4 8 Triangles 4 8 Isoscelesand Equilateral Triangles

  • Slides: 27
Download presentation
4 -8 Triangles 4 -8 Isoscelesand Equilateral Triangles Warm Up Lesson Presentation Lesson Quiz

4 -8 Triangles 4 -8 Isoscelesand Equilateral Triangles Warm Up Lesson Presentation Lesson Quiz Holt Geometry

4 -8 Isosceles and Equilateral Triangles Do Now 1. Find each angle measure. True

4 -8 Isosceles and Equilateral Triangles Do Now 1. Find each angle measure. True or False. If false explain. 2. Every equilateral triangle is isosceles. 3. Every isosceles triangle is equilateral. Holt Geometry

4 -8 Isosceles and Equilateral Triangles Objectives TSW prove theorems about isosceles and equilateral

4 -8 Isosceles and Equilateral Triangles Objectives TSW prove theorems about isosceles and equilateral triangles. TSW apply properties of isosceles and equilateral triangles. Holt Geometry

4 -8 Isosceles and Equilateral Triangles Vocabulary legs of an isosceles triangle vertex angle

4 -8 Isosceles and Equilateral Triangles Vocabulary legs of an isosceles triangle vertex angle base angles Holt Geometry

4 -8 Isosceles and Equilateral Triangles In an Isosceles triangle, the congruent sides are

4 -8 Isosceles and Equilateral Triangles In an Isosceles triangle, 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. 3 is the vertex angle. 1 and 2 are the base angles. Holt Geometry

4 -8 Isosceles and Equilateral Triangles Holt Geometry

4 -8 Isosceles and Equilateral Triangles Holt Geometry

4 -8 Isosceles and Equilateral Triangles Reading Math The Isosceles Triangle Theorem is sometimes

4 -8 Isosceles and Equilateral Triangles Reading Math The Isosceles Triangle Theorem is sometimes stated as “Base angles of an isosceles triangle are congruent. ” Holt Geometry

4 -8 Isosceles and Equilateral Triangles Example 1: Astronomy Application The length of YX

4 -8 Isosceles and Equilateral Triangles Example 1: Astronomy Application The length of YX is 20 feet. Explain why the length of YZ is the same. Holt Geometry

4 -8 Isosceles and Equilateral Triangles Example 2 If the distance from Earth to

4 -8 Isosceles and Equilateral Triangles Example 2 If the distance from Earth to a star in September is 4. 2 1013 km, what is the distance from Earth to the star in March? Explain. 4. 2 1013; since there are 6 months between September and March, the angle measures will be approximately the same between Earth and the star. By the Converse of the Isosceles Triangle Theorem, the triangles created are isosceles, and the distance is the same. Holt Geometry

4 -8 Isosceles and Equilateral Triangles Example 3: Finding the Measure of an Angle

4 -8 Isosceles and Equilateral Triangles Example 3: Finding the Measure of an Angle Find m F. Holt Geometry

4 -8 Isosceles and Equilateral Triangles Example 4: Finding the Measure of an Angle

4 -8 Isosceles and Equilateral Triangles Example 4: Finding the Measure of an Angle Find m G. Holt Geometry

4 -8 Isosceles and Equilateral Triangles Example 5 Find m H. Holt Geometry

4 -8 Isosceles and Equilateral Triangles Example 5 Find m H. Holt Geometry

4 -8 Isosceles and Equilateral Triangles Example 6 Find m N. Holt Geometry

4 -8 Isosceles and Equilateral Triangles Example 6 Find m N. Holt Geometry

4 -8 Isosceles and Equilateral Triangles The following corollary and its converse show the

4 -8 Isosceles and Equilateral Triangles The following corollary and its converse show the connection between equilateral triangles and equiangular triangles. Holt Geometry

4 -8 Isosceles and Equilateral Triangles Holt Geometry

4 -8 Isosceles and Equilateral Triangles Holt Geometry

4 -8 Isosceles and Equilateral Triangles Example 7: Using Properties of Equilateral Triangles Find

4 -8 Isosceles and Equilateral Triangles Example 7: Using Properties of Equilateral Triangles Find the value of x. Holt Geometry

4 -8 Isosceles and Equilateral Triangles Example 8: Using Properties of Equilateral Triangles Find

4 -8 Isosceles and Equilateral Triangles Example 8: Using Properties of Equilateral Triangles Find the value of y. Holt Geometry

4 -8 Isosceles and Equilateral Triangles Example 9 Find the value of JL. Holt

4 -8 Isosceles and Equilateral Triangles Example 9 Find the value of JL. Holt Geometry

4 -8 Isosceles and Equilateral Triangles Holt Geometry

4 -8 Isosceles and Equilateral Triangles Holt Geometry

4 -8 Isosceles and Equilateral Triangles Remember! A coordinate proof may be easier if

4 -8 Isosceles and Equilateral Triangles 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. Holt Geometry

4 -8 Isosceles and Equilateral Triangles Example 4: Using Coordinate Proof Prove that the

4 -8 Isosceles and Equilateral Triangles 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 ∆ABC, X is the mdpt. of AB, and Y is the mdpt. of AC. Prove: XY = Holt Geometry 1 AC. 2

4 -8 Isosceles and Equilateral Triangles Example 4 Continued Proof: Draw a diagram and

4 -8 Isosceles and Equilateral Triangles 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 (3 a, b). By the Distance Formula, XY = √ 4 a 2 = 2 a, and AC = 4 a. 1 Therefore XY = AC. 2 Holt Geometry

4 -8 Isosceles and Equilateral Triangles Check It Out! Example 4 What if. .

4 -8 Isosceles and Equilateral Triangles Check It Out! Example 4 What if. . . ? The coordinates of isosceles ∆ABC are A(0, 2 b), B(-2 a, 0), and C(2 a, 0). X is the midpoint of AB, and Y is the midpoint of AC. Prove ∆XYZ is isosceles. y Proof: A(0, 2 b) Draw a diagram and place the coordinates as shown. X Y Z B(– 2 a, 0) Holt Geometry x C(2 a, 0)

4 -8 Isosceles and Equilateral Triangles Check It Out! Example 4 Continued By the

4 -8 Isosceles and Equilateral Triangles 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 y Formula, XZ = YZ = √a 2+b 2. A(0, 2 b) So XZ YZ and ∆XYZ is isosceles. X Y Z B(– 2 a, 0) Holt Geometry x C(2 a, 0)

4 -8 Isosceles and Equilateral Triangles Holt Geometry

4 -8 Isosceles and Equilateral Triangles Holt Geometry

4 -8 Isosceles and Equilateral Triangles Lesson Quiz: Part I Find each angle measure.

4 -8 Isosceles and Equilateral Triangles Lesson Quiz: Part I Find each angle measure. 1. m R 28° 2. m P 124° Find each value. 3. x 5. x Holt Geometry 20 4. y 26° 6

4 -8 Isosceles and Equilateral Triangles Lesson Quiz: Part II 6. The vertex angle

4 -8 Isosceles and Equilateral Triangles Lesson Quiz: Part II 6. The vertex angle of an isosceles triangle measures (a + 15)°, and one of the base angles measures 7 a°. Find a and each angle measure. a = 11; 26°; 77° Holt Geometry