University:
Missouri State University
Course:
MTH 460 | College Geometry
Academic year:

2022
Views:

442

Pages:

10
Author:

Jamal Simon

Section 2-5: Perpendicular Lines & Proofs Perpendicular Lines – two lines that intersect to form right angles. Symbol: ^ Review: You can write all definitions as biconditional statements… Biconditional: Two lines are perpendicular, if and only if, they intersect to form right angles. A D X C B Given: AB ^ CD Possible Conclusions: DXB is a right angle. Definition of Perpendicular Lines. CXB is a right angle. Definition of Perpendicular Lines. CXA is a right angle. Definition of Perpendicular Lines. AXD is a right angle. Definition of Perpendicular Lines. Once we have said one of these, then we can say… mAXD = 90 Definition of a right angle. A D X C B Given: AXD is a right angle Possible Conclusions: mAXD = 90 Definition of a Right Angle AB ^ DC Definition of a Perpendicular Lines Theorem: If two lines are perpendicular, then they form congruent adjacent angles. Given: Two lines are perpendicular. Prove: The lines form congruent adjacent angles. A D X C Given: AB ^ DC B Prove: AXD @ DXB D Given: AB ^ DC A Prove: AXD @ DXB Statements 1. AB ^ DC X Reasons C 1. Given 2. AXD is a right angle. 2. Definition of DXB is a right angle. Perpendicular Lines 3. mAXD = 90 mDXB = 90 3. Definition of a right angle. 4. mAXD = mDXB AXD @ DXB 4. Substitution B Theorem: 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! A D X C B Given: AXD @ DXB Prove: AB ^ DC D Given: AXD @ DXB Prove: AB ^ DC Statements 1. mAXD = mDXB AXD @ DXB A X Reasons 1. Given B C 2. mAXD + mDXB = 180 2. Angle Addition Postulate 3. mAXD + mAXD = 180 3. Substitution 2mAXD = 180 4. mAXD = 90 4. Division Property 5. AXD is a right angle. 5. Definition of a right angle. 6. AB ^ DC 6. Definition of perpendicular Lines Theorem: If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary. Given: OA ^ OC Prove: AOB and BOC are complementary angles. A B O C Given: OA ^ OC A Prove: AOB and BOC are complementary angles. Statements Reasons O B C 1. OA ^ OC 1. Given 2. AOC is a right angle. 2. Definition of Perpendicular Lines 3. Definition of a right angle. 3. mAOC = 90 4. mAOB + mBOC = mAOC 4. Angle Addition Postulate 5. mAOB + mBOC = 90 5. Substitution 6. AOB and BOC are complementary angles 6. Definition of Complementary Angles Check for Understanding Pg.57 CE #6-11