3.4 Parallel and Perpendicular Lines Objectives: G.CO.9: Prove geometric theorems about lines and angles. G.CO.12: Make formal geometric constructions with a variety of tools and methods. For the Board: You will be able to prove and apply theorems about perpendicular lines. Bell Work 3.4: Solve each inequality. 1. x – 5 < 8 2. 3x + 1 > x Solve each equation. 3. 5y = 90 4. 5x + 15 = 90 Solve the system of equations. 5. 6y = 90 and 8y – 3x = 90 Anticipatory Set: Perpendicular lines are lines that intersect to form right angles. The shortest distance from a point to a line is along the perpendicular to the line from the point. So the distance from a point to a line is defined as the length of the perpendicular segment from the point to the line. Open the book to page 172 and read example 1. C Practice 1: a. Name the shortest segment from point A to BC. P Write and solve an inequality for x. x - 8 AP x – 8 > 12 or x > 20 12 b. Write and solve an inequality for x. 10x - 2 B 7x + 4 < 10x – 2 7x + 4 A -3x < -6 x > 2 Instruction: Congruent Linear Pairs → Perpendicular Lines If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular. Given: <1 <2 and <1 and <2 form a linear pair Prove: l | m 1 2 l Proof: <1 and <2 must be right angles, therefore l | m by the definition of perpendicular. m Parallel Transitive Theorem t If two lines are parallel to the same line, then they are parallel. 1 l 2 Given: l ||m and m||n m Prove: l||n 3 n Perpendicular Transversal Theorem If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. l Given: l | h, h||k Prove: l | k 1 h 3 k Two Perpendiculars Theorem In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. Given: l | h, l | k Prove: h ||k Practice: t a. Given <3 <4 what can be concluded about t and l? 1 2 3 4 l What postulate or theorem applies? 5 6 m t | l, Congruent Linear Pairs Theorem 7 8 b. Given l||n and t | n, what can be concluded about lines t and l? 9 10 n What postulate or theorem applies? 111 2 t | n, Perpendicular Transversal Theorem c. Given l||m and l||n, what can be concluded about lines m and n? What postulate or theorem applies? m||n, Parallel Transitive Theorem d. Given <6 <8 what can be concluded about t and m? What postulate or theorem applies? t | m, Congruent Linear Pairs Theorem e. Given t | l and t | n, what can be concluded about l and n? What postulate or theorem applies? l||n, Two Perpendiculars Theorem f. Given <1 <5 and <7 <10, what can be concluded about lines l and n? What postulate or theorem applies? Since <1 <5, l||m by the Corresp. <’s Post. Since <7 <10, m||n by the Alt. Int. <’s Th. Therefore, l||n by the Parallel Transitive Theorem. g. Given <3 <6 and <1 is a right angle, what can be concluded about t and m? What postulate or theorem applies. Since <3 <6, l||m by the Alt. Int. <’s Th. Since <1 is a rt. <, t | l by the defn. of | . Therefore, t | m by the Perpendicular Transversal Theorem. h. Given <5 and <10 are right angles, what can be concluded m and n? What theorem or postulate applies? Since <5 and <10 are right angles, t | m and t | n by the defn. of | . Therefore m||n by the Two Perpendiculars Theorem. Optional Proofs: Parallel Transitive Theorem t Given: l ||m and m||n 1 Prove: l||n l 2 m Proof: Statements Reasons 1. l ||m 1. Given 3 n 2. <1 <2 2. Corresponding <’s Post. 3. m||n 3. Given 4. <2 <3 4. Corresponding <’s Post. 5. <1 <3 5. Transitive Prop. of 6. l||n 6. Corresponding <’s Converse Post. l Perpendicular Transversal Theorem Given: l | h, h|| k 1 h Prove: l | k 3 k Proof: Statements Reasons 1. l | h, h||k 1. Given 2. <1 is a right angle 2. Definition of Perpendicular Lines 3. <1 <3 3. Corresponding Angles Postulate 4. <3 is a right angle 4. Substitution Property of Equality 5. l | k 5. Definition of Perpendicular Lines l Two Perpendiculars Theorem Given: l | h, l | k 1 h Prove: h ||k Proof: Statements Reasons 3 1. l | h, l | k 1. Given k 2. <1 and <3 are rt angles 2. Definition of Perpendicular Lines 3. <1 <3 3. Right Angle Congruence Theorem 4. h||k 4. Corresponding Angles Converse Theorem t Read example 2 on page 173 then with your partner complete practice 2. Practice 2: Write a two-column proof. 1 3 Given: r||s, <1 <2 4 5 r Prove: r | t 6 2 Proof: s 1. r||s 1. Given 2. <3 <2 2. Corresponding Angles Theorem 3. <1 <2 3. Given 4. <1 <3 4. Transitive Property of Congruence 5. r | t 5. Congruent Linear Pair of Angles form Perpendicular Lines Assessment: Question Student Pairs. Independent Practice: Text: pg. 175-178 prob. 2, 3, 6, 7, 10 – 15 . For a Grade: Handout 3.4 .