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Theorem: If two lines intersect to form a linear pair of congruent angles, then the lines are ___________________. If 1 2 , then __________________. h 1 2 g “Congruent angles of a linear pair are formed by perpendicular lines.” Theorem: If two lines are perpendicular, then they intersect to form four ___________________. If a b, then 1, 2, 3, and 4 are _____________________. “Perpendicular lines intersect to form 4 right angles.” Example 1: Draw Conclusions ⃡ . What can you In the diagram at the right, ⃡𝐴𝐵 ⊥ 𝐵𝐶 conclude about 1 and 2. Theorem: If two sides of two adjacent acute angles are perpendicular, then the angles are ______________________. ⃡ ⊥ 𝐵𝐶 ⃡ , then 1 and 2 are complementary. If 𝐵𝐴 “Two adjacent angles formed by perpendicular lines are complementary.” b 2 1 3 4 a Example 2: Prove IT! 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 STATEMENTS 1. 𝐸𝐷 ⊥ 𝐸𝐹 REASONS 1. 2. 𝑚∠𝐷𝐸𝐹 is a right angle 2. lines intersect to form 4 right ∠𝑠. 3. 𝑚∠𝐷𝐸𝐹 = 90° 3. 4. 𝑚∠7 + 𝑚∠8 = 𝑚∠𝐷𝐸𝐹 4. 5. 𝑚∠7 + 𝑚∠8 = 90° 5. 6. 6. Perpendicular Transversal Theorem If a transversal is perpendicular to one of two parallel lines, then it is ________________________________ to the other. If ℎ ∥ 𝑘 and 𝑗 ⊥ ℎ, then ___________________. Lines Perpendicular to a Transversal Theorem In a plane, if two lines are perpendicular to the same line, then they are ___________________________ to each other. If 𝑚 ⊥ 𝑝 and 𝑛 ⊥ 𝑝, then __________________. Example 3: Draw Conclusions Determine which lines, if any, must be parallel in the diagram. Explain your reasoning. Proofs Prove Perpendicular Transversal Thm. Prove Lines Perpendicular to a Transversal Theorem Given: m p, n p Prove: m n Given: h k, j h Prove: j k 1 j 1 2 2 Statements 1. m p, n p Reasons 1. Given 2. 1 is a right angle 2. 2 is a right angle 3. 1 2 3. 4. m n 4. Statements Reasons 1. h k, j h 1. Given 2. 1 2 2. 3. m 1 = m 2 3. 4. 1 is a right angle 4. 5. m 1 = 90 5. 6. m 2 = 90 6. 7. 2 is a right angle 7. 8. j k 8.