LESSON 4–3 Congruent Triangles

Five-Minute Check (over Lesson 4–2) TEKS Then/Now New Vocabulary Key Concept: Definition of Congruent Polygons Example 1: Identify Corresponding Congruent Parts Example 2: Use Corresponding Parts of Congruent Triangles Theorem 4.3: Third Angles Theorem Example 3: Real-World Example: Use the Third Angles Theorem Example 4: Prove that Two Triangles are Congruent Theorem 4.4: Properties of Triangle Congruence Lesson Menu

Find m1. A. 115 B. 105 C. 75 D. 65 5-Minute Check 1

Find m2. A. 75 B. 72 C. 57 D. 40 5-Minute Check 2

Find m3. A. 75 B. 72 C. 57 D. 40 5-Minute Check 3

Find m4. A. 18 B. 28 C. 50 D. 75 5-Minute Check 4

Find m5. A. 70 B. 90 C. 122 D. 140 5-Minute Check 5

One angle in an isosceles triangle has a measure of 80° One angle in an isosceles triangle has a measure of 80°. What is the measure of one of the other two angles? A. 35 B. 40 C. 50 D. 100 5-Minute Check 6

Mathematical Processes G.1(F), G.1(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(F), G.1(G) TEKS

You identified and used congruent angles. Name and use corresponding parts of congruent polygons. Prove triangles congruent using the definition of congruence. Then/Now

congruent polygons corresponding parts Vocabulary

Concept 1

Identify Corresponding Congruent Parts Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement. Angles: Sides: Answer: All corresponding parts of the two polygons are congruent. Therefore, ABCDE  RTPSQ. Example 1

The support beams on the fence form congruent triangles The support beams on the fence form congruent triangles. In the figure ΔABC  ΔDEF, which of the following congruence statements correctly identifies corresponding angles or sides? A. B. C. D. Example 1

In the diagram, ΔITP  ΔNGO. Find the values of x and y. Use Corresponding Parts of Congruent Triangles In the diagram, ΔITP  ΔNGO. Find the values of x and y. O  P CPCTC mO = mP Definition of congruence 6y – 14 = 40 Substitution Example 2

NG = IT Definition of congruence x – 2y = 7.5 Substitution Use Corresponding Parts of Congruent Triangles 6y = 54 Add 14 to each side. y = 9 Divide each side by 6. CPCTC NG = IT Definition of congruence x – 2y = 7.5 Substitution x – 2(9) = 7.5 y = 9 x – 18 = 7.5 Simplify. x = 25.5 Add 18 to each side. Answer: x = 25.5, y = 9 Example 2

In the diagram, ΔFHJ  ΔHFG. Find the values of x and y. A. x = 4.5, y = 2.75 B. x = 2.75, y = 4.5 C. x = 1.8, y = 19 D. x = 4.5, y = 5.5 Example 2

Concept 2

ΔJIK  ΔJIH Congruent Triangles Use the Third Angles Theorem ARCHITECTURE A drawing of a tower’s roof is composed of congruent triangles all converging at a point at the top. If IJK  IKJ and mIJK = 72, find mJIH. ΔJIK  ΔJIH Congruent Triangles mIJK + mIKJ + mJIK = 180 Triangle Angle-Sum Theorem Example 3

mIJK + mIJK + mJIK = 180 Substitution Use the Third Angles Theorem mIJK + mIJK + mJIK = 180 Substitution 72 + 72 + mJIK = 180 Substitution 144 + mJIK = 180 Simplify. mJIK = 36 Subtract 144 from each side. mJIH = 36 Third Angles Theorem Answer: mJIH = 36 Example 3

TILES A drawing of a tile contains a series of triangles, rectangles, squares, and a circle. If ΔKLM  ΔNJL, KLM  KML, and mKML = 47.5, find mLNJ. A. 85 B. 45 C. 47.5 D. 95 Example 3

Write a two-column proof. Prove That Two Triangles are Congruent Write a two-column proof. Prove: ΔLMN  ΔPON Example 4

2. Vertical Angles Theorem Prove That Two Triangles are Congruent Proof: Statements Reasons 1. Given 1. 2. LNM  PNO 2. Vertical Angles Theorem 3. M  O 3. Third Angles Theorem 4. ΔLMN  ΔPON 4. CPCTC Example 4

Find the missing information in the following proof. Prove: ΔQNP  ΔOPN Proof: Reasons Statements 1. 1. Given 2. 2. Reflexive Property of Congruence 3. Q  O, NPQ  PNO 3. Given 4. _________________ 4. QNP  ONP ? 5. Definition of Congruent Polygons 5. ΔQNP  ΔOPN Example 4

B. Vertical Angles Theorem A. CPCTC B. Vertical Angles Theorem C. Third Angles Theorem D. Definition of Congruent Angles Example 4

Concept 3

LESSON 4–3 Congruent Triangles