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

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

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

Vocabulary Isosceles: Isosceles: at least two congruent sides Legs: 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: leg leg vertex angle base angles base angles base leg leg vertex angle base leg leg vertex angle leg leg

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

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

1) In each triangle, solve for x and y. b) a) A B C Q 80° 2x + 7 R x + 5 x° y° X = y = 50 X = 16 3x – 9 S

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

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

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

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

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

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

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

Example 2 A. Find mT. Example 2a

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

Which of the following is true if a||b? 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 5-Minute Check 5

ALGEBRA If m1 = 9x + 6, m2 = 2(5x – 3), and m3 = 5y + 14, find y. 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 Example 3

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

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

Example 8 Find the length of each side.

Example 9 Solve for x.

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.

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.

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

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

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. Example 3

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