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1-5. Segment and Angles Bisectors. Warm Up. Lesson Presentation. Lesson Quiz. Holt Geometry. 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

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  1. 1-5 Segment and Angles Bisectors 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 1-5 Segments and Angles Bisectors Possible answer: 31 4

  3. 1-5 Segments and Angles Bisectors Objectives Bisect a Segment. Bisect and Angle

  4. 1-5 Segments and Angles Bisectors Vocabulary Midpoint Bisects Segment Bisector Compass Straightedge Construct Construction Midpoint Formula Angle Bisector

  5. 1-5 Segments and Angles Bisectors A segment can be divided into two identical parts using the Midpoint. In geometry we say that the midpoint bisects a segment if it is equidistant from the endpoints. An segment bisector is a segment, ray, line or plane that intersects a segment at its midpoint.

  6. 1-5 Segments and Angles Bisectors Activity 1: Construct Segment Bisector

  7. 1-5 Segments and Angles Bisectors If you know the coordinates of the endpoints of a segment, you can find the coordinates of the midpoint by using the Midpoint Formula:

  8. 1-4Angles and Their Measures 1-5 Segments and Angles Bisectors Example 1: Using the Midpoint Formula

  9. 1-4Angles and Their Measures 1-5 Segments and Angles Bisectors Example 1: Using the Midpoint Formula

  10. 1-5 Segments and Angles Bisectors An Angle Bisector is a ray that divides and angle into two adjacent angles that are congruent.

  11. 1-5 Segments and Angles Bisectors Activity 1: Construct Angle Bisector

  12. 1-5 Segments and Angles Bisectors Example 2: Using Angle Bisectors If you know the measure of the bigger angle, we can use the definition of angle bisectors to find the measure of the smaller angles.

  13. 1-5 Segments and Angles Bisectors Example 2: Using Angle Bisectors

  14. 115  2 1-5 Segments and Angles Bisectors Example 3: Using the Angle Addition Postulate mDEG = 115°, and EF is the angle bisector. Find mFEG mDEG = mDEF + mFEG Angle Additions Postulate. 115= 2(mFEG) Equality property. Divide by 2 on both sides. Simplify. 57.5 = mFEG

  15. 1-5 Segments and Angles Bisectors 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.

  16. KM bisects JKL, mJKM = (4x + 6)°, and mMKL = (7x – 12)°. Find mJKM. 1-5 Segments and Angles Bisectors Example 4: Finding the Measure of an Angle

  17. +12 +12 –4x –4x 1-5 Segments and Angles Bisectors 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.

  18. 1-5 Segments and Angles Bisectors Example 4 Continued Step 2 FindmJKM. mJKM = 4x + 6 = 4(6) + 6 Substitute 6 for x. = 30 Simplify.

  19. QS bisects PQR, mPQS = (5y – 1)°, and mPQR = (8y + 12)°. Find mPQS. 1-5 Segments and Angles Bisectors 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.

  20. 1-5 Segments and Angles Bisectors Check It Out! Example 4a Continued Step 2 FindmPQS. mPQS = 5y – 1 = 5(7) – 1 Substitute 7 for y. = 34 Simplify.

  21. JK bisects LJM, mLJK = (-10x + 3)°, and mKJM = (–x + 21)°. Find mLJM. +x +x –3 –3 1-5 Segments and Angles Bisectors 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.

  22. 1-5 Segments and Angles Bisectors 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°

  23. 1-5 Segments and Angles Bisectors 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°

  24. 4. BD bisects ABC, mABD = , and mDBC = (y + 4)°. Find mABC. 1-5 Segments and Angles Bisectors Lesson Quiz: Part II 32° 5. Use a protractor to draw an angle with a measure of 165°.

  25. 1-5 Segments and Angles Bisectors Lesson Quiz: Part III 6. mWYZ = (2x – 5)° and mXYW = (3x + 10)°. Find the value of x. 35

  26. 1-5 Segments and Angles Bisectors Skills Autonomous Practice Pages: 29, 30, 31, 32

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