1. Chapter 2 Section 5 Perpendicular lines

2. Define: Perpendicular lines (  ) Two lines that intersect to form right or 90 o angles Remember all definitions are Biconditionals: If two lines are perpendicular then they form right angles If two lines form right angles then they are perpendicular The box shows you the right angle

3. Perpendicular Line Theorems If two lines are perpendicular, then they form congruent adjacent angles If two lines form congruent adjacent angles, then they are perpendicular

4. Perpendicular Line Theorems If two lines are perpendicular, then they form congruent adjacent angles 12 r t Given: r  t Prove: <1  <2 Hypothesisconclusionreason Noner  tgiven If r  t If m<1= 90 o & m<2 = 90 o If m<1= m<2 then m<1= 90 o & m<2 = 90 o def of perpendicular lines then m<1 = m<2 transitive prop / substitution then <1  <2 def of congruent angles

5. Define: Linear Pair of Angles Two adjacent angles whose exterior sides are opposite rays. 1 2 Angles 1 and 2 are a linear pair.

6. Perpendicular Line Theorems If two lines form congruent adjacent angles, then they are perpendicular 12 r t Given: R 1  R 2 Prove: r  t Hypothesisconclusionreason None R 1  R 2 or m R 1 = m R 2 given If R 1 and R <2 are a linear pair If m R 1 = m R 2 and m R 1 + m R 2 = 180 o If 2 (m R 1) = 180 o then m R 1 + m R 2 = 180 o Angle addition post then 2 (m R 1) = 180 o substitution then m R 1 = 90 o Division property If m R 1 = 90 o then r  t Def of  lines

7. Define: Perpendicular Pair of Angles Two adjacent acute angles whose exterior sides are perpendicular. 1 2 Angles 1 and 2 are a perpendicular pair.

8. Perpendicular Pair Theorem If two angles are a perpendicular pair, then the angles are complementary 1 2 A B C X Given: R 1 and R 2 are a perpendicular pair Prove: R 1 and R 2 are complementary angles

9. Proof of  Pair Theorem 1 2 A B C X Given: R 1 and R 2 are a perpendicular pair Prove: R 1 and R 2 are complementary angles Hypothesisconclusionreason None R 1 and R 2 are a  pair given If R 1 and R 2 are a  pair If X is in the interior of R ABC Def of  pair then m R ABC = 90 o Def of  lines then m R ABC = m R 1 +m R 2 Angle addition postulate If m R ABC = m R 1 +m R 2 and m R ABC = 90 o then m R 1 +m R 2 = 90 o Transitive property or substitution then BA  BC If BA  BC If m R 1 +m R 2 = 90 o then R 1 & R 2 are comp. R <‘s Def of comp. R ‘s

10. Definition of   90 or right  line theorems  2 lines forming  adjacent R ’s  pair theorem-  pair complementary R ’s summary

11. Practice work P 58 we 1-25all