3-2 Proving Lines Parallel. Objective: Use a transversal in proving lines parallel.
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1 3-2 Proving Lines Parallel Objective: Use a transversal in proving lines parallel.
2 Objectives: 1) Identify angles formed by two lines and a transversal. 2) Prove and use properties of parallel. Page 132 textbook If a transversal intersects two parallel lines, then same-side interior angles are supplementary. STATEMENTS REASONS 1. a b, Given 2. 1 & 2 are supplementary s & 4 are supplementary s. 2. Same-side Interior s Theorem 3. Same-side Interior s Theorem Congruent Supplements Theorem Supplements of angles (or of the same angle) are. STATEMENTS REASONS 1. a b 1. Given & 2 are supplementary & 2 are supplementary. 2. Corresponding s Postulate 3. Linear Pair Postulate 4. Substitution Property Linear pair is a pair of two adjacent s that form a straight line. Linear Pair Postulate: If two s form a linear pair, then they are supplementary.
3 Properties of parallel lines can be summarized as follows: If two parallel lines are cut by a transversal, then corresponding angles are congruent. alternate interior (or alternate exterior) angles are congruent. same-side interior (or same-side exterior) angles are supplementary. We use these statements (postulate or theorem) to prove angles are either congruent or supplementary. The converse of each conditional gives us a new postulate or theorem. We use the converse of these statements (postulate or theorem) to prove lines cut by the transversal are parallel.
4 Corresponding Angles Postulate If a transversal intersects two parallel lines, then the corresponding angles are congruent. then the two lines are parallel. congruent corresponding s, l m Alternate Interior (or Alternate Exterior) Angles Theorem If a transversal intersects two parallel lines, then the alternate interior (or alternate exterior) angles are congruent. then the two lines are parallel. congruent alternate interior s, l m then the two lines are parallel. congruent alternate exterior s, l m
5 Same-Side Interior (or Same-Side Exterior) Angles Theorem If a transversal intersects two parallel lines, then the same-side interior (or same-side exterior) angles are supplementary. then the two lines are parallel. same-side interior s that are supplementary l m then the two lines are parallel. same-side exterior s that are supplementary l m
6 A flow proof uses arrows to show the logical connections between the statements. Reasons are written below the statements. - Converse of the Alternate Interior Angles Theorem If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. Given Vertical s Theorem Transitive Prop. of (or Substitution Prop.) Converse of the Corresponding Angles Postulate
7 Using Postulate Converse of the Corresponding s Postulate Use the diagram at the right. Which lines, if any, must be parallel if 3 and 2 are supplementary? Justify your answer with a theorem or postulate. 4 2 congruent Converse of the Corresponding Angles EC DK alternate interior s congruent congruent 5x 14 5x 14 5x x x 2
8 EC DK ; Converse of the Corresponding Angles Postulate 7x = 180 7x = 126 x = 18 Check:
9 Practice 3-2 Proving Lines Parallel a) same-side interior b) QR c) TS d) same-side interior e) Same-Side Interior Angles f) TS g) 3-5: Converse of the Same-Side Interior Angles Theorem
10 Practice 3-2 Proving Lines Parallel l and m; Converse of Same-Side Interior s Theorem none BC and AD; Converse of Same-Side Interior s Theorem RT and HU; Converse of Corresponding s Postulate BH and CI; Converse of Corresponding s Postulate a and b; Converse of Same-Side Interior s Theorem
11 Practice 3-2 Proving Lines Parallel x = 180 x = 180 x = 43 x x 20 = 180 2x = 180 3x + 2x 10 = 180 5x = 190 x = 90 x = 38 x = x 20 = 180 x 30 = 70 x = 180 x = 100 x = 70
12 Practice 3-2 Proving Lines Parallel
13 Practice 3-2 Proving Lines Parallel
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Name Date Period Geometry Midterm Review 2016-2017 Vocabulary: 1. Points that lie on the same line. 1. 2. Having the same size, same shape 2. 3. These are non-adjacent angles formed by intersecting lines.
You try: What is the definition of an angle bisector? You try: You try: is the bisector of ABC. BD is the bisector of ABC. = /4.MD.
US Geometry 1 What is the definition of a midpoint? midpoint of a line segment is the point that bisects the line segment. That is, M is the midpoint of if M M. 1 What is the definition of an angle bisector?