1. Special Pairs of Angles

3. Two or more angles that have the same angle measure
Congruent Angles 
Two or more angles that have the same angle  measure
(in degrees or radians).
<ABD ≅ < DBC
Arcs with out hash marks do not necessarily indicate congruency. Arcs are used to indicate angles being referenced.

4. Certain angle pairs are given special names based on their relative position to one another or based on the sum of their respective measures. 

5. Adjacent angles.
Adjacent angles are any two angles that share a common side and a common vertex but NO common interior points.
In Figure 1 , ∠1 and ∠2
are adjacent angles ...

6. An Angle Bisector is a ray that divides and angle into two adjacent angles that are congruent.

7. Angle bisector
If YM is the angle bisector,
then
X M Y Z

8. Solve for x
BX bisects A
X Solve for X B
C 2x + 6 = 3x + 2
4 = x

9. You Try: Finding the Measure of an Angle
KM bisects JKL, mJKM = (4x + 6)°, and mMKL = (7x – 12)°. Find mJKM.

10. Activity 1: Construct Angle Bisector
1-5 Segments and Angles Bisectors
Activity 1: Construct Angle Bisector

12. GUIDED PRACTICE
LMN and PQR are complementary angles. Find the measures of the angles if m LMN = (4x – 2)° and m PQR = (9x + 1)°.
SOLUTION
m LMN + m PQR = 90°
Complementary angle
(4x – 2 )° + ( 9x + 1 )° = 90°
Substitute value
13x – 1 = 90
Combine like terms
13x = 91
Add 1 to each side
x = 7
Divide 13 from each side

14. Given: ∠ A and ∠ B are complementary.
∠ C and ∠ B are complementary
Prove: ∠ A = ∠ C
1.) ∠ A and ∠ B are complementary.
∠ C and ∠ B are complementary
1.) Given
2.) ∠ A + ∠ B = 90
∠ C  + ∠ B = 90
2.) Definition of complementary angles.
3.) ∠ A = ∠ B
∠ C = ∠ B
3.) Subtraction Property of Equality
4.) ∠ A = ∠ C
4.) Transitive Property

15. In the previous proof, we proved the
Congruent Complements Theorem
If two angles are complementary to the same angle, or to equal angles, then they are equal to each other.

16. Supplementary angles are two angles whose sum is 180°
Supplementary angles are two angles whose sum is 180°. In Figure , ∠ ABC is a straight angle. Therefore m ∠6 + m ∠7 = 180°, so ∠6 and ∠7 are supplementary
Because m ∠8 + m ∠9 = 180°, ∠8 and ∠9 are supplementary

17. Given: ∠ R and ∠ P are supplementary
∠ P and ∠ Q are supplementary
Prove: ∠ R = ∠ Q
1.) ∠ R and ∠ P are complementary.
∠ P and ∠ Q are complementary
1.) Given
2.) ∠ R + ∠ P = 180
∠ P  + ∠ Q = 180
2.) Definition of complementary angles.
3.) ∠ R = ∠ P
∠ Q = ∠ P
3.) Subtraction Property of Equality
4.) ∠ R = ∠ Q
4.) Transitive Property

19. A linear pair of angles is formed when two lines intersect
A linear pair of angles is formed when two lines intersect. Two angles are said to be linear if they are adjacent angles formed by two intersecting lines. The measure of a straight angle is 180 degrees, so a linear pair of angles must add up to 180 degrees.

22. What you can determine from a given diagram:

24. Given: Line l intersects line m at Q Prove: =
2.) ∠1 and ∠4 are linear pair
∠4 and ∠3 are linear pair
2.) Given (diagram)
3.) ∠1 and ∠4 are supplementary
∠4 and ∠3 are supplementary
3.) Definition of linear pair
4.) ∠1 = ∠3
4.) Congruent Supplements Theorem

25. List all the pairs of supplementary angles
List all the pairs of vertical angles

26. You Try:
Solve for X. Then find the measures of the angles.

28. The following diagram shows an equilateral triangle
The following diagram shows an equilateral triangle. Find the value of X and Z in degrees and explain your thinking.

29. The following diagram shows an isosceles triangle
The following diagram shows an isosceles triangle. Find the value of x in degrees.

31. Find the measure of each angle in degrees.