1 Geometry Section 4-1B Angles Outside the Triangle Pg. 247 Be ready to grade 4-1A Quiz Friday
2 The last section of this chapter referred to angles inside of a triangle. Those are called _____________________. interior angles X Y Z W X, Y, and Z are interior angles. Definition
3 Exterior Angle: An angle of a triangle that is formed by extending one of the sides of the triangle. X Y Z W W is an exterior angle. Definition
4 Adjacent Interior Angle: The angle that forms a linear pair with the exterior angle. X Y Z W Z is an adjacent interior angle with W. Definition
5 Remote Interior Angle: An angle NOT adjacent to the exterior angle. X Y Z W X and Y are remote interior angles to W. Definition
6 1, 2 and 3 4 and a. Exterior Angles b. Remote Interior Angles for Note that 8 is an exterior angle, but 7 is not. 9 9 is not an exterior angle. Why? Try It: Pg. 248 Try It:
7 m1m1m2m2m3m3m4m4 83 o 40 o 84 o 71 o 148 o 54 o o 97 o 32 o 64 o 55 o 125 o 180 – 83 = – ( ) = – 148= – ( ) = – ( ) = – 55= 125 Try It: Pg. 249 Try It:
8 Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. X Y Z W m X + m Y = m W. Pg. 249
9 Exterior Angle Inequality Theorem: The measure of an exterior angle of a triangle is greater than the measure either of its remote interior angles. X Y Z W A theorem that follows immediately FROM another is called a _____________. corollary If we know that m X + m Y = m W, then…. Pg. 249
10 Each angle of an equilateral triangle measures 60 o. 60 o Pg. 249 Theorem
11 The acute angles of a right triangle are complementary. a b m a + m b = 90 o Pg. 249 Theorem
12 What are the remote angles for 1? 3 and 4 2 and 4 2 and 3 #1 – 3 Pg. 250 Exercises
13 Find m m 5 =m 7 = (2x + 31) o m 6 = x o m 8 = (4x - 19) o x + 2x x – 19 = 180 x = – 24 = 156 o #7 Pg. 250 Exercises
14 The brace JK is fastened to the wall and the shelf shown. Which angle has the greater measure, LJK or MKJ? Justify your answer. #10 Pg. 250 MKJ has the greater measure because of the Exterior Angle Inequality Theorem. LJK is a remote interior angle for MKJ. Exercises
15 The measures of the acute angles of a right triangle are in the ratio of 1:2. What is the measure of the smallest angle of the triangle? #11 Pg. 250 We know that the acute angles of a right triangle are complementary therefore: Exercises 1x + 2x = 90 x = 30 o Be sure to pay attention to which angle they are asking you for! Biggest or smallest?
16 If the measures of the exterior angles of the acute angles of a right triangle are (6x + 23) o and (4x + 17) o, find the measures of the acute angles. #12 Pg. 251 Exercises
17 3 C 24 1 DAB If m 1 = 18 o, find m 2. #13 Pg. 251 m 2 = 180 – ( ) = 72 o Exercises
18 3 C D A B If m 4 = 150 o and m 3 = 130 o find m ACD. #14 Pg. 251 Find the measure of CAD. CAD = 180 – 150 = 30 o. ACD = 180 – ( ) = 20 o. Exercises
19 3 C D A B If m 1 = 20 o, find m 3. #15 Pg. 251 Find the measure of 2. 2 = 180 – ( ) = 70 o. 3 = 180 – 70= 110 o. Exercises
20 A 1 2 B D C E If AC || BE, explain how the figure above shows that m 1 + m 2 + m 3 = 180 o. #25 Pg. 252 By the properties of parallel lines, we know that m 1 = m 5 and m 2 = m 4. Since m 3 + m m 5 = 180, we can use substitution to prove that m 1 + m 2 + m 3 = 180 o. Exercises
21 Homework: Practice 4-1B Quiz Friday