* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download 4-2 - Midland ISD
Survey
Document related concepts
Steinitz's theorem wikipedia , lookup
Euler angles wikipedia , lookup
Reuleaux triangle wikipedia , lookup
Atiyah–Singer index theorem wikipedia , lookup
Rational trigonometry wikipedia , lookup
Noether's theorem wikipedia , lookup
Riemann–Roch theorem wikipedia , lookup
Trigonometric functions wikipedia , lookup
Four color theorem wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Euclidean geometry wikipedia , lookup
Integer triangle wikipedia , lookup
Transcript
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 Over Lesson 4–1 Classify ΔRST . A. acute B. equiangular C. obtuse D. right Over Lesson 4–1 Find y___ if ΔRST is an isosceles triangle with ___ RS RT. A. 8 B. 10 C. 12 D. 14 Over Lesson 4–1 Find x if ΔABC is an equilateral triangle. A. 2 B. 4 C. 6 D. 8 Over Lesson 4–1 A. ΔABC B. ΔACB C. ΔADC D. ΔCAB Over Lesson 4–1 Classify ΔMNO as scalene, isosceles, or equilateral if MN = 12, NO = 9, and MO = 15. A. scalene B. isosceles C. equilateral Over Lesson 4–1 Which is not a classification for ΔFGH? A. acute B. scalene C. isosceles D. equiangular 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) You classified triangles by their side or angle measures. • Apply the Triangle Angle-Sum Theorem. • Apply the Exterior Angle Theorem. • auxiliary line • exterior angle • remote interior angles • flow proof • corollary 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. Use the Triangle Angle-Sum Theorem Formulate Find m1 first because the measure of two angles of the triangle are known. Use the Vertical Angles Theorem to find m2. Then you will have enough information to find the measure of 3. Determine Triangle Angle-Sum Theorem Simplify. Subtract 117 from each side. Use the Triangle Angle-Sum Theorem 1 and 2 are congruent vertical angles. So, m2 = 63. Triangle Angle-Sum Theorem Simplify. Subtract 142 from each side. Answer: Therefore, m1 = 63, m2 = 63, and m3 = 38. Justify The sums of the measures of the angles in each triangle should be 180. m1 + 43 + 74 = 63 + 43 + 74 or 180 m2 + m3 + 79 = 63 + 38 + 79 or 180 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. Find the measure of 3. A. 95 B. 75 C. 57 D. 85 Use the Exterior Angle Theorem GARDENING Find the measure of FLW in the fenced flower garden shown. mLOW + mOWL = mFLW x + 32 = 2x – 48 Exterior Angle Theorem Substitution 32 = x – 48 Subtract x from each side. 80 = x Add 48 to each side. Answer: So, mFLW = 2(80) – 48 or 112. 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 Find Angle Measures in Right Triangles Find the measure of each numbered angle. Exterior Angle Theorem m1 = 38 + 32 Simplify. = 70 70 + m2 = 180 110 If 2 s form a linear pair, they are supplementary. Substitution Subtract 70 from each side. 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 78+ m4 = 180 102 Simplify. Subtract 78 from each side. Find Angle Measures in Right Triangles m5 + 102+ 41 = 180 m5 + 143 = 180 37 Triangle Angle-Sum Theorem Simplify. Subtract 143 from each side. m1 = 70, m2 = 110, m3 = 46, m4 = 102, m5 =37 Find m3. A. 50 B. 45 C. 85 D. 130 LESSON 4–2 Angles of Triangles