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3.6 Prove Theorems About Perpendicular Lines Daily Openers October 23 1. What is the distance between the points (2, 3) and (5, 7)? 2. State the postulate, theorem, or converse that proves m || n. 3. State the postulate, theorem, or converse that proves p || q. 34º 115º n 34º m 115º q p 4. State whether the lines are perpendicular, parallel, or neither. Line 1: (11, 7), (1, 3) Line 2: (2, 4), (2, 3) 5. RT bisects ∠PRS. If the m∠PRT = 15x 12 and m∠TRS = 7x + 12 find x, m∠PRT, and m∠TRS. 3.6 Prove Theorems About Perpendicular Lines 3.6 Prove Theorems About Perpendicular Lines Thm 3.8 If two lines intersect to form a linear pair of congruent angles, then g the lines are perpendicular. If ∠1 ≅ ∠2, then g ⊥ h 1 h 2 Thm 3.9 If two lines are perpendicular, then they intersect to form a four right angles. If a ∞ b, then ∠1, ∠2, ∠3, ∠4 are right angles. Thm 3.10 1 2 3 4 b If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. A If BA ⊥ BC, then ∠1 and ∠2 are complementary. 1 2 B C 3.6 Prove Theorems About Perpendicular Lines Example 1: Prove theorem 3.10 Prove that if two sides of two adjacent acute angles are perpendicular, then the angles are complementary. Given: ED ⊥ EF Prove: ∠7 and ∠8 are complementary D 7 8 Statements 1. ED ⊥ EF 2. ∠DEF is a right angle 3. m∠DEF = 90º 4. m∠7 + m∠8 = m∠DEF 5. m∠7 + m∠8 = 90º 6. ∠7 and ∠8 are compl. ∠s E F Reasons 1. Given 2. ⊥ lines intersect to form 4 rt. ∠s (theorem 3.9) 3. Definition of a right angle 4. Angle Addition Postulate 5. Substitution POE 6. Definition of complementary ∠s Homework: Day 1 Pages 194 – 197 #1, 3–7, 22–25, 31, 32 3.6 Prove Theorems About Perpendicular Lines Daily Openers October 26 Name the angles as corresponding, alternate interior, alternate exterior, or consecutive interior angles 1. ∠1 and ∠12 2. ∠4 and ∠8 3. ∠11 and ∠15 4. ∠7 and ∠16 5. ∠2 and ∠9 3 1 2 4 9 11 10 12 5 6 8 7 13 14 15 16 6. Write the equation of the line through point R(2, 3) that is perpendicular to the line through the points Q(3, 8) and S(1, 2) 7. Write the equation of the line through point A(5, 4) that is parallel to the line through the points B(1, 5) and C(2, 1). 3.6 Prove Theorems About Perpendicular Lines Write the theorem that justifies the statement 1. ∠1 and ∠2 are right ∠s 2. r ⊥ s r 1 2 1 2 s Theorem 3.9 Theorem 3.10 3. ∠1 and ∠2 are complementary. Theorem 3.8 1 2 3.6 Prove Theorems About Perpendicular Lines Find m∠1 5. 4. 1 60º 75º 1 75o + m∠1 = 90o (Angle addition Postulate) m∠1 = 15 6. Find the measure of ∠1 ∠6 4 50º 3 5 45º 2 1 6 90o 60o = 30 o m∠1 = 30o (Vertical Angles Congruence Theorem) Since ∠1 and ∠6 form a right angle, then lines a and b are perpendicular lines. m∠1 = 40o (Vertical ∠4) m∠2 = 45o (Vertical ∠5) m∠3 = 45o (Vertical 45o) m∠4 = 40o (90o 50o) m∠5 = 45o (90o 45o) m∠6 = 50o (Vertical 50o) 3.6 Prove Theorems About Perpendicular Lines Thm 3.11 Perpendicular Transversal Theorem If a transversal is perpendicular to one of two parallel lines, then j k it is perpendicular to the other. h If h || k and j ⊥ h, then j ⊥ k Thm 3.12 Lines Perpendicular to a Transversal Theorem In a plane, if two lines are perpendicular to the same line, then m n they are parallel to each other If m ⊥ n then m || n. distance from a point to a line : p the length of the perpendicular segment from the point to the line Day 2 Pages 194 – 197 #9–14, 17–21, 29, 30, 33, 34, 40 3.6 Prove Theorems About Perpendicular Lines 7. Is r || s? r s t m 8. Is m || n? 9. Is r || t? n