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7.1 Triangle Application Theorems and 7.2 Two Proof-Oriented Theorems

7.1 Triangle Application Theorems and 7.2 Two Proof-Oriented Theorems. Objectives : To apply theorems about the interior angles and exterior angles of triangles To recognize and apply properties of the midline of triangles. Index Card!. Definition : Exterior angles (page 296)

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7.1 Triangle Application Theorems and 7.2 Two Proof-Oriented Theorems

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  1. 7.1 Triangle Application Theorems and 7.2 Two Proof-Oriented Theorems Objectives: To apply theorems about the interior angles and exterior angles of triangles To recognize and apply properties of the midline of triangles.

  2. Index Card! Definition: Exterior angles (page 296) Be sure to include a diagram!!!

  3. Index Card! What are . . . Remote Interior Angles? Be sure to include a diagram!!! X • So, • Remote interior angles never include the • interior angle adjacent to the exterior angle. • There are always 2 interior angles located • across from any exterior angle of a triangle!

  4. Theorems – INDEX CARDS! • The sum of the measures of the three angles of a triangles is 180. • The measure of an exterior angles of a triangles is equal to the sum of the measures of the remote interior angles. • If a segment joining the midpoints of two sides of a triangle is parallel to the third side, then its length is one-half the length of the third side (Midline Theorem). • If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are congruent (No-Choice Theorem). See pages 295, 296, and 302 Don’t forget to draw diagrams for each!!!!!

  5. Theorems – INDEX CARDS! B • The sum of the measures of the three angles of a triangles is 180. See page 295 Don’t forget to draw diagrams for each!!!!! B ∡A + ∡B + ∡C = straight angle! A C C A

  6. Theorems – INDEX CARDS! B • The sum of the measures of the three angles of a triangles is 180. See page 295 Don’t forget to draw diagrams for each!!!!! B B A C C A

  7. Theorems – INDEX CARDS! B • The sum of the measures of the three angles of a triangles is 180. See page 295 Don’t forget to draw diagrams for each!!!!! B A A C C A

  8. Theorems – INDEX CARDS! B • The sum of the measures of the three angles of a triangles is 180. See page 295 Don’t forget to draw diagrams for each!!!!! B A C C A C

  9. Theorems – INDEX CARDS! B • The sum of the measures of the three angles of a triangles is 180. See page 295 Don’t forget to draw diagrams for each!!!!! B ∡A + ∡B + ∡C = straight angle! A C A C

  10. Theorems – INDEX CARDS! B • The sum of the measures of the three angles of a triangles is 180. See page 295 Don’t forget to draw diagrams for each!!!!! B B ∡A + ∡B + ∡C = straight angle! ∡A + ∡B + ∡C = 180⁰ A A A C C A C C || lines → alt interior ∡s ≅

  11. Theorems – INDEX CARDS! • The measure of an exterior angles of a triangles is equal to the sum of the measures of the remote interior angles. See pages 295, 296, and 302 Don’t forget to draw diagrams for each!!!!! B Remember! ∡A + ∡B + ∡C = 180⁰ A C D

  12. Theorems – INDEX CARDS! • The measure of an exterior angles of a triangles is equal to the sum of the measures of the remote interior angles. See pages 295, 296, and 302 Don’t forget to draw diagrams for each!!!!! B And now . . . ∡DCB + ∡BCA = 180⁰ And now . . . ∡DCB + ∡BCA = 180⁰ And now . . . ∡DCB + ∡BCA = 180⁰ Remote Interior Remote Interior Exterior Remember! ∡A + ∡B + ∡C = 180⁰ Remember! ∡A + ∡B + ∡C = 180⁰ Remember! ∡A + ∡B + ∡C = 180⁰ ∡DCB = ∡A + ∡B A C D

  13. Theorems – INDEX CARDS! • If a segment joining the midpoints of two sides of a triangle is parallel to the third side, then its length is one-half the length of the third side (Midline Theorem). See pages 295, 296, and 302 Don’t forget to draw diagrams for each!!!!! x 20 10 5 x 40 20 10 2x

  14. Theorems – INDEX CARDS! • If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are congruent. (No-Choice Theorem) • See pages 295, 296, and 302 • Don’t forget to draw diagrams for each!!!!! • If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are congruent. (No-Choice Theorem) See pages 295, 296, and 302 Don’t forget to draw diagrams for each!!!!! DUH! DUH!

  15. G M H J Example 1: In the diagram as marked, if m G = 50, find m M. Solution: 2x + 2y + 50 = 180 2x + 2y = 130 x + y = 65 65 + m M = 180 m M = 115 50⁰ 2 115⁰ x y y x

  16. Example 2: The vertex angle of an isosceles triangle is twice as large as one of the base angles. Find the measure of the vertex angle. Let the measure of the vertex  = 2x Let each base angle = x Now: x + x + 2x = 180 4x = 180 x = 45 m vertex  = 2(45) = 90 2x x x

  17. Example 3: In ΔDEF, the sum of the measures of D and E is 110. The sum of the measures of E and F is 150. Find the sum of the measures of D + F. Solution: D + E + F = 180 110 + F = 180 F = 70 D + 150 = 180 D = 30  D + F = 30 + 70 = 100 30⁰ 70⁰

  18. Example 4: Find GH. E H G Midline ALERT !!! 8 D F 16

  19. Example 5: ProofGiven: A ≅ DProve: E ≅ C DUH! DUH! D A B ? ? C E ;D

  20. End lesson

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