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Warm Up 1. Draw AB and AC , where A , B , and C are noncollinear.

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. 2 x + 3 + x – 4 + 3 x – 5 = 180 4. 5 x + 2 = 8 x – 10. B. A. C. D. E. F. Possible answer:. X = 31. X = 4. Objectives.

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Warm Up 1. Draw AB and AC , where A , B , and C are noncollinear.

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  1. 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: X = 31 X = 4

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

  3. 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

  4. Line l C B A Postulate 1-6 • Segment addition postulate states that, if three points (A, B, C) are collinear and B is between A and C then. AB + BC = AC • Remember Collinear points are points that lie on the same line.

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

  6. The set of all points between the sides of the angle is the interior of an angle. The exterior of an angleis the set of all points outside the angle. Angle Name R, SRT, TRS, or 1 You cannot name an angle just by its vertex if the point is the vertex of more than one angle. In this case, you must use all three points to name the angle, and the middle point is always the vertex.

  7. 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

  8. Teach! Example 1 Write the different ways you can name the angles in the diagram. RTQ, STR, STQ ,1, 2

  9. 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.

  10. 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.

  11. 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.

  12. Teach! 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.

  13. 1 4 3 1 2 2 2 1 • There are Four Types of Angle Pairs. • Vertical Angles • Adjacent Angles • Complimentary Angles • Supplementary Angles

  14. 1 4 3 2 • Vertical angles are two angles whose sides are opposite rays. • 1 and 2 are a pair of Vertical angles • 3 and 4 are also vertical angles

  15. 1 2 • Adjacent angles are two Coplanar angles with a common side. • Share the same plane • Common Side • Common Vertex • No interior points in common

  16. 30o 600 • Complimentary angles are two angles whose measures add-up to 90o. • They do not have to be adjacent. 1 2

  17. Supplementary angles are angles whose measured sum adds-up to 180o • They can be adjacent linear pair • They can also be non adjacent or share a common side or vertex. 1 2

  18. 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.

  19. If angle AOC is a straight angle then m of angle AOB + m angle BOC = 180

  20. –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.

  21. TEACH! 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.

  22. An angle bisector is a ray that divides an angle into two congruent angles. JK bisects LJM; thus LJKKJM.

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

  24. +12 +12 –4x –4x Example 4 Continued Step 1 Find x. mJKM = mMKL Def. of  bisector (4x + 6)° = (7x – 12)° Substitute the given values. Add 12 to both sides. 4x + 18 = 7x Simplify. Subtract 4x from both sides. 18 = 3x Divide both sides by 3. 6 = x Simplify.

  25. Example 4 Continued Step 2 FindmJKM. mJKM = 4x + 6 = 4(6) + 6 Substitute 6 for x. = 30 Simplify.

  26. QS bisects PQR, mPQS = (5y – 1)°, and mPQR = (8y + 12)°. Find mPQS. Check It Out! Example 4a Find the measure of each angle. Step 1 Find y. Def. of  bisector Substitute the given values. 5y – 1 = 4y + 6 Simplify. y – 1 = 6 Subtract 4y from both sides. y = 7 Add 1 to both sides.

  27. Check It Out! Example 4a Continued Step 2 FindmPQS. mPQS = 5y – 1 = 5(7) – 1 Substitute 7 for y. = 34 Simplify.

  28. JK bisects LJM, mLJK = (-10x + 3)°, and mKJM = (–x + 21)°. Find mLJM. +x +x –3 –3 Check It Out! Example 4b Find the measure of each angle. Step 1 Find x. LJK = KJM Def. of  bisector (–10x + 3)° = (–x + 21)° Substitute the given values. Add x to both sides. Simplify. –9x + 3 = 21 Subtract 3 from both sides. –9x = 18 Divide both sides by –9. x = –2 Simplify.

  29. Check It Out! Example 4b Continued Step 2 FindmLJM. mLJM = mLJK + mKJM = (–10x + 3)° + (–x + 21)° = –10(–2) + 3 – (–2) + 21 Substitute –2 for x. = 20 + 3 + 2 + 21 Simplify. = 46°

  30. Lesson Quiz: Part I Classify each angle as acute, right, or obtuse. 1. XTS acute right 2. WTU 3. K is in the interior of LMN, mLMK =52°, and mKMN = 12°. Find mLMN. 64°

  31. 4. BD bisects ABC, mABD = , and mDBC = (y + 4)°. Find mABC. Lesson Quiz: Part II 32° 5. Use a protractor to draw an angle with a measure of 165°.

  32. Lesson Quiz: Part III 6. mWYZ = (2x – 5)° and mXYW = (3x + 10)°. Find the value of x. 35

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