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4-5 Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 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. 4-5 Isosceles and Equilateral Triangles Knowledge: Justify Mathematical Argument (1)(G) A builder using the truss shown at the right claims that οACB will have the same measure as οADB. π¨πͺ and π¨π« represent identical beams, and π¨π© bisects οCAD. Is the builder correct? Justify your answer. Yes. The builder is correct. It is given that π΄πΆ β π΄π· and by definition of angle bisectors, οCAB β οDAB. By the Reflexive Prop. of β, π΄π΅ β π΄π΅. Thus, βACB β βADB by SAS Postulate. οACB β οADB because of CPCTC. 4-5 Isosceles and Equilateral Triangles Knowledge: Making a Conjecture A. Construct congruent segments to make a conjecture about the angles opposite the congruent sides in an isosceles triangle. Step 1: Construct an isosceles βABC on your paper, with π΄πΆ β π΅πΆ. 4-5 Isosceles and Equilateral Triangles Know: Making a Conjecture Construct congruent segments to make a conjecture about the angles opposite the congruent sides in an isosceles triangle. Step 2: Fold the paper so that the two congruent sides fit exactly one on top of the other. Create the paper. Notice that οA and οB appear to be congruent. 4-5 Isosceles and Equilateral Triangles Communicate: Connect Mathematical Ideas (1)(F) Think: How can folding a piece of paper help you tell if two angles are congruent? When folding the paper, congruent angles will fit exactly one on top of the other. 4-5 Isosceles and Equilateral Triangles Knowledge: Making a Conjecture Write a conjecture that you observed for the angles opposite the congruent sides in an isosceles triangle. Angles opposite the congruent sides in an isosceles triangle are congruent. 4-5 Isosceles and Equilateral Triangles Knowledge: Making a Conjecture Write a conjecture that you observed for the sides opposite the congruent angles in an isosceles triangle. Sides opposite the congruent angles in an isosceles triangle are congruent. 4-5 Isosceles and Equilateral Triangles Connect to Math By the end of todayβs lesson, SWBAT 1. Prove theorems about isosceles and equilateral triangles. 2. Apply properties of isosceles and equilateral triangles. 4-5 Isosceles and Equilateral Triangles Vocabulary legs of an isosceles triangle vertex angle base base angles 4-5 Isosceles and Equilateral Triangles 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. ο3 is the vertex angle. ο1 and ο2 are the base angles. 4-5 Isosceles and Equilateral Triangles 4-5 Isosceles and Equilateral Triangles Example 1: Proving the Isosceles Triangle Theorem Begin with isosceles βXYZ with πΏπ β πΏπ. Draw πΏπ©, the bisector of vertex angle οYXZ. Given: ππ β ππ, ππ΅ bisects οYXZ Prove: οY β οZ Statements 1. ππ ο ππ ; ππ΅ bisects οYXZ 2. ο1 ο ο2 3. ππ΅ ο ππ΅ 4. βXYB ο βXZB 5. οY β οZ Reasons 1. Given 2. Definition of angle bisector 3. Reflex. Prop. of ο 4. SAS Postulate Steps 1, 2, 3 5. CPCTC 4-5 Isosceles and Equilateral Triangles Example 2: Proving the Isosceles Triangle Theorem A builder using the truss shown at the right claims that οACB will have the same measure as οADB. π¨πͺ and π¨π« represent identical beams, and π¨π© bisects οCAD. Is the builder correct? Justify your answer. Yes. The builder is correct. It is given that π΄πΆ β π΄π· by the Isosceles Triangle Theorem. 4-5 Isosceles and Equilateral Triangles 4-5 Isosceles and Equilateral Triangles Example 3: Using the Isosceles Triangle Theorem and its Converse A. Is π¨π© congruent to πͺπ© ? Explain. Yes. Since οC β οA, π΄π΅ β πΆπ΅ by the Converse of the Isosceles Triangle Theorem. B. Is οA congruent to οDEA ? Explain. Yes. Since π΄π· β πΈπ·, οA β οDEA by the Isosceles Triangle Theorem. 4-5 Isosceles and Equilateral Triangles Reading Math The Isosceles Triangle Theorem is sometimes stated as βBase angles of an isosceles triangle are congruent.β 4-5 Isosceles and Equilateral Triangles 4-5 Isosceles and Equilateral Triangles Example 4: Using Algebra What is the value of x ? Since π΄π΅ β πΆπ΅, βABD is isosceles β. By the Isosceles β Theorem οA β οC. mοC = 54o Since π΅π· bisects οABC, you know by Theorem 4-5 that π΅π· β₯ π΄πΆ. So, οBDC = 90o. mοC + mοBDC + mοDBC = 180o 54 + 90 + x = 180o x = 36o β Sum Theorem. Substitute. Subtract 144 from each side. 4-5 Isosceles and Equilateral Triangles Example 5: Complete each Statement. Explain why it is true. a. π½π» β ________ ππ Converse of Isosceles β Theorem ππ β ππΏ Converse of Isosceles β Thrm. b. πΌπ» β ________ ππ c. π½πΌ β ________ Converse of Isosceles β Thrm. and Segment Addition Post. οπππ Isosceles β Theorem d. οπ½ππΌ β ________ 4-5 Isosceles and Equilateral Triangles The following corollary and its converse show the connection between equilateral triangles and equiangular triangles. 4-5 Isosceles and Equilateral Triangles Equilateral Triangle Equiangular Triangle 4-5 Isosceles and Equilateral Triangles Example 6: Using Algebra B A. What is the value of x ? 40o Because x is the measure of an angle in an equilateral triangle, x = 60o. F D yo xo A C E G 4-5 Isosceles and Equilateral Triangles Example 6: Using Algebra B B. What is the value of y ? 40o F mοDCE + mοDEC + mοEDC = 180. β Sum Theorem. Substitute. D 60 + 70 + y = 180 Subtract 130 from each side. yo y = 50 xo A C E G 4-5 Isosceles and Equilateral Triangles Example 7: Using Algebra A. What is the value of x ? x + 2y = 180o β Sum Theorem. x + 2(70) = 180 Substitute. x = 40o Subtract 140 from each side. B. What is the value of y ? It is given that the triangle is an isosceles β. Thus, the base angles are congruent. Since 110o and the base angle to y are linear pair. Hence, y = 70o by Linear Pair Postulate. 4-5 Isosceles and Equilateral Triangles Example 8: Using Algebra 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 + 15 + 7a + 7a = 180o β Sum Theorem. 15a + 15 = 180 15a = 165 a = 11 (a + 15)° Combined Like Terms Subtract 15 from each side. Subtract 15 from each side. Therefore, each angle measure is 26°; 77°; 77° 7a° 7a° 4-5 Isosceles and Equilateral Triangles Exit Ticket: Find each angle measure. 1. mοR 2. mοP Find each value. 3. x 5. x 4. y