1. Copyright © Ed2Net Learning, Inc.1 Angle and Angles Measurement Geometry

2. Copyright © Ed2Net Learning, Inc.2 Angles Angles have two sides that share a common endpoint called the vertex of the angle. side vertex

3. Copyright © Ed2Net Learning, Inc.3 Angles The most common measure for angles is the degree. A circle can be separated into 360 equal-sized parts. Each part would make up a one-degree (1°) angle. 1 degree (°) Why 360 degrees? Probably because old calendars (such as the Persian Calendar) used 360 days for a year - when they watched the stars they saw them revolve around the North Star one degree per day.

4. Copyright © Ed2Net Learning, Inc.4 Types of Angles Right angle This symbol indicates a right angle. Exactly 90° Acute angle Less than 90° Obtuse angle between 90° and 180° Straight angle Exactly 180° Quarter of a circle Half of a circle

5. Copyright © Ed2Net Learning, Inc.5 Measure Angles Align the center of the protractor with the vertex of the angle. The angle measures 150°. It is an obtuse angle.

6. Copyright © Ed2Net Learning, Inc.6 Measure Angles Align the center of the protractor with the vertex of the angle. The angle measures 75°. It is an acute angle.

7. Copyright © Ed2Net Learning, Inc.7 Your Turn! Use a protractor to find the measure of each angle. Then classify each angle as acute, obtuse, right, or straight.

8. Copyright © Ed2Net Learning, Inc.8 Pairs of Angles Two angles whose sum is 90° are complementary angles. 1 2 1 2 m 1=30°, m 2=60°, m 1+ m 2 = 90°

9. Copyright © Ed2Net Learning, Inc.9 Pairs of Angles Two angles whose sum is 180° are supplementary angles. 1 2 1 2 m 1=120°, m 2=60°, m 1+ m 2 = 180°

10. Copyright © Ed2Net Learning, Inc.10 Your Turn! Angles M and N are supplementary. If m M = 85° what is the measure of N?

11. Copyright © Ed2Net Learning, Inc.11 Estimate Angle Measures To estimate the measure of an angle, compare it to an angle whose measure you know. About 90° About 60° You could use the corner of a piece of paper as a reference for 90°. You could fold the corner in half for 45°, and use the edge of the paper for 180°.

12. Copyright © Ed2Net Learning, Inc.12 Your Turn! Estimate the measure of each angle.

13. Copyright © Ed2Net Learning, Inc.13 Draw an Angle Step 1: Draw one side of the angle. Then mark the vertex and draw an arrow.

14. Copyright © Ed2Net Learning, Inc.14 Draw an Angle Step 2: Place the center point of protractor on the vertex. Align the mark labeled 0 on the protractor with the line. Find 74°on the correct scale and make a pencil mark.

15. Copyright © Ed2Net Learning, Inc.15 Draw an Angle Step 3: Use a straightedge to draw the side that connects the vertex and the pencil mark.

16. Copyright © Ed2Net Learning, Inc.16 Your Turn! Use a protractor and a ruler to draw the angles:  68°  105°

17. Copyright © Ed2Net Learning, Inc.17 Congruent Segments A line segment is a straight path between two endpoints. To indicate the line segment JK, write JK. Line segments that have the same length are called congruent segments. J K

18. Copyright © Ed2Net Learning, Inc.18 Congruent Segments Step 1: Draw JK. The use a ruler to draw a line segment longer than JK. Label it LM. J K L M

19. Copyright © Ed2Net Learning, Inc.19 Congruent Segments Step 2: Place the compass at J and adjust the compass setting so you can place the pencil tip on K. The compass setting equals the length of JK. J K L M

20. Copyright © Ed2Net Learning, Inc.20 Congruent Segments Step 3: Using this setting, place the compass tip at L. Draw an arc to intersect LM. Label the intersection LM. Label the intersection P. LP is congruent to JK. L M P

21. Copyright © Ed2Net Learning, Inc.21 Congruent Angles Step 1: Draw JKM. Then use a straightedge to draw ST. KM J

22. Copyright © Ed2Net Learning, Inc.22 Congruent Angles Step 2: Place the tip of the compass at K. Draw an arc to intersect both sides of JKM. Label the points of intersection X and Y. KM J X Y

23. Copyright © Ed2Net Learning, Inc.23 Congruent Angles Step 3: Using this setting, place the compass at point S. Draw an arc to intersect ST. Label the intersection W. ST W

24. Copyright © Ed2Net Learning, Inc.24 Congruent Angles Step 4: Place the point of the compass at Y. Adjust so that the pencil tip is on X. KM J X Y

25. Copyright © Ed2Net Learning, Inc.25 Congruent Angles Step 5: Using this setting, place the compass at W. Draw an arc to intersect the arc in Step 3. Label the intersection U. Draw SU. JKM is congruent to UST. ST U W

26. Copyright © Ed2Net Learning, Inc.26 Let us take a Break!

27. Copyright © Ed2Net Learning, Inc.27

28. Copyright © Ed2Net Learning, Inc.28 Bisectors To bisect something means to separate it into two equal parts. You can use a straightedge and a compass to bisect the segments and angles.

29. Copyright © Ed2Net Learning, Inc.29 Bisect a Line Segment Step 1: Draw AB Step 2: Place the compass at point A. Using a setting greater than one half the length of AB, draw two arcs as shown. Step 3: Using the same setting, place the compass at point B. Draw an arc above and below as shown. AB AB The arcs should intersect

30. Copyright © Ed2Net Learning, Inc.30 Bisect a Line Segment Step 4: Use a ruler to align the intersections. Draw a segment that intersects AB. Label the intersection M. AB M The vertical line segment bisects AB at M. The segments AM and MB are congruent. This can be written as AM  MB. This means that the measure of AM is equal to the measure of MB. The line segments are also perpendicular. That is, they meet at right angles.

31. Copyright © Ed2Net Learning, Inc.31 Your Turn! Draw a line segment measuring 6 centimeters. Then use a ruler and compass to bisect the segment.

32. Copyright © Ed2Net Learning, Inc.32 Bisect an Angle Step 1: Draw MNP Step 2: Place the compass at point N and draw an arc that intersects both sides of the angle. Label the points of intersection X and Y. M N P M N P X Y

33. Copyright © Ed2Net Learning, Inc.33 Bisect an Angle Step 3: With the compass at point X, draw an arc as shown. Step 4: Using the same setting, place the compass point at Y and draw another arc as shown. Label the intersection Z. M N P X Y Z

34. Copyright © Ed2Net Learning, Inc.34 Bisect an Angle Step 5: Use a ruler to draw NZ. M N P X Y Z NZ bisects MNP. Therefore, MNZ and ZNP are congruent. This can be written as MNZ  ZNP

35. Copyright © Ed2Net Learning, Inc.35 Your Turn! Draw a 120°angle. Then use a straightedge and a compass to bisect the angle.

36. Copyright © Ed2Net Learning, Inc.36 Assessment 1. Explain the difference between acute angles and obtuse angles. 2. Define bisect. 3. If m A = 108° and A and B are supplementary, find m B. 4. Angles J and K are complementary. Find m J if m K =58°.

37. Copyright © Ed2Net Learning, Inc.37 Assessment 5. Describe how the corner of a textbook can be used to estimate the measure of an angle.