8 2 Triangle Sum Theorem

Triangle Sum Theorem Focus - Prove the Triangle Sum Theorem

If you add up the measures of the interior angles of a triangle, What is the total? 180º 25º 65º

How many lines can be drawn through point B that would become parallel to M B

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.

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

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?

Theorem: Opposite Angles of Quadrilaterals If both pairs of opposite angles of a quadrilateral are equal in measure, … then the quadrilateral is a parallelogram.

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?

Find the missing angles. 120º x º 55º 25º yº wº Order: y w x

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 .

Find the missing angles’ measurements (L and J). 120º 55º 25º J L remote interior angles

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.

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.

For Warmup 70° y° z° w° D E 50° x° (x-30)° x° 70° 60° 50° 110° D E 50° 70° (80-30)° 80°

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 >

Want to spend class time preparing for math WASL questions? Group Work Group Points Three sections will vote. 60% will show favorable response.

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.

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

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.

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

Assignment: pg. 444-448: 14-18a, 26-34, 37-40

8 2 Triangle Sum Theorem

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8 2 Triangle Sum Theorem