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October 13, 2015 2.5 Perpendicular Lines October 13, 2015 2.5 Objectives 1) Apply the definition and theorems about perpendicular lines. October 13, 2015 Theorem 2-4 If two lines are perpendicular, then they form congruent adjacent angles. On a proof you would write: "If 2 lines ⊥, then they form ≅ adj. ∠'s" Theorem 2-5 If two lines form congruent adjacent angles, then the lines are perpendicular. On a proof you would write: "If 2 lines form ≅ adj. ∠'s, then the lines are ⊥ " October 13, 2015 Theorem 2-6 If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary. On a proof you would write: A B O C "If ext. sides of 2 adj. acute ∠'s are ⊥, then the ∠'s are complementary" Copy and complete the proof of Theorem 2-5: If two lines form congruent adjacent angles, then the lines are perpendicular. m 1 Given: ∠1 ≅ ∠2 Prove: m ⊥ n Statement Reason 2 n October 13, 2015 Copy and complete the proof of Theorem 2-5: If two lines for congruent adjacent angles, then the lines are perpendicular. m Given: ∠1 ≅ ∠2 Prove: m ⊥ n 1 2 on i t u Sol Statement Reason 1. ∠1 ≅ ∠2 2. m∠1 + m∠2 = 180 3. m∠2 + m∠2 = 180 4. m∠2 m ⊥ n= 90 5. ___________ Given 1. _________________ Angle Addition Postulate 2. _________________ Substitution 3. _________________ Division Property 4. _________________ 5. Definition of ⊥ Lines GIVEN: AB ⊥ CD Use the diagram to classify each statement as true or false. n A E C D G F 1. AB ⊥ EF 2. 3. 4. 5. 6. 7. F T ∠CGB is a right angle T ∠CGA is a right angle T m∠DGB = 90 ∠EGC and ∠EGA are complements ∠DGF is complementary to ∠DGA ∠EGA is complementary to ∠DGF B T F T October 13, 2015 Name the definition or state the theorem that justifies the statement about the diagram. (page 1 of 3) E D F B A C 1) If ∠EBC is a right angle, then BE ⊥ AC. AN ER SW Def. of | lines 2) If AC ⊥ BE, then ∠ABE is a right angle. AN ER SW Def. of | lines Name the definition or state the theorem that justifies the statement about the diagram. (page 2 of 3) E D F B A 3) If BE ⊥ AC, AN ER SW C (2-6) If the ext. sides of 2 adj. angles are | , then the angles are complementary then ∠ABD and ∠DBE are complementary 4) If ∠ABD and ∠DBE are complementary angles, then m∠ABD + m∠DBE = 90 AN ER SW Def. of complementary angles October 13, 2015 Name the definition or state the theorem that justifies the statement about the diagram. (page 3 of 3) E D F B A C 5) If BE ⊥ AC, then m∠ABE = 90 AN Def. of | lines ER SW 6) If ∠ABE ≅ ∠EBC, then AC ⊥ BE AN ER SW (2-5) If 2 lines form ≅ adjacent angles, then the lines are |