4.6 Isosceles, Equilateral, and Right Triangles

1. 4.6 Isosceles, Equilateral, and Right Triangles Objectives: I will be able to… -Solve for missing angles and sides using properties of iso., equil., and right triangles Vocabulary: Legs, base, base angles, vertex angle

2. Then/Now You identified isosceles and equilateral triangles. • Use properties of isosceles triangles. • Use properties of equilateral triangles.

3. Labeling Triangles Vertex: Adjacent Side: Opposite Side: Vertex: points joining the sides of the triangle. Adjacent Side: sides sharing a common vertex. Opposite Side: side opposite from the vertex.

4. vertex angle vertex angle vertex angle leg leg leg leg leg leg leg leg base angles base angles base base Vocabulary Isosceles: at least two congruent sides Legs: congruent sides Vertex Angle: angle between legs Base: non congruent side Base Angles: angles adjacent to the base Isosceles: Legs: Vertex Angle: Base: Base Angles:

5. 4.6 Isosceles, Equilateral, and Right Triangles Base Angles Theorem: If 2 sides in a triangle are congruent, then the opposite base angles are congruent. B A C

6. 4.6 Isosceles, Equilateral, and Right Triangles Converse of Base Angles Theorem: If 2 base angles in a triangle are congruent, then the opposite sides are congruent. B A C

7. 1) In each triangle, solve for x and y. b) a) Q 2x + 7 80° B R x + 5 y° x° 3x – 9 S A C

8. 2) Solve for x and y. 50 y° y° x° y + y + 110 = 180 x + x + 40 = 180 y = 35 x = 70

9. Equilateral Triangles Corollary to Base Angles Theorem: If a triangle is equilateral, then it is equiangular. B All angles are 60° A C

10. Equilateral Triangles Corollary to Base Angles Theorem: If a triangle is equiangular, then it is equilateral. B All angles are 60° A C

11. 3) Solve for x, y, and z. z° B x° y° A C

12. Vocabulary Right Triangle: contains one right angle Hypotenuse: side across from right angle longest side of the triangle Legs: sides adjacent to the right angle Right Triangle: Hypotenuse:   Legs: hypotenuse leg leg

13. 4) Classify the triangle by sides and angles. A(5, 2) B (5, 6) C (1, 6)

14. Example 1 A. Which statement correctly names two congruent angles? A. PJM PMJ B. JMK JKM C. KJP JKP D. PML PLK Example 1a

15. Example 2 A. Find mT. Example 2a

16. 5-Minute Check 5 Example 3 Find x if ΔLMN is isosceles triangle with vertex angle L. If LM = 2x – 4, MN = 4x + 6, and LN = 3x – 14.

17. 5-Minute Check 5 Example 4 Which of the following is true if a||b? • m1 = m 5 • m1 + m 2 = 180 • C.m3 + m 4 = 180 • D.m3 = m 5 4 5

18. Example 3 Example 5 ALGEBRA If m1 = 9x + 6, m2 = 2(5x – 3), and m3 = 5y + 14, find y. A.y = 14 B.y = 20 C.y = 16 D.y = 24

19. Example 2 Example 6 A. Find mR. B. Find PR.

20. 5-Minute Check 1 containing the point (5, –2) in point-slope form? Example 7

21. Example 8 Find the length of each side.

22. Example 9 Solve for x.

23. Example 10 SPORTS A pennant for the sports teams at Lincoln High School is in the shape of an isosceles triangle. If the measure of the vertex angle is 18°, find the measure of each base angle.

24. Example 11 BRIDGES Every day, cars drive through isosceles triangles when they go over the Leonard Zakim Bridge in Boston. The ten-lane roadway forms the bases of the triangles. a. The angle labeled A in the picture has a measure of 67°. What is the measure of ∠B? b. What is the measure of ∠C? c. Name the two congruent sides.

25. 5-Minute Check 4 containing the point (4, –6) in slope-intercept form? A. B. C. D. Example 12

26. 5-Minute Check 6 A. B. C. D. Example 13

27. Example 3 Example 14 GAMES In the game Tic-Tac-Toe, four lines intersect to form a square with four right angles in the middle of the grid. Is it possible to prove any of the lines parallel or perpendicular? Choose the best answer. • A. The two horizontal lines are parallel. • B. The two vertical lines are parallel. • The vertical lines are perpendicular to the horizontal lines. • All of these statements are true.

28. Homework: • p.289-290 #5, 20-22, 30, 32, 48