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8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
8 2 Triangle Sum Theorem
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8 2 Triangle Sum Theorem

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Proof that the interior angles of triangles add up to 180 degrees

Proof that the interior angles of triangles add up to 180 degrees

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  • 1. Triangle Sum Theorem Focus - Prove the Triangle Sum Theorem
  • 2. If you add up the measures of the interior angles of a triangle, What is the total?
    • 180º
    25º 65º
  • 3. How many lines can be drawn through point B that would become parallel to M B
  • 4. Unique Parallel line Postulate – Through a point not on a given line, there is one and only one line parallel to the given line. m b This postulate will be used in the proof on the next slide.
  • 5. Given: Prove:
    • 1. ABC is a triangle
    • 2. Draw helping line
    • through B so that
    • 3.
    • 4.
    • 5.
    • 6.
    • 7.
    • 1. Given
    • 2. Unique Parallel Postulate (pg. 439)
    • 3. If the sides of an angle form a straight line, then the angle is a straight angle with a measure of 180º
    • 4. Whole-Parts Postulate
    • 5. Substitution
    • 6. Alternate Interior Angle Theorem
    • 7. Substitution Property
    B C A 2 1 3 D E 4 5
  • 6. Quadrilateral Theorem
    • The sum of the measures of the angles of a quadrilateral are:
    • What is the sum of the measures of a rectangle, which is a quadrilateral?
  • 7. Theorem: Opposite Angles of Quadrilaterals
    • If both pairs of opposite angles of a quadrilateral are equal in measure, …
    • then the quadrilateral is a parallelogram.
  • 8. What can we find out?
    • The ladder is leaning on the ground at a 75º angle. At what angle is the top of the ladder touching the building?
  • 9. What can we find out?
    • The ladder is leaning on the ground at a 75º angle. At what angle is the top of the ladder touching the building?
  • 10. Find the missing angle.
    • 47º
  • 11. Find the missing angle.
    • 47º
  • 12. Find the missing angles.
    • 120º
    • x º
    • 55º 25º yº wº
    Order: y w x
  • 13. Exterior Angle Theorem
    • An exterior angle of a triangle…
    • … is equal in measure to the sum of the measures of its two remote interior angles .
  • 14. Find the missing angles’ measurements (L and J).
    • 120º
    • 55º 25º
    J L remote interior angles
  • 15. What is the measure of Angle U?
    • 120º
    • 55º 25º
    J L T U To find the measure of angle T, use the definition of supplemental angles. To find the measure of angle U, use the vertical angle theorem.
  • 16. In today’s lesson we…
    • Proved the Triangle Sum Theorem (180 degrees)
    • Found Quadrilaterals’ Interior Angles Theorem to state the total degrees is 360.
    • Used the above theorems and the Exterior Angle Theorem to find measurements of a Triangle’s exterior angle.
  • 17. Assignment: pg. 443: 2-9, 13
  • 18. For Warmup 70° y° z° w° D E 50° x° (x-30)° x° 70° 60° 50° 110° D E 50° 70° (80-30)° 80°
  • 19. From tdn.com last night.
    • OLYMPIA, Wash. – The state Senate has made good on a promise to help students graduate from high school by passing a bill that would eliminate one diploma obstacle.
    • The measure would eliminate the requirement to keep taking the math Washington Assessment of Student Learning every year until graduation.
    • High school students who fail the 10th grade math test, would still be required to keep earning math credits, however.
    • The bill, which passed on a vote of 43-4, still requires approval in the House. It would affect the class of 2009, if it passes both houses and is signed by the governor.
    • The bill is Senate Bill 5498.
    • * Legislature (weblink) < http:// www.leg.wa.gov >
  • 20. Want to spend class time preparing for math WASL questions?
    • Group Work
    • Group Points
    • Three sections will vote.
    • 60% will show favorable response.
  • 21. Pg. 446: #17 Proof of Theorem If both pairs of opposite angles of a quadrilateral are equal in measure, then the quadrilateral is a parallelogram.
  • 22. Given: JKLM is a quadrilateral Prove: JKLM is a parallelogram
    • 1. JKLM is a quadrilateral
    • 2.
    • 3.
    • 4.
    • 5.
    • 6.
    • are supplementary angles
    • 8.
    • 9.
    • 10. are supplementary angles
    • 11.
    • 12. JKLM is a parallelogram
    • 1. Given
    • 2. Quadrilateral Sum Theor.
    • 3. Given
    • 4. Substitution
    • 5. Distributive Property
    • 6. Division Property of Equality
    • 7. Def. of Supplementary angles
    • 8. Converse of Co-Interior Th.
    • 9. Substitution
    • 10. Def. of Supplemental Angles
    • 11. Chuck Taylor Co-Interior Th.
    • 12. Def. of Parallelogram
    M J L K statements justifications
  • 23. Pg. 447: #34 Proof of Theorem An Exterior Angle of a Triangle is equal in measure to the sum of its two remote interior angles.
  • 24. Given: with exterior Prove:
    • 1. ABC is a triangle with exterior angle 4.
    • 2.
    • 3.
    • 4.
    • 5.
    • 6.
    • 7.
    • 1. Given
    • 2. Straight Angle Postulate
    • 3. Whole-Parts Postulate
    • 4. Substitution
    • 5. Triangle Sum Theorem
    • 6. Substitution Property
    • 7. Subtraction Prop. Of Equality
    C D A B 2 1 3 4 statements justifications
  • 25. Assignment: pg. 444-448: 14-18a, 26-34, 37-40

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