Section 2-5 Perpendicular Lines & Proofs

Perpendicular Lines – two lines that intersect to form right angles. Symbol: ^ Biconditional: Two lines are perpendicular, if and only if, they intersect to form right angles.

Once we have said one of these, then we can say… Given: AB ^ CD A B X Possible Conclusions: C ÐDXB is a right angle. _________ _ ____________ ____ ÐCXB is a right angle. ÐCXA is a right angle. ÐAXD is a right angle. Once we have said one of these, then we can say… mÐAXD = 90 Definition of a right angle.

D A B C Possible Conclusions: mÐAXD = 90 AB ^ DC Given: ÐAXD is a right angle A B X C Possible Conclusions: mÐAXD = 90 Definition of a Right Angle AB ^ DC Definition of a Perpendicular Lines

D A B C Given: AB ^ DC Prove: ÐAXD @ ÐDXB Theorem 2-4: If two lines are perpendicular, then they form congruent adjacent angles. Given: Two lines are perpendicular. Prove: The lines form congruent adjacent angles. D Given: AB ^ DC Prove: ÐAXD @ ÐDXB A B X C

D A B C X Given: AB ^ DC Prove: ÐAXD @ ÐDXB Statements Reasons PROOF OF THEOREM 2-4: D Given: AB ^ DC Prove: ÐAXD @ ÐDXB A B X C Statements Reasons 1. AB ^ DC 1. Given 2. ÐAXD is a right angle. ÐDXB is a right angle. 2. Definition of Perpendicular Lines 3. mÐAXD = 90 mÐDXB = 90 3. Definition of a right angle. 4. mÐAXD = mÐDXB ÐAXD @ ÐDXB 4. Substitution

D A B C Given: ÐAXD @ ÐDXB Prove: AB ^ DC Theorem 2-5: If two lines form congruent adjacent angles, then the lines are perpendicular. What is the relationship between this theorem and the last one? They are converses! D A Given: ÐAXD @ ÐDXB Prove: AB ^ DC B X C

D A B C X Given: ÐAXD @ ÐDXB Prove: AB ^ DC Statements Reasons PROOF OF THEOREM 2-5: D Given: ÐAXD @ ÐDXB Prove: AB ^ DC A B X Statements Reasons C 1. ÐAXD @ ÐDXB mÐAXD = mÐDXB 1. Given 2. mÐAXD + mÐDXB = 180 2. Angle Addition Postulate 3. mÐAXD + mÐAXD = 180 2mÐAXD = 180 3. Substitution 4. mÐAXD = 90 4. Division Property 5. ÐAXD is a right angle. 5. Definition of a right angle. 6. Definition of perpendicular Lines 6. AB ^ DC

Theorem 2-6: If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary. Given: OA ^ OC Prove: ÐAOB and ÐBOC are complementary angles. A O C B

4. Angle Addition Postulate PROOF OF THEOREM 2-6: A B Given: OA ^ OC Prove: ÐAOB and ÐBOC are complementary angles. O C Statements Reasons 1. OA ^ OC 1. Given 2. ÐAOC is a right angle. 2. Definition of Perpendicular Lines 3. mÐAOC = 90 3. Definition of a right angle. 4. mÐAOB + mÐBOC = mÐAOC 4. Angle Addition Postulate 5. mÐAOB + mÐBOC = 90 5. Substitution 6. ÐAOB and ÐBOC are complementary angles 6. Definition of Complementary Angles

EXAMPLE 4: THEOREM 2-6 A 2 O 1 C 3 Statements Reasons 1. AO ^ CO Given: AO ^ CO Prove: Ð1 and Ð3 are complementary angles 2 O 1 C 3 Statements Reasons 1. AO ^ CO 1. Given 2. If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary. 2. Ð1 and Ð2 are complementary angles 3. Definition of Complementary Angles 3. mÐ1 + mÐ2 = 90 4. Ð2 @ Ð3, mÐ2 = mÐ3 4. Vertical Angle Theorem 5. mÐ1 + mÐ3 = 90 5. Substitution 6. Ð1 and Ð3 are complementary angles 6. Definition of Complementary Angles