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Geometry Review Angles and Parallel Lines Objectives: Measure and Classify Angles Describe Angle Pair Relationships Classify Angle Pairs made by Parallel Lines cut by a Transversal Describe Angle Pair Relationships within Parallel Lines Part 1: Angles Vertex An angle consists of two A Sides B different rays (sides) that share a common endpoint (vertex). C This angle can be called: Angle ABC, <ABC, <CBA OR it can be named by the vertex like so: <B Types of Angles Go to the following website to learn about the different types of angles. Don’t forget to press the play button on top of the picture to see the animation. http://www.mathsisfun.com/angles.html Use the diagram to find the measure of the indicated angle. Then classify each angle as either acute, right, obtuse, or straight. 1. KHJ 2. GHK 3. GHJ 4. GHL 5. What is the measure of DOZ?... How did you get to your answer? D G 25 O 40 Z Angle Addition Postulate If P is in the interior of RST, then mRST = mRSP + mPST. 6. Given that mLKN = 151°, find mLKM and mMKN. M L 2x+10 4x-3 K N Congruent Angles Two angles are congruent angles if they have the same measure. Angle Bisector An angle bisector is a ray that divides an angle into two congruent angles. 7. Ray BD bisects <ABC. Solve for x. B A D ( x 2 21)o (10 x ) C 8. In a diagram, YW bisects XYZ. mXYW = (m2 81) ° and m ZYW= (-18m)°. Find mXYW. Hint: Draw the picture Hint: Angle may not be drawn to scale C Comes Before S… Complementary Angles Sum to 90 Degrees Supplementary Angles Sum to 180 m1 m2 90 m3 m4 90 m5 m6 180 m7 m8 180 Linear Pairs of Angles Linear Pairs of Angles Two adjacent angles form a linear pair if their noncommon sides are opposite rays. The angles in a linear pair are supplementary. Vertical Angles Vertical Angles Two nonadjacent angles are vertical angles if their sides form two pairs of opposite rays. Vertical angles are formed by two intersecting lines. S 9. Name a pair of complementary 10. 11. 12. 13. angles. Name a pair of supplementary angles. Y Name a linear pair. Name a pair of vertical angles. Name a pair of congruent angles. U T X W V Part 2: Parallel Lines What would you call two lines which do not intersect? Answer: Parallel Lines Exterior B A Interior D C Exterior A solid arrow placed on two lines of a diagram indicate the lines are parallel. The symbol || is used to indicate parallel lines. AB || CD A slash through the parallel symbol || indicates the lines are not parallel. B AB || CD A D C Transversal Definition: A line, ray, or segment that intersects 2 or more COPLANAR lines, rays, or segments. Exterior Exterior Parallel lines Interior Non-Parallel lines Interior Exterior Exterior transversal transversal Transversal A transversal is a line which intersects two or more lines in a plane. The intersected lines do not have to be parallel. j k m t Lines j, k, and m are intersected by line t. Therefore, line t is a transversal of lines j, k, and m. INTERIOR –The space INSIDE the 2 lines interior EXTERIOR -The space OUTSIDE the 2 lines In the diagram below: Angles 1, 2, 7 and 8 are exterior angles. Angles 3, 4, 5, and 6 are on the interior. exterior Exterior 1 exterior 3 2 4 Interior 5 6 7 8 Exterior When two parallel lines are cut by a transversal special angle relationships form. Exterior 1 3 4 Interior 5 6 7 8 Exterior 2 ♥Alternate Interior Angles are CONGRUENT ♥Alternate Exterior Angles are CONGRUENT ♥Same Side Interior Angles are SUPPLEMENTARY ♥Same Side Exterior Angles are SUPPLEMENTARY Corresponding Angles Corresponding Angles: Two angles that occupy corresponding positions. 2 6, 1 5, 3 7, 4 8 Exterior 1 3 2 4 Interior 5 6 7 8 Exterior Corresponding Angles When two parallel lines are cut by a transversal, pairs of corresponding angles are formed. L Line L GPB = PQE G A D P B Q E F Line M Line N GPA = PQD BPQ = EQF APQ = DQF Four pairs of corresponding angles are formed. Corresponding pairs of angles are congruent. Lines l and m are parallel. l||m. 14. Name all pairs of corresponding angles. 15. Determine all the missing angle measures. 42° c° a° b° d° e° g° f° l m Same Side Interior/Exterior Angles Same Side Interior Angles: Two angles that lie between parallel lines on the same sides of the transversal. APQ + DQP = 1800 Same Side Exterior Angles: Two angles that lie outside parallel lines on the same sides of the transversal. L A G Line L 1200 Exterior P B 600 1200 600 D BPQ + EQP = 1800 Interior E Q F Line M Exterior Line N Same Side Interior or Same Side Exterior angles are supplementary. Alternate Interior/Exterior Angles Alternate Interior Angles: Two angles that lie between parallel lines on opposite sides of the transversal (but not a linear pair). Alternate Exterior Angles: Two angles that lie outside parallel lines on opposite sides of the transversal. L G A D Line L P B Q E Line M Line N BPQ = DQP APQ = EQP F Two pairs of alternate angles are formed. Lines l and m are parallel. l||m 16. Name all pairs of alternate interior angles. 17. Name all pairs of alternate exterior angles. 18. Determine all the missing angle measures. 81° c° a° b° d° e° g° f° l m Make sure you are in slide show view to do this test. Name the pairs of the following angles formed by a transversal. GG G AA 500 Line Line Line LL L P P BBB 1300 D DD Q Q Q EEE FFF Line Line Line MM M Line Line N Line NN 19) Find the missing angles. 70 ° 70 ° b° Hint: The 3 angles in a triangle sum to 180°. d° 65 ° 20) Find the missing angles. 45 ° 50 ° b° Hint: The 3 angles in a triangle sum to 180°. d° 75 ° In the figure a || b. 21. Name the angles congruent to 3. 22. Name all the angles supplementary to 6. 23. If m1 = 105° what is m3? 24. If m5 = 120° what is m2? 25. Final Exercise 40° 60° Find all the missing angle measures, and name the postulate or theorem that gives us permission to make our statements.