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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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### Transcript

• 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