Concept Example 1 Congruent Segments and Angles A. Name two unmarked congruent angles. Answer: BCA and A BCA is opposite BA and A is opposite BC, so BCA A. ___ Example 1 Congruent Segments and Angles B. Name two unmarked congruent segments. Answer: BC BD ___ BC is opposite D and BD is opposite BCD, so BC BD. ___ A.A B.B C.C D.D Example 1a A. PJM PMJ B. JMK JKM C. KJP JKP D. PML PLK A. Which statement correctly names two congruent angles? A.A B.B C.C D.D Example 1b B. Which statement correctly names two congruent segments? A.JP PL B.PM PJ C.JK MK D.PM PK Concept 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. Example 2 Find Missing Measures A. Find m R. Triangle Sum Theorem m Q = 60, m P = m R Simplify. Subtract 60 from each side. Divide each side by 2. Answer: m R = 60 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. Example 2 Find Missing Measures B. Find PR. Answer: PR = 5 cm A.A B.B C.C D.D Example 2a A.30 B.45 C.60 D.65 A. Find m T. A.A B.B C.C D.D Example 2b A.1.5 B.3.5 C.4 D.7 B. Find TS. Example 3 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 Find Missing Values m DFE= 60Definition of equilateral triangle 4x 8 = 60Substitution 4x= 68Add 8 to each side. x= 17Divide 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= FEDefinition of equilateral triangle 6y + 3= 8y 5Substitution 3= 2y 5Subtract 6y from each side. 8= 2yAdd 5 to each side. Example 3 Find Missing Values 4= yDivide each side by 2. Answer: x = 17, y = 4 A.A B.B C.C D.D Example 3 A.x = 20, y = 8 B.x = 20, y = 7 C.x = 30, y = 8 D.x = 30, y = 7 Find the value of each variable. Example 4 Apply Triangle Congruence NATURE Many geometric figures can be found in nature. Some honeycombs are shaped like a regular hexagon. That is, each of the six sides and interior angle measures are the same. Given: HEXAGO is a regular polygon. ONG is equilateral, N is the midpoint of GE, and EX || OG. Prove:ENX is equilateral. ___ Example 4 Apply Triangle Congruence Proof: ReasonsStatements 1.Given1.HEXAGO is a regular polygon. 5.Midpoint Theorem 5.NG NE 6.Given 6.EX || OG 2.Given 2.ONG is equilateral. 3. Definition of a regular hexagon 3. EX XA AG GO OH HE 4. Given 4.N is the midpoint of GE Example 4 Apply Triangle Congruence Proof: ReasonsStatements 7. Alternate Exterior Angles Theorem 7. NEX NGO 8.ONG ENX 8. SAS 9.OG NO GN 9. Definition of Equilateral Triangle 10. NO NX, GN EN 10. CPCTC 11. XE NX EN 11. Substitution 12. ENX is equilateral. 12. Definition of Equilateral Triangle Example 4 Proof: ReasonsStatements 1.Given1.HEXAGO is a regular hexagon. 2.Given 2. NHE HEN NAG AGN ___ Given: HEXAGO is a regular hexagon. NHE HEN NAG AGN Prove: HN EN AN GN ___ 3.HE EX XA AG GO OH 3.Definition of regular hexagon 4.HNE ANG 4.ASA A.A B.B C.C D.D Example 4 Proof: ReasonsStatements 5.HN AN, EN NG 6.HN EN, AN GN 6.Converse of Isosceles Triangle Theorem 7.HN EN AN GN 7.Substitution 5.CPCTE ___ Given: HEXAGO is a regular hexagon. NHE HEN NAG AGN Prove: HN EN AN GN ___