2. Module 5 Lesson 1 Investigating Angles of Triangles (Please print the Learning guide notes that go with this lesson so that you can follow them and use them for reference on the assignments.)

3. Triangles can be classified by sides AND by angles. When you look at the sides, you have 3 types of triangles: – Scalene – no sides are equal – Isosceles – 2 sides are equal – Equilateral – all 3 sides are equal

4. When you look at the angles, you also have 4 types of triangles. Acute triangle – all angles are less than 90° Obtuse triangle – one angle is more than 90° Right triangle – one angle is equal to 90° Equiangular triangle – all angles have the same measure (that is 60°)

5. Classify each triangle (by sides and by angles) 3 4 5 A) B) C)

6. Classify each triangle (by sides and by angles) 3 4 5 A) B) C) Obtuse and IsoscelesRight and scalene Equilateral and equiangular

7. Triangle Angle Sum Theorem The sum of the interior angles of any triangle is 180 °.

8. Example Find x. x46° 54° 54 + 46 + x = 180 100 + x = 180 -100 ------------------- x = 80

9. Another example Find m<G. x + 6 G 3x-1 2x + 13 F H Remember <G + <F + <H = 180 So the set up is like this: 3x – 1 + 2x + 13 + x + 6 = 180 6x + 18 = 180 -18 -18 --------------------- 6x = 162 x = 27 You must plug x = 27 into <G = 3x – 1 3(27) -1 = 71-1 = 70 ANSWER: m<G = 70 WARNING: This is correct but this is NOT the answer.

10. ALWAYS READ THE QUESTION! The two most common mistakes in Geometry are problems with algebra (equations) and not reading the questions. Students will set up the problem and solve correctly for x, but the question asks for something else. Be sure to read the question and PLUG x BACK IN if necessary.

11. Exterior Angle Theorem Exterior means “outside” Remote means “far away” The exterior angle of a triangle equals the sum of the remote interior angles. RULE:

12. Example 164° 37° Find m<1. <1 is an EXTERIOR angle. The rule is <1 = 37 + 64 So <1 = 101°

13. Another example 4x + 20 110° 2x+ 8 Find x. Exterior = sum of two remote interior angles 4x + 20 = 110 + 2x + 8 4x + 20 = 2x + 118 -2x ------------------------------ 2x + 20 = 118 -20 -20 -------------------- 2x = 98 x = 49

14. Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are also congruent. You have to match the congruent sides with the angles that are opposite them. Do that by drawing an arrow.

15. So if you are given AB = AC, then <B = <C. A CB

16. Example: Find x and then find m<R. 6x -6 R 2x + 18 S T Part 1: Find x. <S = <T 2x + 18 = 6x – 6 -2x ---------------------- 18 = 4x – 6 +6 +6 ----------------------- 24 = 4x 6 = x Part 2: Find m<R. Remember <R + <S + <T = 180 since they make a triangle. So plug x = 6 into <S and <T. <S = 2x + 18 = 2(6) + 18 =12 + 18 = 30. <T = 6x – 6 = 6(6) – 6 = 36-6 = 30. <S = 30 and <T = 30 So <R + 30 + 30 = 180 <R + 60 = 180 so that means <R = 120

17. The converse of the Isosceles Triangle Theorem is also true. Remember for the converse of a statement, just flip the order. So, if the two angles of a triangle are congruent, then the two sides opposite those angles are also congruent. So if m<D =m<E, then CE = DC. C ED

18. Example Find GH. G HJ 5x + 9 7x + 3 3x + 15 So the set up is 7x + 3 = 3x + 15 -3x -3x -------------------- 4x + 3 = 15 -3 -3 ---------------- 4x = 12 x = 3 WARNING! READ THE QUESTION! GH = 7x + 3 plug in x = 3 GH = 7(3) + 2 GH = 21 + 2 ANSWER: GH = 23

19. Equilateral Triangles Since all 3 sides of an equilateral triangle are the same, then all 3 angles must be the same. x + x + x = 180 3x = 180 x = 60 Fact: Each angle of an equilateral triangle = 60° xx x