Pearson Unit 1 Topic 3: Parallel & Perpendicular Lines 3-3: Proving Lines Parallel Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

TEKS Focus: (5)(C) Use the constructions of congruent segments, congruent angles, angle bisectors, and perpendicular bisectors to make conjectures about geometric relationships. (1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication. (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. (6)(A) Verify theorems about angles formed by the intersection of lines and line segments, including vertical angles, and angle formed by parallel lines cut by a transversal and prove equidistance between the endpoints of a segment and points on its perpendicular bisector and apply these relationships to solve problems.

Basic Terms Flow Proof—a form of proof in which arrows show the logical connections between the statements

What does this diagram tell you about lines n and m? 72 n m t

Postulate

Recall that the converse of a theorem is found by exchanging the hypothesis and conclusion. The converse of a theorem is not automatically true. If it is true, it must be stated as a postulate or proved as a separate theorem. Remember

What does this diagram tell you about lines n and m? 108 n m t

What does this diagram tell you about lines n and m? 72 n m 108 t

Theorem

What does this diagram tell you about lines n and m? 108 n m t

Proof #1 Given: ℓ || m Prove: 2  7 1. ℓ || m 1. Given 2. 2  6 Use the Converse of the Corresponding Angles Postulate and the given to prove the Alternate Exterior Angles Theorem. Given: ℓ || m Prove: 2  7 Statements Reasons 1. ℓ || m 1. Given 2. 2  6 2. Corr. Angles Post. 3. 6  7 3. Vertical Angles Thm. 4. Transitive Prop. of  4. 2  7

Proof #2 Use the Converse of the Corresponding Angles Postulate and the given to show that ℓ || m. Given: m1 = m3 Prove: ℓ || m Statements Reasons 1. 1 & 3 corresponding angles 1. Def. of Corr. s 2. m1 = m 3 2. Given 3. 1  3 3. Def. of  s. 4. ℓ || m 4. Converse of Corr. s Post.

Proof #3 Refer to the diagram. Use the given information and the theorems you have learned to show that r || s. Given: 4  8 Prove: r || s Statements Reasons 1. 4  8 1. Given 2. 4 and 8 are alt. ext. s 2. Def. of alternate ext. s 3. Converse of Alt. Ext. s Thm. 3. r || s

Example: 4 Use the given information and the theorems you have learned to show that r || s. Given: m2 = (10x + 8)°, m6 = (25x –67)°, x = 5 Prove: r || s m2 = 10x + 8 = 10(5) + 8 = 58 m6 = 25x – 67= 25(5) – 67 = 58 2 & 6 are alternate interior ’s 2  6 r || s by Converse of Alt. Int. s Theorem

Example: 5 A carpenter is creating a woodwork pattern and wants two long pieces to be parallel. m1= (8x + 20)° and m2 = (2x + 10)°. If x = 15, show that pieces A and B are parallel.

Example: 5 continued Thoughts: A line through the center of the horizontal piece forms a transversal to pieces A and B. 1 and 2 are same-side interior angles. If 1 and 2 are supplementary, then pieces A and B are parallel. Substitute 15 for x in each expression.

Example: 5 continued m1 = 8x + 20 = 8(15) + 20 = 140 m1+m2 = 140 + 40 = 180 1 & 2 are same-side interior s A || B by Converse of Same Side Int. s

Proof #6 Converse of the Alternate Exterior Angles Theorem

EXTRA EXAMPLES NOT USED IN COMPOSITION BOOK FOLLOW. ALSO REMEMBER TO LOG-ON TO YOUR PEARSON ACCOUNT TO LOOK AT OTHER EXAMPLES BEFORE BEGINNING THE ON-LINE HW AND THE WRITTEN HW.

Example: 7 What if…? Suppose the corresponding angles on the same side of the boat measure (4y – 2)° and (3y + 6)°, where y = 8. Show that the oars are parallel. 4y – 2 = 4(8) – 2 = 30° 3y + 6 = 3(8) + 6 = 30° The angles are congruent, so the oars are || by the Converse of the Corr. s Post.