1) If a bike trail intersects at a street intersection, and there are two parallel roads going across, it creats a transversal. Using this transversal, Tommy wanted to prove that the angle of the exterior side of one road was equal to the angle of the exterior side of another road since the roads are parallel. Given that one of the angles is 120 degrees prove that the other angle is equal. Draw a 2 column proof and fill it out.
Since that angle is 120, your trying to prove that the other angle is also 120 degrees. Using the vertical angles theorem, you can prove that angle a is also 120 degrees. Then using corresponding angles, you can see that the unknown angle is also 120 since corr. <'s are congruent. Using transitive property, then you can use an equality to equalize the two values.
(There are multiple ways to complete this problem.)
Final Answer~ in two column proof
2) If Ms. Ashmore's room was a coordinate grid, then all the students were seated in desks. Tommy's seat was (3,3) and Elise's seat is at (-1,2). Two other students' desks were at (1,2) and (-3,3). The quadrilateral that this creates was reflected over the y axis and was rotated counterclockwise around the origin. What are the new coordinates of the four kids' seats? (seats/desks interchangeable)
This problem utilizes composition of transformations. A reflection over the y- axis means that the x coordinate becomes negative. For the coordinates,
T'-> (-3,3) E'-> (1,2) A' -> (-1,2) B'-> (3,3)
A counterclockwise 90 degree rotation would mean the mapping would be (x,y)-. (-y,x). This is because all rotations without other instruction are counterclockwise. In addition, it rotates from the origin, meaning flipping the coordinate distance to the origin.
T"-> (-3,-3) E"-> (-2,1) A" -> (-2,-1) B"-> (-3,3) This would be the final answer- the coordinates of the desks.
This is helpful because of the multistep process. It shows a combination of transformations.
Vinny Shirvaikar's Practice Problems: 1. Question- In the city of Plano, Texas, there is a place where there are two exactly-parallel roads. Coincidentally, another road cuts across both of them, making it a transversal. All Johnny knows is that Angle 1 (please refer to the picture at the bottom of the problem) is 47.6 degrees. He wants to know the degrees of all the 7 other angles for a school project. Help him by listing out the measures of each angle and writing the theorem or postulate you used to get there. (Picture not to scale)
Explanation for Each Angle- Angle 1: The problem gives you this information. Angle 2: Angles 1 and 2 form a linear pair, because they make a straight line. Therefore, their measures are supplementary. Since supplementary angles add up to 180 degrees, and 180-47.6 is 132.4, Angle 2 measures 132.4 degrees. Angle 3: This is a more simple one. Vertical angles are congruent, and Angles 1 and 3 are vertical, so Angles 1 and 3 are congruent, or measure the same. Angle 4: The logic for getting this answer is that same as the logic for Angle 2. Angle 5: By definition, Angle 1 is corresponding with Angle 5. The Corresponding Angles Theorem says that corresponding angles are congruent. So, Angles 1 and 5 measure the same, 47.6 degrees. Angle 6: Angles 4 and 6 are alternate interior angles, so, by the Alternate Interior Angles Theorem, they are congruent. That means they have the same measure, so Angle 6 equals 132.4 degrees. Angle 7: Angle 1's measure is given in the problem. It is 47.6 degrees. Angle 7 is its alternate exterior angle, and by the Alternate Exterior Angles Theorem, those are congruent. So, Angle 7 is 47.6 degrees. Angle 8: According to the Linear Pair Theorem, linear pairs are supplementary. Since Angles 7 and 8 are a linear pair, they are supplementary, and since Angle 7 is 47.6 degrees, Angle 8 is 132.4 degrees.
2. Question- Three picnics are being held at a park with 100 evenly-sized squares of land. The park is just like a coordinate grid with every quadrant being 5x5. The first picnic is held at the square (2,3). The second is held at (-4,1), and the last is at (5,-5). As you could already tell, the park owner loves squares, so he told all the picnics to rotate 90 degrees about the park's origin. Which squares (as coordinate points) did each of the picnics end up in?
Answer- First picnic: (-3,2) Second picnic: (-1,-4) Third picnic: (5,5)
Explanation- The mapping for a 90 degree rotation is (x,y) to (-y,x). Since it doesn't specifically state otherwise, the rotation is counter-clockwise, meaning the mapping is still (-y,x). So, for the first picnic, (2,3) goes to (-3,2). For the second picnic, it goes from (-4,1) to (-1,-4), and the last picnic goes from (5,-5) to (-5,-5).