Transcript 4-6
LESSON 4–6 Isosceles and Equilateral Triangles Five-Minute Check (over Lesson 4–5) TEKS Then/Now New Vocabulary Theorems: Isosceles Triangle Example 1: Congruent Segments and Angles Proof: Isosceles-Triangle Theorem Corollaries: Equilateral Triangle Example 2: Find Missing Measures Example 3: Find Missing Values Example 4: Real-World Example: Apply Triangle Congruence Over Lesson 4–5 Refer to the figure. Complete the congruence statement. ? ΔWXY Δ_____ by ASA. A. ΔVXY B. ΔVZY C. ΔWYX D. ΔZYW Over Lesson 4–5 Refer to the figure. Complete the congruence statement. ? ΔWYZ Δ_____ by AAS. A. ΔVYX B. ΔZYW C. ΔZYV D. ΔWYZ Over Lesson 4–5 Refer to the figure. Complete the congruence statement. ? ΔVWZ Δ_____ by SSS. A. ΔWXZ B. ΔVWX C. ΔWVX D. ΔYVX Over Lesson 4–5 What congruence statement is needed to use AAS to prove ΔCAT ΔDOG? A. C D B. A O C. A G D. T G Targeted TEKS G.6(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems. Mathematical Processes G.1(E), G.1(F) You identified isosceles and equilateral triangles. • Use properties of isosceles triangles. • Use properties of equilateral triangles. • legs of an isosceles triangle • vertex angle • base angles Congruent Segments and Angles A. Name two unmarked congruent angles. ___ opposite ___BA BCA is and A is opposite BC, so BCA A. Answer: BCA and A 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 A. Which statement correctly names two congruent angles? A. PJM PMJ B. JMK JKM C. KJP JKP D. PML PLK B. Which statement correctly names two congruent segments? A. JP PL B. PM PJ C. JK MK D. PM PK Find Missing Measures A. Find mR. Since QP = QR, QP QR. By the Isosceles Triangle Theorem, base angles P and R are congruent, so mP = mR . Use the Triangle Sum Theorem to write and solve an equation to find mR. Triangle Sum Theorem mQ = 60, mP = mR Answer: mR = 60 Simplify. Subtract 60 from each side. Divide each side by 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 A. Find mT. A. 30° B. 45° C. 60° D. 65° B. Find TS. A. 1.5 B. 3.5 C. 4 D. 7 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°. Find Missing Values mDFE = 60 4x – 8 = 60 4x = 68 x = 17 Definition of equilateral triangle Substitution Add 8 to each side. 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. Find Missing Values 4 =y Answer: x = 17, y = 4 Divide each side by 2. 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 Apply Triangle Congruence Given: HEXAGO is a regular polygon. ___ ΔONG is equilateral, N is the midpoint of GE, and EX || OG. Prove: ΔENX is equilateral. Apply Triangle Congruence Proof: Statements Reasons 1. HEXAGO is a regular polygon. 1. Given 2. ΔONG is equilateral. 2. Given 3. EX XA AG GO OH HE 3. Definition of a regular hexagon 4. N is the midpoint of GE. 4. Given 5. NG NE 5. Midpoint Theorem 6. EX || OG 6. Given Apply Triangle Congruence Proof: Statements 7. NEX NGO 8. ΔONG ΔENX Reasons 7. Alternate Exterior Angles Theorem 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. ΔENX is equilateral. 12. Definition of Equilateral Triangle Given: HEXAGO is a regular hexagon. NHE HEN NAG AGN ___ ___ ___ ___ Prove: HN EN AN GN Proof: Statements Reasons 1. HEXAGO is a regular hexagon. 1. Given 2. NHE HEN NAG AGN 2. Given 3. HE EX XA AG GO OH 3. Definition of regular hexagon 4. ΔHNE ΔANG 4. ASA Proof: Statements Reasons 5. HN AN, EN NG ? 5. ___________ 6. HN EN, AN GN 6. Converse of Isosceles Triangle Theorem 7. HN EN AN GN 7. Substitution A. Definition of isosceles triangle B. Midpoint Theorem C. CPCTC D. Transitive Property LESSON 4–6 Isosceles and Equilateral Triangles