1. 1.4 Angle Measures

2. Objectives: How to label, measure, and classify angles How to label, measure, and classify angles Identifying and using congruent angles Identifying and using congruent angles Creating and utilizing an angle bisector Creating and utilizing an angle bisector

3. Rays To create angles we must first define a ray. A ray is part of a line which has one endpoint and extends infinitely in one direction. Rays are named stating the endpoint first and then any other point on the ray. To create angles we must first define a ray. A ray is part of a line which has one endpoint and extends infinitely in one direction. Rays are named stating the endpoint first and then any other point on the ray.

4. Labeling Rays We could label this ray as AB, AC, or AD but not CA. We could label this ray as AB, AC, or AD but not CA. A D C B

5. More about Rays If you choose a point on a line, that point determines exactly two rays called opposite rays. If you choose a point on a line, that point determines exactly two rays called opposite rays. Q R P QP and QR are opposite rays.

6. Angles and Their Parts An angle is formed by two noncollinear rays that have a common endpoint. The rays are called sides and the common endpoint is the vertex. An angle is formed by two noncollinear rays that have a common endpoint. The rays are called sides and the common endpoint is the vertex.angle angle B C A Side AB Vertex A Side AC

7. Labeling Angles We label angles any of the following ways: We label angles any of the following ways: BAC, CAB, A, or 1 BAC, CAB, A, or 1 C B A 1

8. More about Angles An angle divides a plane into three distinct parts. Points A, B, and C lie on the angle. Points D and E line in the interior of the angle. Points F and G lie in the exterior of the angle. An angle divides a plane into three distinct parts. Points A, B, and C lie on the angle. Points D and E line in the interior of the angle. Points F and G lie in the exterior of the angle. C B A 1 F D E G

9. Name all angles that have B as a vertex. Answer:  5,  6,  7, and  ABG Example 1a:

10. Name the sides of  5. Answer: and or are the sides of  5. Example 1b:

11. Write another name for  6. Answer:  EBD,  FBD,  DBF, and  DBE are other names for  6. Example 1c:

12. a. Name all angles that have X as a vertex. b. Name the sides of  3. c. Write another name for  3. Answer:  1,  2,  3, and  RXB or  RXN Answer:  AXB,  AXN,  NXA,  BXA Answer: Your Turn:

13. Measuring Angles To measure an angle we use a protractor. Place the center of the protractor on the vertex and one side of the angle on either side of the 0° line of the protractor. The protractor will have two scales running from 0° to 180° in opposite directions. Read the measure of the angle by viewing the alignment of the other side of the angle with the proper scale. To measure an angle we use a protractor. Place the center of the protractor on the vertex and one side of the angle on either side of the 0° line of the protractor. The protractor will have two scales running from 0° to 180° in opposite directions. Read the measure of the angle by viewing the alignment of the other side of the angle with the proper scale.

14. Classifying Angles There are four types of angles. There are four types of angles.angles Acute angles measure < 90°. Right angles measure 90°. Right angles measure 90°. Obtuse angles measure > 90° but 90° but < 180°. Straight angles measure 180°. Straight angles measure 180°.

15. Measure  TYV and classify it as right, acute, or obtuse.  TYV is marked with a right angle symbol, so measuring is not necessary. Answer: is a right angle. Example 2a:

16. Measure  WYT and classify it as right, acute, or obtuse. Use a protractor to find that. Answer: > is an obtuse angle. Example 2b:

17. Measure  TYU and classify it as right, acute, or obtuse. Use a protractor to find that m. Answer: is an acute angle. Example 2c:

18. Measure each angle named and classify it as right, acute, or obtuse. a.  CZD b.  CZE c.  DZX Answer: 150, obtuse Answer: 90, right Answer: 30, acute Your Turn:

19. Congruent Angles Just as segments that have equal measures are congruent, angles that have the same measures are congruent. To label angles congruent we use tic marks just like we used for segments. Just as segments that have equal measures are congruent, angles that have the same measures are congruent. To label angles congruent we use tic marks just like we used for segments. BAC  YXZ A Y B X Z C

20. More about Congruent Angles A ray that divides an angle into two congruent angles is called an angle bisector. If AD bisects BAC then BAD is congruent to CAD. A ray that divides an angle into two congruent angles is called an angle bisector. If AD bisects BAC then BAD is congruent to CAD. C B A D

21. INTERIOR DESIGN Wall stickers of standard shapes are often used to provide a stimulating environment for a young child’s room. A five-pointed star sticker is shown with vertices labeled. Find m  GBH and m  HCI if  GBH  HCI, m  GBH 2x + 5, and m  HCI 3x – 10. Example 3:

22. Given Definition of congruent angles Substitution Add 10 to each side. Subtract 2x from each side. Example 3:

23. Given Simplify. Use the value of x to find the measure of one angle. Since. or 35 Answer: Both measure 35. Example 3:

24. SIGNS A railroad crossing sign forms congruent angles. In the figure,  WVX  ZVY. If m  WVX 7a + 13 and m  ZVY 10a – 20, find the actual measurements of  WVX and  ZVY. Answer: Your Turn:

25. Assignment: Geometry: Pg. 34 – 35, #12 – 37, 42 - 43 Geometry: Pg. 34 – 35, #12 – 37, 42 - 43 Pre-AP Geometry: Pg. 34 – 35, #12 – 39, 45 - 48 Pre-AP Geometry: Pg. 34 – 35, #12 – 39, 45 - 48