Warm Up Find each angle measure: 𝑚∠𝐴𝑋𝐵= 𝑚∠𝐶𝑋𝐵= E 𝑚∠𝐸𝑋𝐷= F 𝑚∠𝐹𝑋𝐸= 𝑚∠𝐸𝑋𝐵= 𝑚∠𝐹𝑋𝐷= A X B C D E F 52o 31o

4.1 Definitions and Theorems (write on a separate sheet of paper) Picture/Example Vertical Angles: Def: Theorem: Complementary Angles: Supplementary Angles: Pairs of opposite angles formed when two lines intersect Vertical angles are congruent Two angles whose measures add to 90o Two angles whose measures add to 180o (”linear pair”)

Vertical Angles Find each indicated angle without using “vertical angles are congruent”. 91o x y

Vertical Angles Find each indicated angle without using “vertical angles are congruent”. 72o x y

Vertical Angles 1 2 3 Writing Prompt: What is the relationship between angle 1 and angle 2? What is the relationship between angle 3 and angle 2? In your own words, why do you think 𝑚∠1=𝑚∠3? 3 1 2

Proving the Vertical Angles Theorem Prove m∠1=𝑚∠3 1 2 3 Proof (quick version) m∠1 + m ∠2 = 180 (linear pairs are supplementary) m∠3 + m ∠2 = 180 (linear pairs are supplementary) m∠1 + m ∠2 = m∠3 + m ∠2 (substitution or transitive property) m∠1 = m ∠3 (subtraction property)

Practice Problems Solve for x in #1 and #2: 1) 2) 3) 𝑚∠2 𝑖𝑠 𝑡ℎ𝑟𝑒𝑒 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑠𝑖𝑧𝑒 𝑜𝑓 𝑚∠1. 𝑊ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑎𝑛𝑔𝑙𝑒? (2x2 – 10x) o (4x2 + 12)o (2x + 23) o (x - 3) o 2x + 23 + x – 3 = 180 3x + 20 = 180 3x = 160 x = 160/3 ≈ 53.3 4x2 + 12 = 2x2 – 10x 2x2 + 10x +12 = 0 x2 + 5x + 6 = 0 (x + 2)(x + 3) = 0 x = -2, -3 let m∠1 = x, m∠2 = 3x 3x + x = 90 4x = 90 x = 22.5 m∠1 = 22.5, m∠2 = 67.5 1 2

4.2 Transversals and Parallel lines Sketch this picture in your notes: Come up with 2 pairs of angles for each relationship Corresponding Angles: Alternate Interior Angles: Alternate Exterior Angles: Same Side Interior Angles: 11 6 9 7 13 3 19 17

Practice Problems p Given that n//m and p //q, find the missing angles. m 10 a) Suppose 𝑚∠3= 79 𝑜 , 𝑓𝑖𝑛𝑑 𝑚∠6: 9 n 12 11 q 2 14 b) Suppose 𝑚∠11= 122 𝑜 , 𝑓𝑖𝑛𝑑 𝑚∠2: 1 3 13 15 4 16 6 5 c) Suppose 𝑚∠8= 63 𝑜 , 𝑓𝑖𝑛𝑑 𝑚∠15: 8 7 c) Suppose 𝑚∠7= 145 𝑜 , 𝑓𝑖𝑛𝑑 𝑚∠11:

Classwork: p. 169 #1-15, 17 p. 180 #1-2, 3-15 odd, 16-19 Homework: Take notes on 4.3 and 4.4 using files on rosenmath.com

Proving the Vertical Angles Theorem Prove m∠1=𝑚∠3 1 2 3 Proof (quick version) m∠1 + m ∠2 = 180 (linear pairs are supplementary) m∠3 + m ∠2 = 180 (linear pairs are supplementary) m∠1 + m ∠2 = m∠3 + m ∠2 (substitution or transitive property) m∠1 = m ∠3 (subtraction property)