1. 4.1 Apply Triangle Sum Properties

2. Objectives  Identify and classify triangles by angles or sides  Apply the Angle Sum Theorem  Apply the Exterior Angle Theorem

3. Parts of a Triangle  A triangle is a 3-sided polygon  The sides of ∆ABC are AB, BC, and AC  The vertices of ∆ABC are A, B, and C  Two sides sharing a common vertex are adjacent sides  The third side is called the opposite side  All sides can be adjacent or opposite (it just depends which vertex is being used) adjacent Side opposite  A C A B

4. Classifying Triangles by Angles Obtuse Obtuse 1 angle is obtuse (measure > 90°) Right 1 angle is right (measure = 90°) One way to classify triangles is by their angles… Acute Acute all 3 angles are acute (measure < 90°) An acute ∆ with all angles  is an equiangular ∆.

5. ARCHITECTURE The triangular truss below is modeled for steel construction. Classify  JMN,  JKO, and  OLN as acute, equiangular, obtuse, or right. Example 1: 60°

6. Answer:  JMN has one angle with measure greater than 90, so it is an obtuse triangle.  JKO has one angle with measure equal to 90, so it is a right triangle.  OLN is an acute triangle with all angles congruent, so it is an equiangular triangle. Example 1:

7. Classifying Triangles by Sides Isosceles Isosceles 2 congruent sides Scalene Scalene no congruent sides Another way to classify triangles is by their sides… Equilateral Equilateral 3 congruent sides

8. Answer:  UTX and  UVX are isosceles. Identify the isosceles triangles in the figure if Isosceles triangles have at least two sides congruent. Example 2a:

9. Identify the scalene triangles in the figure if Answer:  VYX,  ZTX,  VZU,  YTU,  VWX,  ZUX, and  YXU are scalene. Scalene triangles have no congruent sides. Example 2b:

10. Identify the indicated triangles in the figure. a. isosceles triangles b. scalene triangles Answer:  ABC,  EBC,  DEB,  DCE,  ADC,  ABD Answer:  ADE,  ABE Example 2c:

11. ALGEBRA Find d and the measure of each side of equilateral triangle KLM if and Since  KLM is equilateral, each side has the same length. So 5 = d Example 3:

12. Next, substitute to find the length of each side. Answer: For  KLM, and the measure of each side is 7. Example 3: KL = 7 LM = 7KM = 7

13. COORDINATE GEOMETRY Find the measures of the sides of  RST. Classify the triangle by sides. Example 4:

14. Answer: ; since all 3 sides have different lengths,  RST is scalene. Use the distance formula to find the lengths of each side. Example 4:

15. Exterior Angles and Triangles  An exterior angle is formed by one side of a triangle and the extension of another side (i.e.  1 ).  The interior angles of the triangle not adjacent to a given exterior angle are called the remote interior angles (i.e.  2 and  3). 1 2 3 4

16. Theorem 4.1 – Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is 180°. m  X + m  Y + m  Z = 180° m  X + m  Y + m  Z = 180° X Y Z

17. Find the missing angle measures. Find first because the measure of two angles of the triangle are known. Angle Sum Theorem Simplify. Subtract 117 from each side. Example 5:

18. Answer: Angle Sum Theorem Simplify. Subtract 142 from each side. Example 5:

19. Find the missing angle measures. Answer: Your Turn:

20. Theorem 4.2 – Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. m  1 = m  2 + m  3 The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. m  1 = m  2 + m  3 1 2 3 4

21. Find the measure of each numbered angle in the figure. Exterior Angle Theorem Simplify. Substitution Subtract 70 from each side. If 2 s form a linear pair, they are supplementary. Example 6:

22. Exterior Angle Theorem Subtract 64 from each side. Substitution Subtract 78 from each side. If 2 s form a linear pair, they are supplementary. Substitution Simplify. Example 6:

23. Subtract 143 from each side. Angle Sum Theorem Substitution Simplify. Answer: Example 6:

24. Find the measure of each numbered angle in the figure. Answer: Your Turn:

25. Corollaries  A corollary is a statement that can be easily proven using a theorem.  Corollary 4.1 – The acute  s of a right ∆ are complementary.  Corollary 4.2 – There can be at most one right or obtuse  in a ∆.

26. Corollary 4.1 Substitution Subtract 20 from each side. Answer: GARDENING The flower bed shown is in the shape of a right triangle. Find if is 20. Example 3:

27. Answer: The piece of quilt fabric is in the shape of a right triangle. Find if is 62. Your Turn:

28. Assignment  Pre-AP Geometry: Pgs. 221-224 #1 – 6, 14 – 19, 21 – 26, 31 – 37