Angle Pairs


Angle Pairs

 

Definitions

  • 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.

 

 

Example

Finding an Angle Measure

∠1 and ∠2 are complementary, and m∠2 = 32. Find m∠1.

Solution

  • 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 

 


Practice

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



Vertical Angles

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

Example

Find m∠2, m∠3, and m∠4.                 


Solution 

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


 

Perpendicular Lines

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.


 

Practice

Find the measures of the numbered angles.

5.       

 

6.   

  

 

 

Parallel Lines

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

Example

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


Example

Use the diagram to find m∠1.     


Solution

∠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



Practice

Find the angle measure.   

  

7.  m∠2

 


8.  m∠3

 

 

9.  m∠4

 

 

10.  m∠6



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.

15.   

       

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.





Answers

1.  supplementary

2.  neither

3.  supplementary

4.  complementary

5.  m∠9 = 126o; m∠10 = 54o; m∠11 = 126o

6.  m∠6 = 43o; m∠7 = 137o; m∠8 = 43o

7.  85o

8.  95o

9.  85o

10.  85o

11.  supplementary

12.  perpendicular

13.  supplementary

14.  complementary

15.  Vertical angles are congruent, so m∠2 = 112o.

16.  5

17.  13

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.