Midterm Review Project [NAME REMOVED]

December 9, 2014

3rd Period

Angle Pairs

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

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

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

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.

•

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.

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.

Example Problem 1

Example Problem 2

Practice Problem 1

Practice Problem 2

Practice Problem 3

Practice Problem 4

Practice Problem 5

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

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+conversev

http://www.wyzant.com/resources/lessons/math/geometry/lines_and_angles/angle_theorems

http://dictionary.reference.com/browse/parallel

Trapezoid and Triangle Mid-

segments

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

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

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

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.

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.

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.

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.

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.

Practice Problem 1

Practice Problem 2

Practice Problem 3

ll

l

lll

Practice Problem 4

Practice Problem 5

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).

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