Isosceles and Equilateral Triangles LESSON 4–6

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

Over Lesson 4–5 5-Minute Check 2 A.ΔVYX B.ΔZYW C.ΔZYV D.ΔWYZ Refer to the figure. Complete the congruence statement. ΔWYZ  Δ_____ by AAS. ?

Over Lesson 4–5 5-Minute Check 3 A.ΔWXZ B.ΔVWX C.ΔWVX D.ΔYVX Refer to the figure. Complete the congruence statement. ΔVWZ  Δ_____ by SSS. ?

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

TEKS 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, mid- segments, and medians, and apply these relationships to solve problems. Mathematical Processes G.1(E), G.1(F)

Then/Now You identified isosceles and equilateral triangles. Use properties of isosceles triangles. Use properties of equilateral triangles.

Vocabulary legs of an isosceles triangle vertex angle base angles

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. ___

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

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

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

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

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 Given: HEXAGO is a regular polygon. ΔONG is equilateral, N is the midpoint of GE, and EX || OG. Prove:ΔENX 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.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.

Example 4 Apply Triangle Congruence Proof: ReasonsStatements 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 Proof: ReasonsStatements 1.Given1.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 ___ Given: HEXAGO is a regular hexagon.  NHE   HEN   NAG   AGN Prove: HN  EN  AN  GN ___

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

Isosceles and Equilateral Triangles LESSON 4–6