Geometry unit 4.5
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This document covers properties of isosceles and equilateral triangles. It defines key terms like legs, vertex angle, and base of an isosceles triangle. It presents theorems like if a triangle is isosceles, the vertex angle is equal to the base angles, and the bisector of the vertex angle is the perpendicular bisector of the base. Examples demonstrate using these properties to find missing angle measures. The connection between equilateral and equiangular triangles is also explained. Coordinate proofs may be used to show triangles are isosceles. Practice problems are included to assess understanding.
Geometry unit 4.5
- 1. UNIT 4.5 ISOSCELES AANNDD EEQQUUIILLAATTEERRAALL TTRRIIAANNLLEESS Warm Up 1. Find each angle measure. 60°; 60°; 60° True or False. If false explain. 2. Every equilateral triangle is isosceles. True 3. Every isosceles triangle is equilateral. False; an isosceles triangle can have only two congruent sides.
- 2. Objectives Prove theorems about isosceles and equilateral triangles. Apply properties of isosceles and equilateral triangles.
- 3. Vocabulary legs of an isosceles triangle vertex angle base base angles
- 4. Recall that an isosceles triangle has at least two congruent sides. The congruent sides are called the legs. The vertex angle is the angle formed by the legs. The side opposite the vertex angle is called the base, and the base angles are the two angles that have the base as a side. Ð3 is the vertex angle. Ð1 and Ð2 are the base angles.
- 5. If AB @ AC, then ÐB @ ÐC. If ÐE @ ÐF, then DE @ DF.
- 6. Bisector of the Vertex The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. AB @ BC BD Ð AB BD ^ AC BD AC If and bisects C, then and bisects .
- 7. Example 1: Finding the Measure of an Angle Find mÐF. mÐF = mÐD = x° Isosc. Δ Thm. mÐF + mÐD + mÐA = 180 Δ Sum Thm. x + x + 22 = 180 Thus mÐF = 79° Substitute the given values. 2x = 158 Simplify and subtract 22 from both sides. x = 79° Divide both sides by 2.
- 8. Example 2: Finding the Measure of an Angle Find mÐG. mÐJ = mÐG Isosc. Δ Thm. (x + 44)° = 3x° Substitute the given values. 44 = 2x Simplify x from both sides. x = 22° Divide both sides by 2. Thus mÐG = 22° + 44° = 66°.
- 9. TEACH! Example 1 Find mÐH. mÐH = mÐG = x° Isosc. Δ Thm. mÐH + mÐG + mÐF = 180 Δ Sum Thm. x + x + 48 = 180 Substitute the given values. 2x = 132 Simplify and subtract 48 from both sides. x = 66° Divide both sides by 2. Thus mÐH = 66°
- 10. TEACH! Example 2 Find mÐN. mÐP = mÐN Isosc. Δ Thm. (8y – 16)° = 6y° Substitute the given values. 2y = 16 Subtract 6y and add 16 to both sides. y = 8° Divide both sides by 2. Thus mÐN = 6(8) = 48°.
- 11. The following corollary and its converse show the connection between equilateral triangles and equiangular triangles.
- 13. Ex. 3A: Using Properties of Equilateral Triangles Find the value of x. ΔLKM is equilateral. Equilateral Δ equiangular Δ (2x + 32)° = 60° The measure of each Ð of an equiangular Δ is 60°. 2x = 28 Subtract 32 both sides. x = 14 Divide both sides by 2.
- 14. Ex. 3B: Using Properties of Equilateral Triangles Find the value of y. ΔNPO is equiangular. Equiangular Δ equilateral Δ 5y – 6 = 4y + 12 Definition of equilateral Δ. y = 18 Subtract 4y and add 6 to both sides.
- 15. TEACH! Example 3 Find the value of JL. ΔJKL is equiangular. Equiangular Δ equilateral Δ 4t – 8 = 2t + 1 Definition of equilateral Δ. 2t = 9 Subtract 4y and add 6 to both sides. t = 4.5 Divide both sides by 2. Thus JL = 2(4.5) + 1 = 10.
- 16. Remember! A coordinate proof may be easier if you place one side of the triangle along the x-axis and locate a vertex at the origin or on the y-axis.
- 17. Example 4: Using Coordinate Proof Prove that the segment joining the midpoints of two sides of an isosceles triangle is half the base. Given: In isosceles ΔABC, X is the mdpt. of AB, and Y is the mdpt. of BC. 1 2 Prove: XY = AC.
- 18. Example 4 Continued Proof: Draw a diagram and place the coordinates as shown. By the Midpoint Formula, the coordinates of X are (a, b), and Y are (3a, b). By the Distance Formula, XY = √4a2 = 2a, and AC = 4a. Therefore XY = AC. 1 2
- 19. TEACH! Example 4 The coordinates of isosceles ΔABC are A(0, 2b), B(-2a, 0), and C(2a, 0). X is the midpoint of AB, Y is the midpoint of AC, and Z(0, 0), . Prove ΔXYZ is isosceles. x A(0, 2b) y X Y B (–2a, 0) Z C (2a, 0) Proof: Draw a diagram and place the coordinates as shown.
- 20. Check It Out! Example 4 Continued By the Midpoint Formula, the coordinates. of X are (–a, b), the coordinates. of Y are (a, b), and the coordinates of Z are (0, 0) . By the Distance Formula, XZ = YZ = √a2+b2 . So XZ @ YZ and ΔXYZ is isosceles. x A(0, 2b) y X Y Z B(–2a, 0) C(2a, 0)
- 21. Lesson Quiz: Part I Find each angle measure. 1. mÐR 2. mÐP Find each value. 3. x 4. y 5. x 28° 124° 20 6 26°
- 22. Lesson Quiz: Part II 6. The vertex angle of an isosceles triangle measures (a + 15)°, and one of the base angles measures 7a°. Find a and each angle measure. a = 11; 26°; 77°; 77°
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