Find m1. A. 115 B. 105 C. 75 D. 65 5-Minute Check 1

Find m2. A. 75 B. 72 C. 57 D. 40 5-Minute Check 2

Find m3. A. 75 B. 72 C. 57 D. 40 5-Minute Check 3

Find m4. A. 18 B. 28 C. 50 D. 75 5-Minute Check 4

Find m5. A. 70 B. 90 C. 122 D. 140 5-Minute Check 5

Use properties of isosceles triangles. Use properties of equilateral triangles. Then/Now

legs of an isosceles triangle vertex angle base angles Vocabulary

Concept

A. Name two unmarked congruent angles. Congruent Segments and Angles A. Name two unmarked congruent angles. BCA is opposite BA and A is opposite BC, so BCA  A. ___ Answer: BCA and A Example 1

B. Name two unmarked congruent segments. Congruent Segments and Angles B. Name two unmarked congruent segments. ___ BC is opposite D and BD is opposite BCD, so BC  BD. Answer: BC  BD Example 1

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

B. Which statement correctly names two congruent segments? A. JP  PL B. PM  PJ C. JK  MK D. PM  PK Example 1b

Concept

Subtract 60 from each side. Answer: mR = 60 Divide each side by 2. Find Missing Measures A. Find mR. Since QP = QR, QP  QR. By the Isosceles Triangle Theorem, base angles P and R are congruent, so mP = mR . Use the Triangle Sum Theorem to write and solve an equation to find mR. Triangle Sum Theorem mQ = 60, mP = mR Simplify. Subtract 60 from each side. Answer: mR = 60 Divide each side by 2. Example 2

Find Missing Measures B. Find PR. Since all three angles measure 60, the triangle is equiangular. Because an equiangular triangle is also equilateral, QP = QR = PR. Since QP = 5, PR = 5 by substitution. Answer: PR = 5 cm Example 2

ALGEBRA Find the value of each variable. Find Missing Values ALGEBRA Find the value of each variable. Since E = F, DE  FE by the Converse of the Isosceles Triangle Theorem. DF  FE, so all of the sides of the triangle are congruent. The triangle is equilateral. Each angle of an equilateral triangle measures 60°. Example 3

mDFE = 60 Definition of equilateral triangle 4x – 8 = 60 Substitution Find Missing Values mDFE = 60 Definition of equilateral triangle 4x – 8 = 60 Substitution 4x = 68 Add 8 to each side. x = 17 Divide each side by 4. The triangle is equilateral, so all the sides are congruent, and the lengths of all of the sides are equal. DF = FE Definition of equilateral triangle 6y + 3 = 8y – 5 Substitution 3 = 2y – 5 Subtract 6y from each side. 8 = 2y Add 5 to each side. Example 3

4 = y Divide each side by 2. Answer: x = 17, y = 4 Find Missing Values Example 3

Find the value of each variable. A. x = 20, y = 8 B. x = 20, y = 7 C. x = 30, y = 8 D. x = 30, y = 7 Example 3