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Tuesday, February 12th 1) Simplify: 3x + 5(9x + 4) - 10 2) The coordinates of the endpoints of the diameter of a circle are H (5, -1) and K (-9, 13). What are the coordinates of the center of the circle? 3) If OF bisects ∡COP, what angles have to be congruent? Answers to Triangle Wks. 1) 75 9) 29 2) 63 10) 147 3) 20 11) 60 4) 55 12) 27 5) 23 13) 50 6) 17 14) 130 7) 85 15) 112 8) 30 16) 32 17) 22 18) 120 19) 31 20) 65 21) 112 22) 68 23)90 24) 90 25) 22 26) 158 Parallel Lines Vertical Angles two angles whose sides form two pairs of opposite rays (make an X) vertical angles are congruent 1 4 3 2 Example: Find the value of x. a) (2x + 3)o (4x -101)o Transversal Line that intersects two coplanar lines at two distinct points transversal Corresponding Angles Formed by the intersection of two coplanar lines and a transversal Angles that are in the same spot at each location 1 3 2 4 Alternate Interior Angles Formed by the intersection of two coplanar lines and a transversal Inside parallel lines and on opposite sides of transversal 5 8 7 6 Same-Side Interior Formed by the intersection of two coplanar lines and a transversal On the inside of the parallel lines and on the same-side on the transversal 9 11 10 12 Example: Identify the following angles as corresponding, alternateinterior, or same-side interior. Name the transversal. a) Angles 10 and 16 b) Angles 1 and 9 Example: Identify the following angles as corresponding, alternateinterior, or same-side interior. Name the transversal. c) Angles 8 and 11 d) Angles5 and 13 Play-doh Activity Work with your partner and complete the worksheet. When you are done, make sure I check your paper. Homework: Parallel Lines Worksheet ONLY IF LINES ARE PARALLEL… Alternate-interior angles are congruent Same-side interior are supplementary Corresponding angles are congruent 1 5 6 2 n 3 m Example: Find the measure of each angles given that m ll n and that angle 2 measures 50o a) 1 7 4 8 b) 3 c) 4 d) 5 e) 6 f) 7 g) 8 Example: Find the values of a, b, and c a b c 65o 40o Example: Find the values of x and y xo 50o yo 70o Example: Find the values of x and y. Then find the measures of the angles. (2x)o 90o yo (y – 50)o Tuesday, September 11 Have your homework out! Book p. 121 #37-41 Proving Lines Parallel Converse of the Corresponding Angles Postulate If two lines are cut by a transversal and congruent corresponding angles, then the two lines are parallel. Flow Proof arrows show the logical connectors between the statements Proof of Converse AlternateInterior Angles Given: Prove: l ll m 3 2 1 l m Converse of the AlternateInterior Angles Theorem If two lines are cut by a transversal and congruent alternate-interior angles, then the two lines are parallel. Converse of Same-side Interior Angles Theorem If two lines are cut by a transversal and supplementary same-side interior angles, then the two lines are parallel. Example: Identify which lines must be parallel if the following angles are congruent. Justify your answer with a theorem or postulate. a) Angles 10 and 16 b) Angles 1 and 9 Example: Identify which lines must be parallel if the following angles are congruent. Justify your answer with a theorem or postulate. c) Angles 8 and 11 d) Angles5 and 13 E Given : a ll b Prove: 4 8 2 1 A 3 7 4 6 5 C 8 F a ll b given B 4 8 If parallel lines are cut by a transversal, then corresponding angles are congruent. D Given: AB ll CD Prove: 2 7 E 2 1 A 3 6 5 C 7 F 8 B 4 D Given: 1 2 Prove: m ll n p 1 Statement m Reason 1 2 given m ll n 2 n If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel Given:5 and 3 are supplementary Prove: AB ll CD Statement Reason Given: r p, t p Prove: r ll t 1 p r Statement Reason 2 t Theorem 3.5 If two lines are parallel to the same line, then they are parallel to each other. Theorem 3.6 In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. Example: Find the value of x for which l ll m. (14 + 3x)o (5x – 66)o Example: Find the value of x for which l ll m. (7x – 8)o 62o Class Work Proof Stations Homework: Finish Proof Worksheet 48) Wednesday, September 12th 1) Simplify: 3x + 5(9x + 4) - 10 2) The coordinates of the endpoints of the diameter of a circle are H (5, -1) and K (-9, 13). What are the coordinates of the center of the circle? 3) If OF bisects ∡COP, what angles have to be congruent? POP QUIZ Angle Type Vertical Corresponding Same-side interior Alternate interior What What else letter do do you they form? know? On the Back of Your Quiz Rate your partners from today! Example: Joe – 95 because he worked hard Sue – 50 because she only did half of the stations Class Work Proof Stations