LESSON 4–2 Angles of Triangles

Five-Minute Check (over Lesson 4–1) TEKS Then/Now New Vocabulary Theorem 4.1: Triangle Angle-Sum Theorem Proof: Triangle Angle-Sum Theorem Example 1: Real-World Example: Use the Triangle Angle-Sum Theorem Theorem 4.2: Exterior Angle Theorem Proof: Exterior Angle Theorem Example 2: Real-World Example: Use the Exterior Angle Theorem Corollaries: Triangle Angle-Sum Corollaries Example 3: Find Angle Measures in Right Triangles Lesson Menu

Classify ΔRST . A. acute B. equiangular C. obtuse D. right 5-Minute Check 1

Find y if ΔRST is an isosceles triangle with RS  RT. ___ A. 8 B. 10 C. 12 D. 14 5-Minute Check 2

Find x if ΔABC is an equilateral triangle. 5-Minute Check 3

A. ΔABC B. ΔACB C. ΔADC D. ΔCAB 5-Minute Check 4

Classify ΔMNO as scalene, isosceles, or equilateral if MN = 12, NO = 9, and MO = 15. A. scalene B. isosceles C. equilateral 5-Minute Check 5

Which is not a classification for ΔFGH? A. acute B. scalene C. isosceles D. equiangular 5-Minute Check 6

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

You classified triangles by their side or angle measures. Apply the Triangle Angle-Sum Theorem. Apply the Exterior Angle Theorem. Then/Now

remote interior angles flow proof corollary auxiliary line exterior angle remote interior angles flow proof corollary Vocabulary

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Use the Triangle Angle-Sum Theorem SOFTBALL The diagram shows the path of the softball in a drill developed by four players. Find the measure of each numbered angle. Analyze Examine the information in the diagram. You know the measures of two angles of one triangle and only one measure of another. You also know that 1 and 2 are vertical angles. Example 1

Triangle Angle-Sum Theorem Use the Triangle Angle-Sum Theorem Formulate Find m1 first because the measure of two angles of the triangle are known. Use the Vertical Angles Theorem to find m2. Then you will have enough information to find the measure of 3. Determine Triangle Angle-Sum Theorem Simplify. Subtract 117 from each side. Example 1

1 and 2 are congruent vertical angles. So, m2 = 63. Use the Triangle Angle-Sum Theorem 1 and 2 are congruent vertical angles. So, m2 = 63. Triangle Angle-Sum Theorem Simplify. Subtract 142 from each side. Answer: Therefore, m1 = 63, m2 = 63, and m3 = 38. Justify The sums of the measures of the angles in each triangle should be 180. m1 + 43 + 74 = 63 + 43 + 74 or 180 m2 + m3 + 79 = 63 + 38 + 79 or 180 Example 1

used to check the reasonableness of the answers found. Use the Triangle Angle-Sum Theorem Evaluate By identifying each part of the problem, this complex problem could be separated into three manageable pieces. The properties of triangles were used to check the reasonableness of the answers found. Example 1

Find the measure of 3. A. 95 B. 75 C. 57 D. 85 Example 1

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GARDENING Find the measure of FLW in the fenced flower garden shown. Use the Exterior Angle Theorem GARDENING Find the measure of FLW in the fenced flower garden shown. mLOW + mOWL = mFLW Exterior Angle Theorem x + 32 = 2x – 48 Substitution 32 = x – 48 Subtract x from each side. 80 = x Add 48 to each side. Answer: So, mFLW = 2(80) – 48 or 112. Example 2

The piece of quilt fabric is in the shape of a right triangle The piece of quilt fabric is in the shape of a right triangle. Find the measure of ACD. A. 30 B. 40 C. 50 D. 130 Example 2

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Find the measure of each numbered angle. Find Angle Measures in Right Triangles Find the measure of each numbered angle. Exterior Angle Theorem m1 = 38 + 32 Simplify. = 70 If 2 s form a linear pair, they are supplementary. Substitution 70 + m2 = 180 Subtract 70 from each side. 110 Example 3

Exterior Angle Theorem Find Angle Measures in Right Triangles m 3 + 64 = 110 Exterior Angle Theorem Simplify. = 46 If 3 s form a linear pair, they are supplementary 46+ 32+ m 4 = 180 Simplify. 78+ m4 = 180 Subtract 78 from each side. 102 Example 3

Triangle Angle-Sum Theorem m5 + 102+ 41 = 180 Find Angle Measures in Right Triangles Triangle Angle-Sum Theorem m5 + 102+ 41 = 180 Simplify. m5 + 143 = 180 Subtract 143 from each side. 37 m1 = 70, m2 = 110, m3 = 46, m4 = 102, m5 =37 Example 3

Find m3. A. 50 B. 45 C. 85 D. 130 Example 3

LESSON 4–2 Angles of Triangles