Two angles are supplementary if the sum of their measures is 180.
Two angles are complementary if the sum of their measures is 90.
You can write "the measure of angle 1" as m∠1.
Finding an Angle Measure
∠1 and ∠2 are complementary, and m∠2 = 32. Find m∠1.
- m∠1 + 32 = 90 Substitute 32 for m∠2.
- m∠1 = 58 Subtract 32 from each side.
m∠1 + m∠ 2 = 90 Definition of complementary angles
Are ∠1 and ∠2 are complementary, supplementary, or neither?
1. m∠1 = 79
m∠2 = 101
2. m∠1 = 53
m∠2 = 47
3. m∠1 = 95
m∠2 = 85
4. m∠1 = 52
m∠2 = 38
When two lines intersect at a point, they form two pairs of angles that do not share a side. These pairs are called vertical angles, and they always have the same measure.
∠1 and ∠3 are vertical angles.
m∠1 = m∠3
∠2 and ∠4 are vertical angles.
m∠2 = m∠4
Using Vertical Angles
Find m∠2, m∠3, and m∠4.
The diagram shows that m∠1 = 90.
∠1 and ∠3 are vertical angles. Their measures are equal, so m∠3 = 90.
∠1 and ∠2 are supplementary.
m∠1 + m∠2 = 180 Definition of supplementary angles
90 + m∠2 = 180 Substitute 90 for m∠1.
m∠2 = 90 Subtract 90 from each side.
∠2 and ∠4 are vertical angles. Their measures are equal, so m∠4 = 90.
ANSWER: m∠2 = m∠3 = m∠4 = 90
When two lines intersect to form one right angle, they form four right angles. Two lines that intersect at a right angle are called perpendicular lines.
Find the measures of the numbered angles.
Two lines in the same plane that do not intersect are called parallel lines. When a line intersects two parallel lines, several pairs of angles that are formed have equal measures.
Angles and Parallel Lines
Corresponding Angles: m∠1 = m∠5; m∠2 = m∠6;
m∠3 = m∠7; m∠4 = m∠8
Alternate Interior Angles: m∠3 = m∠6; m∠4 = m∠5
Alternate Exterior Angles: m∠1 = m∠8; m∠2 = m∠7
Using Parallel Lines
Use the diagram to find m∠1.
∠1 and ∠5 are corresponding angles, so they have equal measures.
Find m∠5. The angle with measure 125 and ∠5 are supplementary.
m∠5 + 125 = 180 Definition of supplementary angles
m∠5 = 55 Subtract 125 from each side.
∠1 and ∠5 have equal measures.
ANSWER: m∠1 = 55
Find the angle measure.
Complete the statement.
11. The sum of the measures of two _?_ angles is 180.
12. Two lines that intersect to form a right angle are called _?_.
Are the angles are complementary, supplementary, or neither?
13. m∠1 = 62, m∠2 = 118
14. m∠1 = 51, m∠2 = 39
Describe and correct the error in the solution.
m∠2 = 68, because vertical angles add up to 180.
Find the value of the variable and the angle measures.
16. m∠1= (5x + 15) and m∠2 = 28x
17. m∠4 = (7n + 39) and m∠5 = (11n - 13)
18. Find the angle measures in the weaving if m∠1 = 122.
19. A student designed the stationery border shown here. Explain how to find m∠2 if m∠1 = 135.
5. m∠9 = 126o; m∠10 = 54o; m∠11 = 126o
6. m∠6 = 43o; m∠7 = 137o; m∠8 = 43o
15. Vertical angles are congruent, so m∠2 = 112o.
18. m∠2 = 58o; m∠3 = 122o; m∠4 = 58o
19. Angles 1 and 2 form a linear pair and are supplementary. So to find m∠2, subtract m∠1 from 180o. m∠2 = 180o - 135o = 45o.