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1. Measuring and Constructing Angles 1-3 Warm Up Lesson Presentation Lesson Quiz Holt Geometry

2. Warm Up • 1. Draw AB and AC, where A, B, and C are noncollinear. • 2. Draw opposite rays DE and DF. • Solve each equation. • 3. 2x + 3 + x – 4 + 3x – 5 = 180 • 4.5x + 2 = 8x – 10 B A C D E F Possible answer: 31 4

3. Objectives Name and classify angles. Measure and construct angles and angle bisectors.

4. Vocabulary angle right angle vertex obtuse angle interior of an angle straight angle exterior of an angle congruent angles measure angle bisector degree acute angle

5. A transit is a tool for measuring angles. It consists of a telescope that swivels horizontally and vertically. Using a transit, a survey or can measure the angle formed by his or her location and two distant points.

6. An angleis a figure formed by two rays, or sides, with a common endpoint called the vertex(plural: vertices). You can name an angle several ways: by its vertex, by a point on each ray and the vertex, or by a number.

7. Angle and Points • An Angle is a figure formed by two rays with a common endpoint, called the vertex. ray vertex ray • Angles can have points in the interior, in the exterior or on the angle. A E D B C Points A, B and C are on the angle. D is in the interior and E is in the exterior. B is the vertex. Lesson 1-4: Angles

8. Naming an angle:(1) Using 3 points (2) Using 1 point (3) Using a number – next slide Using 3 points: vertex must be the middle letter This angle can be named as Using 1 point: using only vertex letter *Use this method is permitted when the vertex point is the vertex of one and only one angle. Since B is the vertex of only this angle, this can also be called . A C B Lesson 1-4: Angles

9. Naming an Angle - continued Using a number: A number (without a degree symbol) may be used as the label or name of the angle. This number is placed in the interior of the angle near its vertex. The angle to the left can be named as . A B 2 C * The “1 letter” name is unacceptable when … more than one angle has the same vertex point. In this case, use the three letter name or a number if it is present. Lesson 1-4: Angles

10. Example 1: Naming Angles A surveyor recorded the angles formed by a transit (point A) and three distant points, B, C, and D. Name three of the angles. Possible answer: BAC CAD BAD

11. Check It Out! Example 1 Write the different ways you can name the angles in the diagram. RTQ, T, STR, 1, 2

12. The measureof an angle is usually given in degrees. Since there are 360° in a circle, one degreeis of a circle. When you use a protractor to measure angles, you are applying the following postulate.

13. 4 Types of Angles Acute Angle: an angle whose measure is less than 90. Right Angle: an angle whose measure is exactly 90 . Obtuse Angle: an angle whose measure is between 90 and 180. Straight Angle: an angle that is exactly 180 . Lesson 1-4: Angles

14. Measuring Angles • Just as we can measure segments, we can also measure angles. • We use units called degrees to measure angles. • A circle measures _____ • A (semi) half-circle measures _____ • A quarter-circle measures _____ • One degree is the angle measure of 1/360th of a circle. 360º ? 180º ? ? 90º Lesson 1-4: Angles

15. Adding Angles • When you want to add angles, use the notation m1, meaning the measure of 1. • If you add m1 + m2, what is your result? m1 + m2 = 58. m1 + m2 = mADC also. Therefore, mADC = 58. Lesson 1-4: Angles

16. Angle Addition Postulate Postulate: The sum of the two smaller angles will always equal the measure of the larger angle. Complete: m  ____ + m ____ = m  _____ MRK KRW MRW Lesson 1-4: Angles

17. Example: Angle Addition K is interior to MRW, m  MRK = (3x), m KRW = (x + 6) and mMRW = 90º. Find mMRK. First, draw it! 3x + x + 6 = 90 4x + 6 = 90 – 6 = –6 4x = 84 x = 21 3x x+6 Are we done? mMRK = 3x = 3•21 = 63º Lesson 1-4: Angles

18. Congruent Angles Definition: If two angles have the same measure, then they are congruent. Congruent angles are marked with the same number of “arcs”. The symbol for congruence is  3 5 Example: 3   5. Lesson 1-4: Angles

19. Angle Bisector An angle bisector is a ray in the interior of an angle that splits the angle into two congruent angles. Example: Since 4   6, is an angle bisector. 5 3 Lesson 1-4: Angles

20. If OC corresponds with c and OD corresponds with d, mDOC = |d– c| or |c– d|. You can use the Protractor Postulate to help you classify angles by their measure. The measure of an angle is the absolute value of the difference of the real numbers that the rays correspond with on a protractor.

21. Example 2: Measuring and Classifying Angles Find the measure of each angle. Then classify each as acute, right, or obtuse. A. WXV mWXV = 30° WXV is acute. B. ZXW mZXW = |130° - 30°| = 100° ZXW = is obtuse.

22. Check It Out! Example 2 Use the diagram to find the measure of each angle. Then classify each as acute, right, or obtuse. a. BOA b. DOB c. EOC mBOA = 40° BOA is acute. mDOB = 125° DOB is obtuse. mEOC = 105° EOC is obtuse.

23. Congruent angles are angles that have the same measure. In the diagram, mABC = mDEF, so you can write ABC  DEF. This is read as “angle ABC is congruent to angle DEF.” Arc marks are used to show that the two angles are congruent. The Angle Addition Postulate is very similar to the Segment Addition Postulate that you learned in the previous lesson.

25. –48°–48° Example 3: Using the Angle Addition Postulate mDEG = 115°, and mDEF = 48°. Find mFEG mDEG = mDEF + mFEG  Add. Post. 115= 48+ mFEG Substitute the given values. Subtract 48 from both sides. 67= mFEG Simplify.

26. Check It Out! Example 3 mXWZ = 121° and mXWY = 59°. Find mYWZ. mYWZ = mXWZ – mXWY  Add. Post. mYWZ= 121– 59 Substitute the given values. mYWZ= 62 Subtract.