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