Notes 2.5

Warm-Up #5 10/6/09 • The measure of a supplement of an angle is 12 more than twice the measure of the angle. Find the measures of the angle and its supplement. • Write and solve an equation!

Homework Answers 1.

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70, 160 13. 35 19. 25 17.5, 107.5 14. 155 20. 10 90-x, 180-x 15. 25 21. 25 90-2y, 180-2y 16. 120 22. 153 45, 45 17. 60 b. 180-x 90, 90 18. 85 28. 120 ∠AFD 29. 30 ∠AFE and ∠EFD or ∠AFE and ∠BFC ∠AFD and ∠AFB ∠BFE and ∠EFD, ∠CFA and ∠AFE, ∠DFC and ∠CFB ∠BFC and ∠EFD ∠BFE and ∠CFD

2.5 Perpendicular Lines Objective: Find measures using perpendicular lines.

Perpendicular-definition • Two lines that intersect to form 90º angles. m

l

• l⊥m

Definition of perpendicular: • Given: l ⊥ m • Statement: m∠1 = 90 and/or m∠2 = 90 OR m • Given: m∠1 = 90 • Statement: l ⊥ m 1 2

• Both reasons are def. of ⊥

l

m

Theorem

1 2

• If two lines are perpendicular, then they form congruent, adjacent angles.

• Given: l ⊥ m • Statement: ∠1 ≅ ∠2 • Reason in proof: ⊥  ≅ adjacent ∠’s

l

m

Theorem

1 2

• If two lines form congruent adjacent angles, then the lines are perpendicular.

• Given: ∠1 ≅ ∠2 • Statement: l ⊥ m • Reason in proof: ≅ adjacent ∠’s  ⊥

l

Theorem • If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary.

• Reason: exterior sides of two adj. ∠’s are ⊥  exterior ∠’s are comp.

exterior

How to use it in a proof:

a

• Given: a ⊥ b • Statement: ∠1 and ∠2 are complements 1 2 • Then you will probably say…

– Statement: m∠1 + m∠2 = 90 by the def. of comp.

b

Example 1 • BE ⊥ AC and BD ⊥ BF. Find x if: • A. m∠DBE = 3x and m∠EBF = 4x – 1 • B. m∠ABD = 3x – 12, m∠DBE = 2x + 2, and m∠EBF = 2x + 8 E

D

F

A

B

C

Homework…

• Page 58-59; 1-12 all and 19-25 odd