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# Download Mod 1 - Aim #5 - Manhasset Schools

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Transcript

Aim #5: How do we do write proofs of angle theorems? Do Now: 1. Solve for the variables if AB ll CD. A 720 a B b e d c f g C h 1120 i j l k CC Geometry H a =_____ b =_____ c =_____ d =_____ e =_____ f =_____ g =_____ h =_____ i =_____ j =_____ k =_____ l =______ D 2. How many lines are there passing through A parallel to CD?_____ Complete: Through a point not on a line, there is _________________________ _____________________________________________________ (This is known as thePARALLEL POSTULATE.) A postulate is a mathematical statement accepted as true without proof. A theorem is a mathematical statement that can be proven. Once a theorem has been proven, it can be added to our list of known facts and used in proofs. We previously accepted the following: • Vertical angles are of equal measure. (theorem) • If a transversal intersects two parallel lines, alternate interior angles are of equal measure. (postulate) Let's prove: If transversal intersects two parallel lines, corresponding angles are congruent. Given: AB ll CD Prove: x = w Statements Reasons You have available the following facts: • Vertical angles are equal in measure. (theorem) • Alternate interior angles are equal in measure if lines are parallel. (postulate) • Corresponding angles are equal in measure if lines are parallel. (theorem) Let's prove: If a transversal intersects two parallel lines, same-side interior angles are supplementary. Given: AB ll CD, transveral EF Prove: m ≮1 + m≮2 = 180 G 1 H Statements Let's prove: 2 3 Reasons The three angles of a triangle sum to 180°. You will need to draw an auxiliary line parallel to one of the triangle‛s sides and passing through the vertex opposite that side. Add any necessary labels and write out your proof. yxoo Given: Triangle ABC o Prove: x + y + z = 180 yxoo Statements Reasons zo Each of the three parallel line theorems has aCONVERSE (or reversing) theorem as follows: Original If two parallel lines are cut by a transversal, then alternate interior angles are congruent. a 1 2 2 b If a ll b, then _______ Original If two parallel lines are cut by a transversal, then corresponding angles are congruent. a 1 2 b If two lines are cut by a transversal such that alternate interior angles are congruent, then the lines are parallel. a 1 b Converse If m 1 = m 2, then __________ Converse If two lines are cut by a transversal such that corresponding angles are congruent, then the lines are parallel. a 1 2 b If a ll b, then _______ If m 1 = m 2, then __________ Original If two parallel lines are cut by a transversal, then interior angles on the same side of the transversal add up to 1800. Converse If two lines are cut by a transversal such that interior angles on the same side of the transversal add up to 1800, then the lines are parallel. a a 1 1 b 2 If a ll b, then ______________ b 2 If m 1 + m 2 = 180, then ____________ Let's prove: If two lines are perpendicular to the same line, they are parallel to each other. State the given facts and the conjecture to be proven: Given: Prove: Statements a b l Reasons Prove the "converse" of the theorem we proved on page 1: If a transversal intersects two lines so that corresponding angles are equal, then the lines are parallel. Given: x = y Prove: AB ll EF zo Statements Reasons Let's Sum it up!! If a transversal intersects 2 parallel lines, 180 supplementary • same side interior angles are ___________________ or add up to ______ equal • alternate interior angles are _______________ equal • corresponding angles are ________________. 0 . CC Geometry H Name ______________________ Date ______________________ HW #5 1) Is it possible to prove lines a and b parallel? Explain your answer. b) a b a) a b 114 0 48 66 c) a 50 70 b d) 0 a 0 120 0 0 29 0 0 b 60 0 C B 2) Given: m≮C + m≮D = 180 m≮B = m≮D Prove: AB ll CD D A OVER 0 3) Prove: d + e - a = 180 (Suggested auxiliary lines have been drawn.) Statements Reasons 1. d + y = 180 1. 2. e = x + y 2. b0 a0 x00 0 c d 3. x = x 3. 4. y = e - x 4. 5. d + e - x = 180 5. 6. a = x 6. 7. d + e - a = 180 7. 0 y 0 e 4) Prove: If a line is perpendicular to one of two parallellines and intersects the other, n then it is perpendicular to the otherparallel line. Complete. Given: Prove: Statements Review: 6) Find the measure of the smaller angle. a b Reasons 7) Find m≮x and m≮y. 26 (6x + 4) 0 2x o y 116 0 0 38 0 x