The document discusses several geometry theorems including: 1. The Triangle Sum Theorem which states that the sum of the interior angles of any triangle is 180 degrees. 2. The theorem that the sum of the interior angles of any quadrilateral is 360 degrees. 3. The Exterior Angle Theorem which states that an exterior angle of a triangle is equal to the sum of the two remote interior angles. The document provides examples of using these theorems to prove other statements and find missing angle measures in geometric figures. It also assigns practice problems for students.

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°

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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