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GEOMETRY LESSON 4-5 (For help, go to Lesson 3-4.) 1.Name the angle opposite AB. 2.Name the angle opposite BC. 3.Name the side opposite A. 4.Name the side opposite C. 5.Find the value of x. Isosceles and Equilateral Triangles 4-5 Check Skills Youll Need By the Triangle Exterior Angle Theorem, x = 75 + 30 = 105°.

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1.What does CPCTC stand for? Use the diagram for Exercises 2 and 3. 2.Tell how you would show ABM ACM. 3.Tell what other parts are congruent by CPCTC. Use the diagram for Exercises 4 and 5. 4.Tell how you would show RUQ TUS. 5.Tell what other parts are congruent by CPCTC. Using Congruent Triangles: CPCTC GEOMETRY LESSON 4-4 Corresponding parts of congruent triangles are congruent. RQ TS, UQ US, R T You are given two pairs of s, and AM AM by the Reflexive Prop., so ABM ACM by ASA. 4-4 You are given a pair of s and a pair of sides, and RUQ TUS because vertical angles are, so RUQ TUS by AAS. AB AC, BM CM, B C

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Isosceles and Equilateral Triangles GEOMETRY LESSON 4-5 4-5 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. 3 is the vertex angle. 1 and 2 are the base angles.

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Isosceles and Equilateral Triangles GEOMETRY LESSON 4-5 4-5

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Isosceles and Equilateral Triangles GEOMETRY LESSON 4-5 4-5 The Isosceles Triangle Theorem is sometimes stated as Base angles of an isosceles triangle are congruent. Reading Math

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Isosceles and Equilateral Triangles GEOMETRY LESSON 4-5 4-5

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Isosceles and Equilateral Triangles GEOMETRY LESSON 4-5 4-5 A corollary is a statement that follows immediately from a theorem.

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Explain why ABC is isosceles. By the definition of an isosceles triangle, ABC is isosceles. Isosceles and Equilateral Triangles GEOMETRY LESSON 4-5 ABC and XAB are alternate interior angles formed by XA, BC, and the transversal AB. Because XA || BC, ABC XAB. The diagram shows that XAB ACB. By the Transitive Property of Congruence, ABC ACB. You can use the Converse of the Isosceles Triangle Theorem to conclude that AB AC. 4-5 Quick Check Using the Isosceles Triangle Theorems

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Suppose that m L = y. Find the values of x and y. m N=m LIsosceles Triangle Theorem m L=yGiven m N + m NMO + m MON=180Triangle Angle-Sum Theorem m N=yTransitive Property of Equality y + y + 90=180Substitute. 2y + 90=180Simplify. 2y=90Subtract 90 from each side. y=45Divide each side by 2. Therefore, x = 90 and y = 45. MOLNThe bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. x=90Definition of perpendicular Isosceles and Equilateral Triangles GEOMETRY LESSON 4-5 4-5 Quick Check Using Algebra

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Because the garden is a regular hexagon, the sides have equal length, so the triangle is isosceles. By the Isosceles Triangle Theorem, the unknown angles are congruent. Example 4 found that the measure of the angle marked x is 120. The sum of the angle measures of a triangle is 180. If you label each unknown angle y, 120 + y + y = 180. 120 + 2y =180 2y =60 y =30 So the angle measures in the triangle are 120, 30 and 30. Isosceles and Equilateral Triangles GEOMETRY LESSON 4-5 Suppose the raised garden bed is a regular hexagon. Suppose that a segment is drawn between the endpoints of the angle marked x. Find the angle measures of the triangle that is formed. 4-5 Quick Check Real-World Connection

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Use the diagram for Exercises 1–3. 1.If m BAC = 38, find m C. 2.If m BAM = m CAM = 23, find m BMA. 3.If m B = 3x and m BAC = 2x – 20, find x. 4.Find the values of x and y. 5.ABCDEF is a regular hexagon. Find m BAC. 71 90 25 x = 60 y = 9 30 4-5 Isosceles and Equilateral Triangles GEOMETRY LESSON 4-5 4-5

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