by-step solution | Explanation |
|---|---|
| m∠1 + m∠3 = 180° | by the Exterior Angle Theorem |
| 120° + m∠3 = 180° | Substitute the value of m∠1 = 120 |
| 120° + m∠3 - 120° = 180° - 120° | Subtraction Property of Equality |
| m∠3 = 60° | Simplify the Equation |
Therefore, the measure of the adjacent interior angle is 60°.
Example 2
If m∠2 = 38°, find the measure of m∠5, given that ∠2 is an exterior angle and ∠5 is its adjacent interior angle.
Solution: ∠2 and ∠5 are supplementary angles so their measures must add up to 180°.
m∠2 + m∠5 = 180°
38° + m∠5 = 180°
38° + m∠5 - 38° = 180° - 38°
m∠5 = 142°
Therefore, the measure of ∠5 is 142°.
Example 3
If m∠4 = 75°, where ∠4 is the exterior angle and ∠6 is its adjacent interior angle. Calculate the measure of m∠6.
Solution: ∠4 and ∠6 are supplementary angles so their measures must add up to 180°.
m∠4 + m∠6 = 180°
75° + m∠6 = 180°
75° + m∠6 - 75° = 180° - 75°
m∠6 = 105°
Therefore, the measure of ∠6 is 105°.
Example 4
Given m∠7 = 56°, determine m∠8, where ∠7 is an exterior angle and ∠8 is the adjacent interior angle.
Solution: Since ∠7 and ∠8 are supplementary angles, their measures add up to 180°.
m∠7 + m∠8 = 180°
56° + m∠8 = 180°
56° + m∠8 - 56° = 180° - 56°
m∠8 = 124°
Therefore, the measure of ∠8 is 124°.
Example 5
If m∠9 = 40°, find the measure of m∠10, given that ∠9 is an exterior angle and ∠10 is an adjacent angle.
Solution: Since ∠9 and ∠10 are supplementary angles, they must add up to 180°.
m∠9 + m∠10 = 180°
40° + m∠10 = 180°
40° + m∠10 - 40° = 180° - 40°
m∠10 = 140°
Therefore, the measure of ∠10 is 140°.
Example 6
Find the measure of m∠12 if m∠11 = 63°, with ∠11 as an exterior angle and ∠12 as the adjacent interior angle.
Solution: Since ∠11 and ∠12 are supplementary angles, their measures add up to 180°.
m∠11 + m∠12 = 180°
63° + m∠12 = 180°
63° + m∠12 - 63° = 180° - 63°
m∠12 = 117°
Therefore, the measure of ∠12 is 117°.
Activity
Try doing this activity. Apply the Exterior Angle Theorem or the other Angle Pair relationships that we already covered.
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