Lesson 2-5 Perpendicular Lines (page 56) Essential Question Can you justify the conclusion of a conditional statement?

Perpendicular Lines

Perpendicular Lines (⊥-lines): … two lines that intersect to form right angles. Example: m n

then _____, _____, _____, and _____ are right angles. Example: If m ⊥n, then _____, _____, _____, and _____ are right angles. ∠1 ∠2 ∠3 ∠4 1 2 m 4 3 n How many angles must be right angles in order for the lines to be perpendicular? _____ 1

Theorem 2-4 ⊥-lines ⇒ ≅ adj∠’s If two lines are perpendicular, then they form congruent adjacent angles. Given: l ⊥ n Prove: ∠1, ∠2, ∠3, & ∠4 are congruent angles. l 2 1 n 3 4

See page 57 Classroom Exercises #1. Given: l ⊥ n Prove: ∠1, ∠2, ∠3, & ∠4 are congruent angles. 2 1 n 3 4 See page 57 Classroom Exercises #1. Statements Reasons ___________________ ___________________ ___________________ ___________________

Note: This is the converse of Theorem 2-4. 2 lines form ≅ adj∠’s ⇒⊥-lines Theorem 2-5 If two lines form congruent adjacent angles, then they are perpendicular . Given: ∠1 ≅ ∠2 Prove: l ⊥ n l 2 1 n Note: This is the converse of Theorem 2-4.

Given: ∠1 ≅ ∠2 Prove: l ⊥ n ___________________ ___________________ See page 58 Written Exercises #2! Statements Reasons ___________________ ___________________ ______________________ ___________________

Theorem 2-6 Ext S 2 adj A∠’s ⊥ ⇒ comp ∠’s If the exterior sides of two adjacent acute angles are perpendicular , then the angles are complementary. A • Given: Prove: ∠AOB and ∠BOC are comp. ∠‘s B • • O C

Given: Prove: ∠AOB and ∠BOC ___________________ ___________________ • Given: Prove: ∠AOB and ∠BOC are comp. ∠‘s B • • O C See page 59 Written Exercises #13. Statements Reasons ___________________ ___________________ ______________________ ___________________

Example:. Name the definition or theorem that justifies Example: Name the definition or theorem that justifies each statement about the diagram. B 1 2 3 4 A C 5 6 8 7 D (a) _________________________________ Def. of ⊥-lines

Example:. Name the definition or theorem that justifies Example: Name the definition or theorem that justifies each statement about the diagram. B 1 2 3 4 A C 5 6 8 7 D (b) _________________________________ ⊥-lines ⇒ ≅ adj∠’s

Ext S 2 adj A∠’s ⊥ ⇒ comp ∠’s Example: Name the definition or theorem that justifies each statement about the diagram. B 1 2 3 4 A C 5 6 8 7 D (c) _________________________________ Ext S 2 adj A∠’s ⊥ ⇒ comp ∠’s

Def. of comp ∠’s then m∠7 + m∠8 = 90º. Example: Name the definition or theorem that justifies each statement about the diagram. B 1 2 3 4 A C 5 6 8 7 D (d) If ∠7 & ∠8 are complementary, then m∠7 + m∠8 = 90º. _________________________________ Def. of comp ∠’s

2 lines form ≅ adj∠’s ⇒⊥-lines Example: Name the definition or theorem that justifies each statement about the diagram. B 1 2 3 4 A C 5 6 8 7 D (e) _________________________________ 2 lines form ≅ adj∠’s ⇒⊥-lines

Example:. Name the definition or theorem that justifies Example: Name the definition or theorem that justifies each statement about the diagram. B 1 2 3 4 A C 5 6 8 7 D (f) ∠4 ≅ ∠5. _________________________________ Vertical ∠’s R ≅

Example:. Name the definition or theorem that justifies Example: Name the definition or theorem that justifies each statement about the diagram. B 1 2 3 4 A C 5 6 8 7 D (g) If ∠ABC is a right angle, then m∠ABC = 90º. _________________________________ Def. of Rt. ∠

65º+25º = 90º 13 Example. If ZW ⊥ ZY, m∠1 = 5x, and m∠2 = 2x - 1, find the value of x. 13 x = ______ W • V • 1 2 • Z Y 65º+25º = 90º

Can you justify the conclusion of a conditional statement? Assignment Written Exercises on pages 58 & 59 GRADED: 3 to 12 ALL numbers GROUP WORK: 19 to 25 odd numbers Can you justify the conclusion of a conditional statement? Prepare for a quiz on Lesson 2-2 to 2-5: Justifications AFTER lesson 2-6!