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Geometry Lesson 1.6 Angle Pair Relationships Warm-Up: Angle Types (review) Name the type of angle and state how many degrees it can have Right angle (exactly 90°) Obtuse angle (> 90° and < 180°) Acute angle < 90° Straight angle (exactly 180°) 1. Vertical Angles Vertical Angles: Two angles whose sides form two pairs of opposite rays In this context, “vertical” means “shared vertex”, not “straight up” 4 1 3 2 1 and 3 are vertical angles 2 and 4 are vertical angles 2. Linear Pairs Linear Pair: Two adjacent angles whose non-common sides are opposite rays 1 2 Adjacent angles have a common side Non-common sides are opposite rays 1 and 2 are a linear pair Example 1: Identifying Angle Pairs Are 1 & 2 adjacent? Yes: Common side Are 1 & 2 a linear pair? Yes: Adjacent & opposite rays Are 3 & 4 a linear pair? No: Adjacent, but not opposite rays Are 2 & 5 vertical angles? Yes: Two pairs of opposite rays Are 1 & 4 vertical angles? No: Sides are NOT opposite rays Are 3 & 5 vertical angles? No: Sides are NOT opposite rays Practice 1: Identify Angle Pairs Answer the questions for each figure (a) (b) No No Yes No Yes No Yes No 3. Properties of Angle Pairs The sum of the angle measures in a linear pair is always 180° 1 m1 + m2 = 180° 2 Vertical Angles are always congruent (equal measures) 4 1 3 2 m1 = m3 m2 = m4 Simulations Vertical angles animation Linear pair animation Example 2a: Finding Angle Measures 129° Linear pair: 51° + m7 = 180° Vertical angles are congruent Linear pair: 136° + m8 = 180° Vertical angles are congruent 103° 44° 53° Example 2b: Finding Angle Measures This time, let’s use algebra… Find the value of x and use it to find the angle measures 115° FHI GHJ (vertical s) 115° mFHI mGHJ (7x – 25)° = (5x + 15)° mFHI = 7(20) – 25 2x = 40 x = 20 mFHI = 115° Now, substitute x… mGHJ = 5(20) + 15 mGHJ = 115° Practice 2: Finding Angle Measures Find x or y and then evaluate the angle measures (b) (a) 4. Complementary Angles Complementary Angles: Two angles whose measures total 90° 1 2 Adjacent or 1 2 Non-adjacent 1 and 2 are complementary m1 + m2 = 90° 5. Supplementary Angles Supplementary Angles: Two angles whose measures total 180° 5 6 Adjacent or 5 6 Non-adjacent 5 and 6 are supplementary m1 + m2 = 180° Example 3a: Complements & Supplements E is a complement of F If mE = 68°, find mF E F mE + mF = 90° 68° + mF = 90° mF = 22° G is a supplement of H If mG = 152°, find mH G H mG + mH = 180° 152° + mH = 180° mH = 28° Example 3b: Comp & Supp w/Algebra A is supplementary to B A is complementary to C A B mA = x°; mB = (x + 40)° mC = (x – 50)° A C Find all angle measures mA + mB = 180° 2x + 40 = 180 x = 70 mA = 70° mB = (70 + 40)° = 110° mC = (70 – 50)° = 20° Practice 3: Comp & Supp s (a) A is a complement of B and mA = 81° Find mB (b) C is a supplement of D and mC = 27° Find mD (c) Repeat example 3b with the following: mA = x°; mB = (2x)°; and mC = (x – 30)° Closure: Angle Pairs Angle measures in a linear pair add up to ________° Angle measures in vertical angles are _____________ Complementary angle measures add up to _______° Supplementary angle measures add up to _______° For some cool computer animations, go to http://www.mathopenref.com/tocs/anglestoc.html Complementary Angles You’re sooo acute!! Homework 1.6 w/s