1. 1.5 Angles and Their Measures
Geometry
Angles and Their Measures
Essential Question: How do you bisect an angle.

2. Geometry 1.5 Angles and Their Measure
Topics/Objectives:
Know what an angle is.
Use Angle Postulates.
Classify angles according to size.
To measure angles
To Bisect angles.
April 20, 2018
Geometry 1.5 Angles and Their Measure

3. Geometry 1.5 Angles and Their Measure
What is an angle?
An ANGLE consists of two different rays that have the same initial point.
Sides
The rays are the sides of the angle.
Initial Point
Vertex
The initial point is the VERTEX of the angle.
April 20, 2018
Geometry 1.5 Angles and Their Measure

4. Geometry 1.5 Angles and Their Measure
Interior/Exterior
R is in the interior of A R A S
S is in the exterior of A
April 20, 2018
Geometry 1.5 Angles and Their Measure

5. Geometry 1.5 Angles and Their Measure
Naming Angles
The angle formed by BA and BC is called ABC or CBA, or just B.
B is the vertex. A C B
Important point: when using three letters in the name of an angle, the vertex must be the middle one.
April 20, 2018
Geometry 1.5 Angles and Their Measure

6. Geometry 1.5 Angles and Their Measure
Angle Measure
The measure of ABC is denoted mABC.
Angles are measured in degrees, using the symbol °.
For example, if the measure of D is 24 degrees, we would write m D = 24°.
Angles are measured using a tool called a protractor.
April 20, 2018
Geometry 1.5 Angles and Their Measure

7. Geometry 1.5 Angles and Their Measure
Congruent Angles
Angles that have the same measure are congruent.
The symbol for congruent is
If TUF and LUK each measure 50°, then they are congruent.
Angles are congruent
Measures are equal
April 20, 2018
Geometry 1.5 Angles and Their Measure

8. Postulate 1.3 Protractor Postulate
The measure of an angle is a real number from 0 to 180.
(This is hugely simplified from the book’s version.)
April 20, 2018
Geometry 1.5 Angles and Their Measure

9. Geometry 1.5 Angles and Their Measure
Angle Addition
70°
40°
30°
April 20, 2018
Geometry 1.5 Angles and Their Measure

10. Geometry 1.5 Angles and Their Measure
Postulate 1.4: AAP A B C D
If B is in the interior of ADC,
then mADB + mBDC = mADC
April 20, 2018
Geometry 1.5 Angles and Their Measure

11. Geometry 1.5 Angles and Their Measure
Angle Classification
Acute A
April 20, 2018
Geometry 1.5 Angles and Their Measure

12. Geometry 1.5 Angles and Their Measure
Angle Classification
RIGHT
Use a square at the vertex to indicate this is a right angle. A
April 20, 2018
Geometry 1.5 Angles and Their Measure

13. Geometry 1.5 Angles and Their Measure
Angle Classification
Obtuse A
April 20, 2018
Geometry 1.5 Angles and Their Measure

14. Geometry 1.5 Angles and Their Measure
Angle Classification
straight A
April 20, 2018
Geometry 1.5 Angles and Their Measure

15. Geometry 1.5 Segment and Angle Bisectors
A ray that divides an angle into two adjacent angles that are congruent. A D B C
Use congruence marks to indicate congruent angles.
April 20, 2018
Geometry 1.5 Segment and Angle Bisectors

16. Angle Bisector ExampleU G M
mBUM = 110°
Ray UG is the angle bisector of BUM.
Find mGUM
and mBUG
110°
55°
55°
Think: Ray UG divides BUM into two equal parts, so each is 55°.
April 20, 2018
Geometry 1.5 Segment and Angle Bisectors

17. Another Angle Bisector ExampleU G M
mBUG = 43°
Ray UG is the angle bisector of BUM.
Find mGUM
and mBUM
86°
43°
43°
Think: Ray UG divides BUM into two equal parts, so each is 43°. The total is 86.
April 20, 2018
Geometry 1.5 Segment and Angle Bisectors

18. Geometry 1.5 Segment and Angle Bisectors
Another Example
RT bisects ARC.
mART = (7x + 12)°
mTRC = (4x + 30)°
Find the measure of each angle.
April 20, 2018
Geometry 1.5 Segment and Angle Bisectors

19. Geometry 1.5 Segment and Angle Bisectors
Solution
7(6) + 12 = 54°
RT bisects ARC.
mART = (7x + 12)°
mTRC = (4x + 30)°
Find the measure of each angle.
mART = 54° A T
4(6) + 30 = 54
7x + 12
mTRC = 54°
4x + 30 C
108°
1. DRAW the picture. R
2. LABEL each angle.
3. WRITE an equation.
7x + 12 = 4x + 30
4. SOLVE.
7x = 4x + 18
3x = 18
x = 6
5. ANSWER the question.
April 20, 2018
Geometry 1.5 Segment and Angle Bisectors

20. Geometry 1.5 Segment and Angle Bisectors
Try it. F
Ray AF bisects CAB
mFAB = 8x + 10
mFAC = 3x + 25
Find mBAF and mCAB. B C A
April 20, 2018
Geometry 1.5 Segment and Angle Bisectors

21. Geometry 1.5 Segment and Angle Bisectors
Solution F B
8x + 10 = 3x + 25
5x = 15
x = 3
8x + 10
3x + 25 C A
mBAF = 8(3) + 10 = 34°
mCAB = 2(34) = 68
April 20, 2018
Geometry 1.5 Segment and Angle Bisectors

22. Geometry 1.5 Angles and Their Measure
Bisecting angles.
An angle bisector is a ray that divides an angle into two angles that are congruent.
April 20, 2018
Geometry 1.5 Angles and Their Measure

23. Summarize what we’ve learned:
What an angle is.
How to name angles.
Angle classifications.
Using a bisector to find angles
Symbols for angle and congruent.
April 20, 2018
Geometry 1.5 Angles and Their Measure