8 2 Triangle Sum Theorem

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

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8 2 Triangle Sum Theorem

  1. 1. Triangle Sum Theorem Focus - Prove the Triangle Sum Theorem
  2. 2. If you add up the measures of the interior angles of a triangle, What is the total? <ul><li>180º </li></ul>25º 65º
  3. 3. How many lines can be drawn through point B that would become parallel to M B
  4. 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. 5. Given: Prove: <ul><li>1. ABC is a triangle </li></ul><ul><li>2. Draw helping line </li></ul><ul><li>through B so that </li></ul><ul><li>3. </li></ul><ul><li>4. </li></ul><ul><li>5. </li></ul><ul><li>6. </li></ul><ul><li>7. </li></ul><ul><li>1. Given </li></ul><ul><li>2. Unique Parallel Postulate (pg. 439) </li></ul><ul><li>3. If the sides of an angle form a straight line, then the angle is a straight angle with a measure of 180º </li></ul><ul><li>4. Whole-Parts Postulate </li></ul><ul><li>5. Substitution </li></ul><ul><li>6. Alternate Interior Angle Theorem </li></ul><ul><li>7. Substitution Property </li></ul>B C A 2 1 3 D E 4 5
  6. 6. Quadrilateral Theorem <ul><li>The sum of the measures of the angles of a quadrilateral are: </li></ul><ul><li>What is the sum of the measures of a rectangle, which is a quadrilateral? </li></ul>
  7. 7. Theorem: Opposite Angles of Quadrilaterals <ul><li>If both pairs of opposite angles of a quadrilateral are equal in measure, … </li></ul><ul><li>then the quadrilateral is a parallelogram. </li></ul>
  8. 8. What can we find out? <ul><li>The ladder is leaning on the ground at a 75º angle. At what angle is the top of the ladder touching the building? </li></ul>
  9. 9. What can we find out? <ul><li>The ladder is leaning on the ground at a 75º angle. At what angle is the top of the ladder touching the building? </li></ul>
  10. 10. Find the missing angle. <ul><li>47º </li></ul>
  11. 11. Find the missing angle. <ul><li>47º </li></ul>
  12. 12. Find the missing angles. <ul><li>120º </li></ul><ul><li>x º </li></ul><ul><li>55º 25º yº wº </li></ul>Order: y w x
  13. 13. Exterior Angle Theorem <ul><li>An exterior angle of a triangle… </li></ul><ul><li>… is equal in measure to the sum of the measures of its two remote interior angles . </li></ul>
  14. 14. Find the missing angles’ measurements (L and J). <ul><li>120º </li></ul><ul><li>55º 25º </li></ul>J L remote interior angles
  15. 15. What is the measure of Angle U? <ul><li>120º </li></ul><ul><li>55º 25º </li></ul>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. 16. In today’s lesson we… <ul><li>Proved the Triangle Sum Theorem (180 degrees) </li></ul><ul><li>Found Quadrilaterals’ Interior Angles Theorem to state the total degrees is 360. </li></ul><ul><li>Used the above theorems and the Exterior Angle Theorem to find measurements of a Triangle’s exterior angle. </li></ul>
  17. 17. Assignment: pg. 443: 2-9, 13
  18. 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. 19. From tdn.com last night. <ul><li>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. </li></ul><ul><li>The measure would eliminate the requirement to keep taking the math Washington Assessment of Student Learning every year until graduation. </li></ul><ul><li>High school students who fail the 10th grade math test, would still be required to keep earning math credits, however. </li></ul><ul><li>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. </li></ul><ul><li>The bill is Senate Bill 5498. </li></ul><ul><li>* Legislature (weblink) < http:// www.leg.wa.gov > </li></ul>
  20. 20. Want to spend class time preparing for math WASL questions? <ul><li>Group Work </li></ul><ul><li>Group Points </li></ul><ul><li>Three sections will vote. </li></ul><ul><li>60% will show favorable response. </li></ul>
  21. 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. 22. Given: JKLM is a quadrilateral Prove: JKLM is a parallelogram <ul><li>1. JKLM is a quadrilateral </li></ul><ul><li>2. </li></ul><ul><li>3. </li></ul><ul><li>4. </li></ul><ul><li>5. </li></ul><ul><li>6. </li></ul><ul><li>are supplementary angles </li></ul><ul><li>8. </li></ul><ul><li>9. </li></ul><ul><li>10. are supplementary angles </li></ul><ul><li>11. </li></ul><ul><li>12. JKLM is a parallelogram </li></ul><ul><li>1. Given </li></ul><ul><li>2. Quadrilateral Sum Theor. </li></ul><ul><li>3. Given </li></ul><ul><li>4. Substitution </li></ul><ul><li>5. Distributive Property </li></ul><ul><li>6. Division Property of Equality </li></ul><ul><li>7. Def. of Supplementary angles </li></ul><ul><li>8. Converse of Co-Interior Th. </li></ul><ul><li>9. Substitution </li></ul><ul><li>10. Def. of Supplemental Angles </li></ul><ul><li>11. Chuck Taylor Co-Interior Th. </li></ul><ul><li>12. Def. of Parallelogram </li></ul>M J L K statements justifications
  23. 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. 24. Given: with exterior Prove: <ul><li>1. ABC is a triangle with exterior angle 4. </li></ul><ul><li>2. </li></ul><ul><li>3. </li></ul><ul><li>4. </li></ul><ul><li>5. </li></ul><ul><li>6. </li></ul><ul><li>7. </li></ul><ul><li>1. Given </li></ul><ul><li>2. Straight Angle Postulate </li></ul><ul><li>3. Whole-Parts Postulate </li></ul><ul><li>4. Substitution </li></ul><ul><li>5. Triangle Sum Theorem </li></ul><ul><li>6. Substitution Property </li></ul><ul><li>7. Subtraction Prop. Of Equality </li></ul>C D A B 2 1 3 4 statements justifications
  25. 25. Assignment: pg. 444-448: 14-18a, 26-34, 37-40

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