Splash Screen

Five-Minute Check (over Lesson 4–5) CCSS 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 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 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 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 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 5-Minute Check 4

G.CO.10 Prove theorems about triangles. Content Standards G.CO.10 Prove theorems about triangles. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Mathematical Practices 2 Reason abstractly and quantitatively. 3 Construct viable arguments and critique the reasoning of others. CCSS

You identified isosceles and equilateral triangles. 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

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° Example 2a

B. Find TS. A. 1.5 B. 3.5 C. 4 D. 7 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

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

Prove: ΔENX is equilateral. 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. ___ Example 4

Proof: Reasons Statements 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. ΔONG is equilateral. 3. Definition of a regular hexagon 3. EX  XA  AG  GO  OH  HE 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 7. NEX  NGO Apply Triangle Congruence Proof: Reasons Statements 7. Alternate Exterior Angles Theorem 7. NEX  NGO 8. ΔONG  ΔENX 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 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. Definition of isosceles triangle B. Midpoint Theorem C. CPCTC Proof: Reasons Statements 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. Definition of isosceles triangle B. Midpoint Theorem C. CPCTC D. Transitive Property Example 4

End of the Lesson