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Five-Minute Check (over Lesson 4–5) NGSSS Then/Now New Vocabulary Theorems: Isosceles Triangle Example 1: Congruent Segments and Angles Corollaries: Equilateral Triangle Example 2: Find Missing Measures Example 3: Find Missing Values Example 4: Real-World Example: Apply Triangle Congruence Lesson Menu

Refer to the figure. Complete the congruence statement Refer to the figure. Complete the congruence statement. ΔWXY  Δ_____ by ASA. ? A. ΔVXY B. ΔVZY C. ΔWYX D. ΔZYW A B C D 5-Minute Check 1

Refer to the figure. Complete the congruence statement Refer to the figure. Complete the congruence statement. ΔWYZ  Δ_____ by AAS. ? A. ΔVYX B. ΔZYW C. ΔZYV D. ΔWYZ A B C D 5-Minute Check 2

Refer to the figure. Complete the congruence statement Refer to the figure. Complete the congruence statement. ΔVWZ  Δ_____ by SSS. ? A. ΔWXZ B. ΔVWX C. ΔWVX D. ΔYVX A B C D 5-Minute Check 3

What congruence statement is needed to use AAS to prove ΔCAT  ΔDOG? A. C  D B. A  O C. A  G D. T  G A B C D 5-Minute Check 4

LA.910.1.6.5 The student will relate new vocabulary to familiar words. MA.912.G.4.1 Classify, construct, and describe triangles that are right, acute, obtuse, scalene, isosceles, equilateral, and equiangular. NGSSS

You identified isosceles and equilateral triangles. (Lesson 4–1) 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 B C D A. Which statement correctly names two congruent angles? A. PJM  PMJ B. JMK  JKM C. KJP  JKP D. PML  PLK A B C D Example 1a

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

A. Find mT. A. 30° B. 45° C. 60° D. 65° A B C D Example 2a

B. Find TS. A. 1.5 B. 3.5 C. 4 D. 7 A B C D Example 2b

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

A B C D 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 A B C D Example 3

Prove: ΔENX is equilateral. 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: ΔENX is equilateral. Example 4

Proof: Reasons Statements 3. 1. Given 1. HEXAGO is a regular polygon. Apply Triangle Congruence Proof: Reasons Statements 1. Given 1. HEXAGO is a regular polygon. 2. Given 2. ∆ONG is equilateral 3. Definition of a regular hexagon EX  XA  AG  GO  OH  HE 3. 4. Given 4. N is the midpoint of GE 5. Midpoint Theorem 5. NG  NE 6. Given 6. EX || OG Example 4

Proof: Reasons Statements 8. ∆ONG  ∆ENX Apply Triangle Congruence Proof: Reasons Statements 7. Alternate Exterior Angles Theorem 7. NEX  NGO 8. SAS 8. ∆ONG  ∆ENX 9 Definition of Equilateral Triangle 9. OG  NO  GN 10. CPCTC 10. NO  NX, GN  EN 11. Substitution 11. XE  NX  EN 12. ΔENX is equilateral. 12. Definition of Equilateral Triangle Example 4

Given: HEXAGO is a regular hexagon. NHE  HEN  NAG  AGN ___ Given: HEXAGO is a regular hexagon. NHE  HEN  NAG  AGN Prove: HN  EN  AN  GN Proof: Reasons Statements 1. Given 1. HEXAGO is a regular hexagon. 2. Given 2. NHE  HEN  NAG  AGN 3. Definition of regular hexagon 4. ASA 3. HE  EX  XA  AG  GO  OH 4. ΔHNE  ΔANG Example 4

A B C D Proof: Reasons Statements A. Definition of isosceles triangle 5. ___________ 5. HN  AN, EN  NG 6. Converse of Isosceles Triangle Theorem 6. HN  EN, AN  GN 7. Substitution 7. HN  EN  AN  GN ? A B C D A. Definition of isosceles triangle B. Midpoint Theorem C. CPCTC D. Transitive Property Example 4

End of the Lesson