# Lesson HOW TO solve problems on supplementary, complementary or vertical angles - Examples

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## How to solve problems on supplementary, complementary or vertical angles - Examples

In this lesson we present the solutions of the typical problems on supplementary, complementary or vertical angles.
For the definitions and basic properties of these angles see the lesson Angles basics under the topic Angles, complementary, supplementary angles of the section Geometry in this site.

 Problem 1 In the Figure 1 the straight line AB and two                                 supplementary angles ACD and BCD are shown. The angle measure of the angle ACD is 110°. What are the angle measures of the angle BCD?. Solution Since the angles ACD and BCD are supplementary, their sum is the straight angle: LACD + LBCD = 180°. Figure 1. To the Problem 1
Therefore, LBCD = 180° - LACD = 180° - 110° = 70°.

Answer. The angle measure of the angle BCD is 70°.

 Problem 2 In the Figure 2 the straight line AB and                                        supplementary angles ACD and BCD are shown. The angle measure of the angle ACD is in 20° greater than that of the angle BCD. Find the angle measures of the angles ACD and BCD. Solution Let x be the angle measure of the angle BCD (in degrees). Then the angle measure of the angle BCD is equal to x + 20. Since the angles ACD and BCD are supplementary, their sum is the straight angle. This gives you the equation Figure 2. To the Problem 2
x + (x+20) = 180.
Simplify this equation step by step and solve it:
2x + 20 = 180,
2x = 180 - 20,
2x = 160,
x = 80.

Thus, you get that the angle measure of the angle BCD is 80°.
Then the angle measure of the angle BCD is 80° + 20° = 100° in accrdance with the problem condition.
You can check that the sum of the angles ACD and BCD is equal to the straight angle: 100° + 80° = 180°.

So, you get the
Answer. The angle measure of the angle ACD is 100° and the angle measure of the angle BCD is 80°.

 Problem 3 Figure 3 shows two adjacent supplementary angles                    ACD and BCD. The angle measure of the angle ACD is in three times greater than that of the angle BCD. Find the angle measures of the angles ACD and BCD. Solution Let x be the angle measure of the angle BCD (in degrees). Then the angle measure of the angle BCD is equal to 3x. Since the angles ACD and BCD are supplementary, their sum is the straight angle. This gives you the equation Figure 3. To the Problem 3
3x + x = 180.
Simplify this equation step by step and solve it:
4x = 180,
x = 45

Thus, you get that the angle measure of the angle BCD is 45°.
Then the angle measure of the angle BCD is 3*45° = 135° in accrdance with the problem condition.
You can check that the sum of the angles ACD and BCD is equal to the straight angle: 135° + 45° = 180°.

So, you get the
Answer. The angle measure of the angle ACD is 135° and the angle measure of the angle BCD is 45°.

 Problem 4 In the Figure 4 the right angle BAC and the                                 complementary angles BAD and DAC are shown. The angle measure of the angle BAD is 50°. What are the angle measures of the angle DAC?. Solution Since the angles BAD and DAC are complementary, their sum is the right angle: LBAD + LDAC = 90°. Figure 4. To the Problem 4
Therefore, LDAC = 90° - LBAD = 90° - 50° = 40°.

Answer. The angle measure of the angle DAC is 40°.

Problem 5
Find the angle, if its complementary angle is in 20° greater.

Solution
Let x be the angle measure of the given angle (in degrees).
Then the angle measure of the complementary angle is equal to x + 20 in accrdance with the problem condition.
Since the sum of the given angle and its complementary is the right angle, you can write the equation
x + (x+20) = 90.
Simplify this equation step by step and solve it:
2x + 20 = 90,
2x = 90 - 20,
2x = 70,
x = 35.

Thus, you get that the angle measure of the given angle is 35°.
Then the angle measure of the complementary angle is 35° + 20° = 55° in accrdance with the problem condition.
You can check that the sum of the given angle and its complementary is equal to the right angle: 35° + 55° = 90°.

So, you get the
Answer. The angle measure of the given angle is 35° and the angle measure of the complementary angle is 55°.

Problem 6
Find the angle, if its complementary angle is in four times greater.

Solution
Let x be the angle measure of the given angle (in degrees).
Then the angle measure of the complementary angle is equal to 4x in accrdance with the problem condition.
Since the sum of the given angle and its complementary is the right angle, you can write the equation
x + 4x = 90.
Simplify this equation step by step and solve it:
5x = 90,
x = 18.

Thus, you get that the angle measure of the given angle is 18°.
Then the angle measure of the complementary angle is 4*18° = 72° in accrdance with the problem condition.
You can check that the sum of the given angle and its complementary is equal to the right angle: 18° + 72° = 90°.

Answer. The angle measure of the given angle is 18° and the angle measure of the complementary angle is 72°.

 Problem 7 In the Figure 5 the straight lines AB and CD                               intersect at the point O. The angle measure of the angle DOB is 63°. What is the angle measures of the angle AOC?. Solution Since the angles AOC and DOB are vertical angles, the angle AOC is congruent to the angle DOB and their angle measures are the same. Figure 5. To the Problem 7
Therefore, the angle measure of the angle AOC is 63°.

Answer. The angle measure of the angle AOC is 63°.

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