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Simplify each expression. 1. 90 – ( x + 20) 2. 180 – (3 x – 10)

Drill: Monday, 9/8. Simplify each expression. 1. 90 – ( x + 20) 2. 180 – (3 x – 10) Write an algebraic expression for each of the following. 3. 4 more than twice a number 4. 6 less than half a number. 70 – x. 190 – 3 x. 2 n + 4.

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Simplify each expression. 1. 90 – ( x + 20) 2. 180 – (3 x – 10)

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  1. Drill: Monday, 9/8 Simplify each expression. 1.90 –(x + 20) 2. 180 – (3x – 10) Write an algebraic expression for each of the following. 3. 4 more than twice a number 4. 6 less than half a number 70 –x 190 –3x 2n + 4 OBJ: SWBAT identify adjacent, vertical, complementary, and supplementary angles in order to find angle measures.

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

  3. 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. 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.

  4. 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.

  5. 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

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

  7. 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.

  8. 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.

  9. Check It Out! Example 3 mXWZ = 121° and mXWY = 2x + 1° and mZWY = 3x + 10°. Find mYWZ. mYWZ = mXWZ – mXWY  Add. Post.

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

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

  12. 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.

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

  14. 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.

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

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

  17. 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°

  18. 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°

  19. 4. BD bisects ABC, mABD = , and mDBC = (y + 4)°. Find mABC. Lesson Quiz: Part II 32°

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

  21. Drill: Monday, 9/8 Simplify each expression. 1.90 –(x + 20) 2. 180 – (3x – 10) Write an algebraic expression for each of the following. 3. 4 more than twice a number 4. 6 less than half a number 70 –x 190 –3x 2n + 4 OBJ: SWBAT identify adjacent, vertical, complementary, and supplementary angles in order to find angle measures.

  22. Motivation

  23. What are Adjacent Angles?

  24. Example 1A: Identifying Angle Pairs AEB and BED have a common vertex, E, a common side, EB, and no common interior points. Their noncommon sides, EA and ED, are opposite rays. Therefore, AEB and BED are adjacent angles and form a linear pair. Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. AEB and BED

  25. Example 1B: Identifying Angle Pairs AEB and BEC have a common vertex, E, a common side, EB, and no common interior points. Therefore, AEB and BEC are only adjacent angles. Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. AEB and BEC

  26. Example 1C: Identifying Angle Pairs Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. DEC and AEB DEC and AEB share E but do not have a common side, so DEC and AEB are not adjacent angles.

  27. Check It Out! Example 1a 5and 6are adjacent angles. Their noncommon sides, EA and ED, are opposite rays, so 5and 6also form a linear pair. Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. 5 and 6

  28. Check It Out! Example 1b Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. 7 and SPU 7and SPU have a common vertex, P, but do not have a common side. So 7and SPU are not adjacent angles.

  29. Check It Out! Example 1c Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. 7 and 8 7and 8have a common vertex, P, but do not have a common side. So 7and 8are not adjacent angles.

  30. Complementary & Supplementary Angle Match-Up

  31. Geo Sketch for Vertical Angles

  32. Another angle pair relationship exists between two angles whose sides form two pairs of opposite rays. Vertical anglesare two nonadjacent angles formed by two intersecting lines. 1and 3are vertical angles, as are 2and 4.

  33. Algebra and Angles

  34. Algebra and Angles

  35. Algebra and Angles

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