Understanding Vertical and Supplementary Angles

What Are Vertical Angles?

Vertical angles (also known as opposite angles) are the angles that are opposite each other when two lines cross. These angles are always equal. For example, when two lines intersect, four angles are formed, and the pairs of opposite angles are vertical angles. If one vertical angle is known, the other vertical angle can be easily determined as they are congruent (equal in measure).

Example:

If two lines intersect and form a vertical angle of $$70^\circ$$, the angle opposite it is also $$70^\circ$$.

What Are Supplementary Angles?

Supplementary angles are two angles whose measures add up to $$180^\circ$$. These angles don’t have to be adjacent (next to each other), but if they are, they form a straight line. When two angles are supplementary, you can find one angle if the other is known by subtracting the known angle from $$180^\circ$$.

Example:

If one angle measures $$110^\circ$$, the supplementary angle is:
$$
180^\circ – 110^\circ = 70^\circ
$$

Key Concepts:

• Vertical angles are equal.
• Supplementary angles add up to $$180^\circ$$.

Practice Questions: Finding Measures of Vertical and Supplementary Angles

Easy Level Questions

1. If one vertical angle is $$45^\circ$$, what is the measure of the opposite angle?
2. Two angles are supplementary, and one angle measures $$100^\circ$$. What is the other angle?
3. What is the measure of the vertical angle if the other angle is $$80^\circ$$?
4. If one angle is $$120^\circ$$, what is the measure of its supplementary angle?
5. Two angles are supplementary. If one angle is $$90^\circ$$, what is the other angle?
6. Find the measure of the vertical angle if one angle is $$30^\circ$$.
7. Two angles are supplementary, and one angle is $$70^\circ$$. What is the other angle?
8. If a vertical angle is $$60^\circ$$, what is the measure of its opposite angle?
9. Two angles are supplementary. If one angle is $$130^\circ$$, what is the other angle?
10. If one vertical angle is $$50^\circ$$, what is the measure of the angle opposite it?
11. Two angles form a straight line, and one angle is $$85^\circ$$. What is the measure of the other angle?
12. If two vertical angles are formed, and one angle is $$95^\circ$$, what is the measure of the other vertical angle?
13. What is the supplementary angle of $$65^\circ$$?
14. If one vertical angle measures $$110^\circ$$, what is the measure of the opposite angle?
15. Two angles are supplementary. If one angle is $$140^\circ$$, what is the other angle?
16. Find the measure of the vertical angle if one angle is $$25^\circ$$.
17. Two angles form a straight line, and one angle measures $$125^\circ$$. What is the measure of the other angle?
18. If a vertical angle is $$75^\circ$$, what is the measure of its opposite angle?
19. Two angles are supplementary, and one angle is $$45^\circ$$. What is the other angle?
20. If one vertical angle is $$15^\circ$$, what is the measure of the opposite angle?

Medium Level Questions

1. Two supplementary angles add up to $$180^\circ$$. If one angle is $$37^\circ$$, what is the other angle?
2. If two vertical angles are formed, and one angle is $$125^\circ$$, what is the measure of the other angle?
3. Two angles are supplementary, and one angle is $$110^\circ$$. What is the measure of the other angle?
4. Find the measure of the vertical angle if one angle is $$150^\circ$$.
5. Two angles form a straight line, and one angle measures $$65^\circ$$. What is the measure of the other angle?
6. Two vertical angles are formed. If one angle is $$47^\circ$$, what is the measure of the opposite angle?
7. If two angles are supplementary, and one angle is $$78^\circ$$, what is the other angle?
8. What is the measure of the vertical angle if one angle is $$125^\circ$$?
9. Two angles form a straight line, and one angle is $$95^\circ$$. What is the measure of the other angle?
10. If two vertical angles are formed and one is $$112^\circ$$, what is the measure of the other angle?
11. Two angles are supplementary. If one angle is $$62^\circ$$, what is the other angle?
12. If two vertical angles are formed, and one angle is $$108^\circ$$, what is the measure of the other angle?
13. Find the measure of the supplementary angle of $$73^\circ$$.
14. If one vertical angle is $$132^\circ$$, what is the measure of its opposite angle?
15. Two angles form a straight line, and one angle is $$115^\circ$$. What is the measure of the other angle?
16. What is the measure of the vertical angle if one angle is $$29^\circ$$?
17. Two angles are supplementary, and one angle is $$85^\circ$$. What is the measure of the other angle?
18. If one vertical angle is $$145^\circ$$, what is the measure of its opposite angle?
19. Two angles form a straight line, and one angle measures $$160^\circ$$. What is the measure of the other angle?
20. What is the supplementary angle of $$98^\circ$$?

Hard Level Questions

1. Two vertical angles are formed, and one angle is $$57^\circ$$. What is the measure of the other angle?
2. Find the measure of the supplementary angle of $$119^\circ$$.
3. If one vertical angle is $$134^\circ$$, what is the measure of its opposite angle?
4. Two angles are supplementary. If one angle is $$88^\circ$$, what is the other angle?
5. Two angles form a straight line. If one angle is $$121^\circ$$, what is the measure of the other angle?
6. Find the measure of the vertical angle if one angle is $$149^\circ$$.
7. Two supplementary angles add up to $$180^\circ$$. If one angle is $$133^\circ$$, what is the other angle?
8. Two vertical angles are formed. If one angle is $$104^\circ$$, what is the measure of the opposite angle?
9. What is the measure of the supplementary angle of $$109^\circ$$?
10. Two angles form a straight line. If one angle is $$117^\circ$$, what is the measure of the other angle?
11. Two angles are supplementary. If one angle is $$135^\circ$$, what is the other angle?
12. If one vertical angle is $$125^\circ$$, what is the measure of the opposite angle?
13. Two supplementary angles add up to $$180^\circ$$. If one angle is $$43^\circ$$, what is the other angle?
14. If one vertical angle is $$67^\circ$$, what is the measure of the opposite angle?
15. Two angles are supplementary, and one angle is $$94^\circ$$. What is the measure of the other angle?
16. Two vertical angles are formed. If one angle is $$140^\circ$$, what is the measure of the other angle?
17. Two angles form a straight line. If one angle is $$150^\circ$$, what is the measure of the other angle?
18. What is the measure of the supplementary angle of $$171^\circ$$?
19. Two vertical angles are formed. If one angle is $$123^\circ$$, what is the measure of the other angle?
20. Two angles form a straight line. If one angle is $$99^\circ$$, what is the measure of the other angle?

Answers with Explanation

Easy Level Answers

1. Answer: $$45^\circ$$
   Explanation: Vertical angles are equal. If one angle is $$45^\circ$$, its opposite angle must also be $$45^\circ$$.
2. Answer: $$80^\circ$$
   Explanation: Supplementary angles add up to $$180^\circ$$.
   $$180^\circ – 100^\circ = 80^\circ$$
3. Answer: $$80^\circ$$
   Explanation: Vertical angles are equal. If one angle is $$80^\circ$$, the opposite angle is also $$80^\circ$$.
4. Answer: $$60^\circ$$
   Explanation: Supplementary angles add up to $$180^\circ$$.
   $$180^\circ – 120^\circ = 60^\circ$$
5. Answer: $$90^\circ$$
   Explanation: Supplementary angles add up to $$180^\circ$$.
   $$180^\circ – 90^\circ = 90^\circ$$
6. Answer: $$30^\circ$$
   Explanation: Vertical angles are equal. If one angle is $$30^\circ$$, its opposite angle is also $$30^\circ$$.
7. Answer: $$110^\circ$$
   Explanation: Supplementary angles add up to $$180^\circ$$.
   $$180^\circ – 70^\circ = 110^\circ$$
8. Answer: $$60^\circ$$
   Explanation: Vertical angles are equal. If one angle is $$60^\circ$$, the opposite angle is also $$60^\circ$$.
9. Answer: $$50^\circ$$
   Explanation: Supplementary angles add up to $$180^\circ$$.
   $$180^\circ – 130^\circ = 50^\circ$$
10. Answer: $$50^\circ$$
    Explanation: Vertical angles are equal. If one angle is $$50^\circ$$, the opposite angle is also $$50^\circ$$.
11. Answer: $$95^\circ$$
    Explanation: Supplementary angles add up to $$180^\circ$$.
    $$180^\circ – 85^\circ = 95^\circ$$
12. Answer: $$95^\circ$$
    Explanation: Vertical angles are equal. If one angle is $$95^\circ$$, the opposite angle is also $$95^\circ$$.
13. Answer: $$115^\circ$$
    Explanation: Supplementary angles add up to $$180^\circ$$.
    $$180^\circ – 65^\circ = 115^\circ$$
14. Answer: $$110^\circ$$
    Explanation: Vertical angles are equal. If one angle is $$110^\circ$$, the opposite angle is also $$110^\circ$$.
15. Answer: $$40^\circ$$
    Explanation: Supplementary angles add up to $$180^\circ$$.
    $$180^\circ – 140^\circ = 40^\circ$$
16. Answer: $$25^\circ$$
    Explanation: Vertical angles are equal. If one angle is $$25^\circ$$, the opposite angle is also $$25^\circ$$.
17. Answer: $$55^\circ$$
    Explanation: Supplementary angles add up to $$180^\circ$$.
    $$180^\circ – 125^\circ = 55^\circ$$
18. Answer: $$75^\circ$$
    Explanation: Vertical angles are equal. If one angle is $$75^\circ$$, the opposite angle is also $$75^\circ$$.
19. Answer: $$135^\circ$$
    Explanation: Supplementary angles add up to $$180^\circ$$.
    $$180^\circ – 45^\circ = 135^\circ$$
20. Answer: $$15^\circ$$
    Explanation: Vertical angles are equal. If one angle is $$15^\circ$$, the opposite angle is also $$15^\circ$$.

Medium Level Answers

1. Answer: $$143^\circ$$
   Explanation: Supplementary angles add up to $$180^\circ$$.
   $$180^\circ – 37^\circ = 143^\circ$$
2. Answer: $$125^\circ$$
   Explanation: Vertical angles are equal. If one angle is $$125^\circ$$, the opposite angle is also $$125^\circ$$.
3. Answer: $$70^\circ$$
   Explanation: Supplementary angles add up to $$180^\circ$$.
   $$180^\circ – 110^\circ = 70^\circ$$
4. Answer: $$150^\circ$$
   Explanation: Vertical angles are equal. If one angle is $$150^\circ$$, the opposite angle is also $$150^\circ$$.
5. Answer: $$115^\circ$$
   Explanation: Supplementary angles add up to $$180^\circ$$.
   $$180^\circ – 65^\circ = 115^\circ$$
6. Answer: $$47^\circ$$
   Explanation: Vertical angles are equal. If one angle is $$47^\circ$$, the opposite angle is also $$47^\circ$$.
7. Answer: $$102^\circ$$
   Explanation: Supplementary angles add up to $$180^\circ$$.
   $$180^\circ – 78^\circ = 102^\circ$$
8. Answer: $$125^\circ$$
   Explanation: Vertical angles are equal. If one angle is $$125^\circ$$, the opposite angle is also $$125^\circ$$.
9. Answer: $$85^\circ$$
   Explanation: Supplementary angles add up to $$180^\circ$$.
   $$180^\circ – 95^\circ = 85^\circ$$
10. Answer: $$112^\circ$$
    Explanation: Vertical angles are equal. If one angle is $$112^\circ$$, the opposite angle is also $$112^\circ$$.
11. Answer: $$118^\circ$$
    Explanation: Supplementary angles add up to $$180^\circ$$.
    $$180^\circ – 62^\circ = 118^\circ$$
12. Answer: $$108^\circ$$
    Explanation: Vertical angles are equal. If one angle is $$108^\circ$$, the opposite angle is also $$108^\circ$$.
13. Answer: $$107^\circ$$
    Explanation: Supplementary angles add up to $$180^\circ$$.
    $$180^\circ – 73^\circ = 107^\circ$$
14. Answer: $$132^\circ$$
    Explanation: Vertical angles are equal. If one angle is $$132^\circ$$, the opposite angle is also $$132^\circ$$.
15. Answer: $$65^\circ$$
    Explanation: Supplementary angles add up to $$180^\circ$$.
    $$180^\circ – 115^\circ = 65^\circ$$
16. Answer: $$29^\circ$$
    Explanation: Vertical angles are equal. If one angle is $$29^\circ$$, the opposite angle is also $$29^\circ$$.
17. Answer: $$95^\circ$$
    Explanation: Supplementary angles add up to $$180^\circ$$.
    $$180^\circ – 85^\circ = 95^\circ$$
18. Answer: $$145^\circ$$
    Explanation: Vertical angles are equal. If one angle is $$145^\circ$$, the opposite angle is also $$145^\circ$$.
19. Answer: $$20^\circ$$
    Explanation: Supplementary angles add up to $$180^\circ$$.
    $$180^\circ – 160^\circ = 20^\circ$$
20. Answer: $$82^\circ$$
    Explanation: Supplementary angles add up to $$180^\circ$$.
    $$180^\circ – 98^\circ = 82^\circ$$

Hard Level Answers

1. Answer: $$57^\circ$$
   Explanation: Vertical angles are equal. If one angle is $$57^\circ$$, the opposite angle is also $$57^\circ$$.
2. Answer: $$61^\circ$$
   Explanation: Supplementary angles add up to $$180^\circ$$.
   $$180^\circ – 119^\circ = 61^\circ$$
3. Answer: $$134^\circ$$
   Explanation: Vertical angles are equal. If one angle is $$134^\circ$$, the opposite angle is also $$134^\circ$$.
4. Answer: $$92^\circ$$
   Explanation: Supplementary angles add up to $$180^\circ$$.
   $$180^\circ – 88^\circ = 92^\circ$$
5. Answer: $$59^\circ$$
   Explanation: Supplementary angles add up to $$180^\circ$$.
   $$180^\circ – 121^\circ = 59^\circ$$
6. Answer: $$149^\circ$$
   Explanation: Vertical angles are equal. If one angle is $$149^\circ$$, the opposite angle is also $$149^\circ$$.
7. Answer: $$47^\circ$$
   Explanation: Supplementary angles add up to $$180^\circ$$.
   $$180^\circ – 133^\circ = 47^\

circ$$

8. Answer: $$104^\circ$$
   Explanation: Vertical angles are equal. If one angle is $$104^\circ$$, the opposite angle is also $$104^\circ$$.
9. Answer: $$71^\circ$$
   Explanation: Supplementary angles add up to $$180^\circ$$.
   $$180^\circ – 109^\circ = 71^\circ$$
10. Answer: $$63^\circ$$
    Explanation: Supplementary angles add up to $$180^\circ$$.
    $$180^\circ – 117^\circ = 63^\circ$$
11. Answer: $$45^\circ$$
    Explanation: Supplementary angles add up to $$180^\circ$$.
    $$180^\circ – 135^\circ = 45^\circ$$
12. Answer: $$125^\circ$$
    Explanation: Vertical angles are equal. If one angle is $$125^\circ$$, the opposite angle is also $$125^\circ$$.
13. Answer: $$137^\circ$$
    Explanation: Supplementary angles add up to $$180^\circ$$.
    $$180^\circ – 43^\circ = 137^\circ$$
14. Answer: $$67^\circ$$
    Explanation: Vertical angles are equal. If one angle is $$67^\circ$$, the opposite angle is also $$67^\circ$$.
15. Answer: $$86^\circ$$
    Explanation: Supplementary angles add up to $$180^\circ$$.
    $$180^\circ – 94^\circ = 86^\circ$$
16. Answer: $$140^\circ$$
    Explanation: Vertical angles are equal. If one angle is $$140^\circ$$, the opposite angle is also $$140^\circ$$.
17. Answer: $$30^\circ$$
    Explanation: Supplementary angles add up to $$180^\circ$$.
    $$180^\circ – 150^\circ = 30^\circ$$
18. Answer: $$9^\circ$$
    Explanation: Supplementary angles add up to $$180^\circ$$.
    $$180^\circ – 171^\circ = 9^\circ$$
19. Answer: $$123^\circ$$
    Explanation: Vertical angles are equal. If one angle is $$123^\circ$$, the opposite angle is also $$123^\circ$$.
20. Answer: $$81^\circ$$
    Explanation: Supplementary angles add up to $$180^\circ$$.
    $$180^\circ – 99^\circ = 81^\circ$$

This set of questions and answers covers a comprehensive range of problems involving vertical and supplementary angles, with clear explanations based on the key properties of these angles.