1. Midterm Review Project [NAME REMOVED] December 9, 2014 3 rd Period

2. Angle Pairs

3. Definitions Parallel: extending the same direction, equal distance at all points, never converging or diverging Transversal the line that crosses two or more other lines

4. Definitions Linear Pair Angles: two adjacent angles that share a leg and are supplementary. Vertical Angles: each pair of opposite angles created by two intersecting lines Corresponding Angles: two lines that are crossed with a transversal line, the angles that are located in the same spot at different intersections

5. Definitions Alternate Exterior Angles: two exterior angles, located on opposite sides of the transversal Alternate Interior Angles: two interior angles, located on opposite sides of the transversal Consecutive Interior Angles: two interior angles located on the same side of the transversal: supplementary

6. Angle Pair Theorems If a transversal line intersects a pair of linear angle pairs, then the pair of angles are supplementary. If a transversal line intersects a pair of vertical angles, then the pair of vertical angles are congruent. If a transversal line intersects a pair of corresponding angles, then the corresponding angles are congruent.

7. Angle Pair Theorems If a transversal line intersects a pair of alternate exterior angles, then the alternate exterior angles are congruent. If a transversal line intersects a pair of alternate interior angles, then the alternate interior angles are congruent. If a transversal line intersects a pair of consecutive interior angles, then the consecutive interior angles are supplementary.

8. Tips and Instructions Corresponding angles RELATE. They are the same angle, translated to a different location. Consecutive interior angles are always on the same side of the transversal line and always inside the parallel lines. Alternate exterior angles are always on the opposite sides of the transversal line, and always on the outside of the parallel lines. Alternate interior angles are always on the opposite sides of the transversal line, and always on the inside of the parallel lines.

9. Example Problem 1

10. Example Problem 2

11. Practice Problem 1

12. Practice Problem 2

13. Practice Problem 3

14. Practice Problem 4

15. Practice Problem 5

16. Solutions for Angle Pairs Practice Problem 1: angle 3 & angle 5; angle 4 & angle 6 Practice Problem 2: 6 & 2; 5 & 1; 8 & 4; 7 & 3 second picture: same as the first Practice Problem 3: An alternate Exterior Angle is when two angles are on the opposite sides of the transversal line, outside of the parallel lines; 2 are shown; 3 & 8, 2 & 7 Practice Problem 4: An alternate Interior Angle Pair is when the two angles are on the opposite sides of the transversal line, but inside the parallel lines. Practice Problem 5: A= Linear pair, B= Vertical pair, C= Alternate Exterior pair, D= Alternate Interior pair, E= Consecutive Interior pair, F= Corresponding pair

17. Sources http://twt.wm.edu/vavocab/printdefs.php?ws=91 http://www.mathopenref.com/linearpair.html http://www.freemathhelp.com/vertical-angles.html http://www.mathopenref.com/anglescorresponding.html http://stageometrych3.wikispaces.com/alternate+interior+and+alternate+exterior+angles+co nversev http://www.wyzant.com/resources/lessons/math/geometry/lines_and_angles/angle_theorem s http://dictionary.reference.com/browse/parallel

18. Trapezoid and Triangle Mid-segments

19. Definitions Parallel: extending the same direction, equal distance at all points, never converging or diverging Mid-segment a line joining the midpoints of two sides

20. Trapezoid Mid-segment Theorems The mid-segment of a trapezoid is parallel to the bases of the trapezoid The length of the mid-segment of a trapezoid is equal to the average of the lengths of the bases X= (a+b)/ 2

21. Triangle Mid-segment Theorems A mid-segment of a triangle is parallel to the third side of the triangle A mid-segment of a triangle is half the length of the third side (side it is parallel to or not touching) The three mid-segments of a triangle divide in the triangle into four congruent triangles

22. Tips and Instructions Remember that the midpoint of each mid-segment is a “half- way” point between point A and B. This means that if point D is your midpoint, line AD would be congruent to line DB. These lines, line AD and DB, are still parallel and congruent to the mid-segment inside of the outer triangle. So AD is the same length as line EF. There are always three possible mid-segments.

23. Tips and Instructions To find the length of the mid-segment, you have to remember to find the average of the two bases. There is only one mid-segment in a trapezoid. This mid-segment is parallel to both the bases.

24. Tips and Instructions The distance between the mid-segment and one of the bases is the same as the distance between the mid-segment and the other base. The distance between line AM is the same length as the distance from mid-segment MN to line BC.

25. Example Problem 1 Explanation: Because of the Trapezoid Mid-segment Theorem, in order to find the mid-segment’s lengths, you must, as shown above, add the two bases together, then divide by two, creating an average and solving x.

26. Example Problem 2 Explanation: Each triangle mid-segment is half of its opposite, parallel side. Each point cuts the outside triangle line in half (the mid-point), so each half created by the mid-point are congruent. So when you add all of the lengths of the “inside” triangle, triangle DEF, you get 15.

27. Practice Problem 1

28. Practice Problem 2

29. Practice Problem 3 ll l l

30. Practice Problem 4

31. Practice Problem 5

32. Solutions Practice Problem 1: x= 17.25; 12 + 22.5 = 34.5, then you have to divide by 2, equaling 17.25 Practice Problem 2: first triangle= 2.5; second triangle= 1.5 Practice Problem 3: a= 16 because the distance between the mid-segment and each base is the same, so 20-18 is 2, you would subtract 2 form 18= 16. Practice Problem 4: 19.5in Practice Problem 5: Trapezoid Mid-segment: The mid-segment is parallel to the bases (one example). Triangle Mid-segment: The mid-segment is half of the third side, or the side is parallel to (one example).

33. Sources http://dictionary.reference.com/browse/midsegment http://hotmath.com/hotmath_help/topics/midsegment-of-a-trapezoid.html http://www.ck12.org/book/CK-12-Geometry-Second-Edition/r4/section/5.1/ http://www.ck12.org/geometry/Trapezoids/lesson/Trapezoids-Intermediate/ http://www.regentsprep.org/regents/math/geometry/gp10/midlinel.htm http://quizlet.com/23999592/geometry-regents-review-flash-cards/ http://pixgood.com/define-midsegment-of-a-trapezoid.html