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Understanding Triangle Geometry: Angle-Sum Theorem and Alternate Exterior Angles

In this lesson, we explore the Triangle Angle-Sum Theorem, which states that the sum of the angles in a triangle is 180°. Students will engage in practical proofs and understand related concepts like alternate exterior angles formed by a transversal. We will derive corollaries from the theorem and learn the Strong Exterior Angle Theorem, illustrating deeper geometric relationships. Homework will involve exercises reinforcing these concepts, along with reflective journal questions for personal assessment and growth.

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Understanding Triangle Geometry: Angle-Sum Theorem and Alternate Exterior Angles

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Presentation Transcript

  1. GBK Geometry Jordan Johnson

  2. Today’s plan – Proof Practice • Greeting • Warm-up Proof • Triangle Angle Sum Theorem • A Practical Proof • Homework / Questions • Clean-up

  3. Prove – In Pairs • Definition: • Two angles formed by a transversal are alternate exterior angles iff they lie outside of the two lines traversed and are on opposite sides of the transversal. • Prove: • Alternate exterior angles on parallel lines are equal.

  4. Theorem 20 • The Triangle Angle-Sum Theorem: • The sum of the angles of a triangle is 180°. • Given: ABC. • Prove: A + B + C = 180°. • Beginning: • Construct line DE, parallel to AC and passing through B. • (What rule permits this?) • Pair up and prove it.

  5. Corollaries to theTriangle Angle-Sum Theorem • If two angles of one triangle are equal to two angles of another triangle, then the third pair of angles are equal. • The acute angles of a right triangle are complementary. • Each angle of an equilateral triangle is 60°.

  6. Going further... • Triangle Angle-Sum Theorem Strong Exterior Angle Theorem • Theorem 21: The StrongExterior Angle Theorem: • An exterior angle of a triangle is equal to the sum of the remote interior angles. • This is “strong” because it makes a statement that implies the “weak” theorem and more.

  7. What are these?

  8. A MiterGauge • Set to the angle desired for the cut. • Push along a slot parallel to the blade.

  9. A MiterGauge • Sketch a diagram and use it to prove that the angle of the cut will equal the angle of the gauge.

  10. Work • Finish Journal #15: • What did you do here? How did you use your time? What did you learn? • Were there any surprises this week? (Meaning, any events or solutions or discoveries you found surprising?) • What words / concepts were new to you this week? What do they mean? • What was good about this week’s classes for you? • What can you and I do better next week? • Do you have any other comments? • Due Thursday (per. 1-3) or Friday (per. 7). • Asg #50: Read Ch. 6 Lesson 5 (pp. 238-241) and do: • Set I Exercises 1-18. • Set II Exercises 31-45. • Bonus: Set III. • Due Monday, 2/11.

  11. Clean-up / Reminders • Pick up all trash / items. • Push in chairs (at front and back tables). • See you tomorrow!

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