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4.2 Angles of Triangles

4.2 Angles of Triangles. Objectives. Apply the Angle Sum Theorem Apply the Exterior Angle Theorem. X. Y. Z. Theorem 4.1 – Angle Sum Theorem. The sum of the measures of the angles of a triangle is 180 °. m X + m Y + mZ = 180 °.

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4.2 Angles of Triangles

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  1. 4.2 Angles of Triangles

  2. Objectives • Apply the Angle Sum Theorem • Apply the Exterior Angle Theorem

  3. X Y Z Theorem 4.1 – Angle Sum Theorem The sum of the measures of the angles of a triangle is 180°. mX + mY + mZ = 180°

  4. Find first because the measure of two angles of the triangle are known. Example 1: Find the missing angle measures. Angle Sum Theorem Simplify. Subtract 117 from each side.

  5. Answer: Example 1: Angle Sum Theorem Simplify. Subtract 142 from each side.

  6. Your Turn: Find the missing angle measures. Answer:

  7. Theorem 4.2 – Third Angle Theorem If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangle are congruent. Abbreviation: If 2 s of one Δ are  to 2 s of another Δ, then third s are .

  8. 2 1 3 4 Exterior Angles and Triangles • An exterior angle is formed by one side of a triangle and the extension of another side (i.e. 1 ). • The interior angles of the triangle not adjacent to a given exterior angle are called the remote interior angles (i.e. 2 and 3).

  9. 2 1 3 4 Theorem 4.3 – Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. m 1 = m2 + m 3

  10. Example 2: Find the measure of each numbered angle in the figure. Exterior Angle Theorem Simplify. If 2 s form a linear pair, they are supplementary. Substitution Subtract 70 from each side.

  11. Example 2: Exterior Angle Theorem Substitution Subtract 64 from each side. If 2 s form a linear pair, they are supplementary. Substitution Simplify. Subtract 78 from each side.

  12. Answer: Example 2: Angle Sum Theorem Substitution Simplify. Subtract 143 from each side.

  13. Your Turn: Find the measure of each numbered angle in the figure. Answer:

  14. Corollaries • A corollary is a statement that can be easily proven using a theorem. • Corollary 4.1 – The acute s of a right ∆ are complementary. • Corollary 4.2 – There can be at most one right or obtuse  in a∆.

  15. GARDENING The flower bed shown is in the shape of a right triangle. Find if is 20. Example 3: Corollary 4.1 Substitution Subtract 20 from each side. Answer:

  16. The piece of quilt fabric is in the shape of a right triangle. Find if is 62. Your Turn: Answer:

  17. Assignment • Geometry: Pg. 189 #3 – 27, 32 - 35 • Pre-AP Geometry: Pg. 189 #11 – 38, 45

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