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Grade 8
Domain Geometry
Cluster Understand congruence and similarity using physical models, transparencies, or geometry software.
Standard Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. <span class='clarification'>For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.</span>
Task Street Intersections

Street Intersections


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Alignments to Content Standards: 8.G.A.5
Student View

Task

Market Street runs parallel to Main Street and both are intersected by 5th Avenue as shown below: 

 

If a car traveling northeast on Market Street turns right to go east on 5th Avenue, it turns (clockwise) through a 35 degree angle as indicated in the picture.

  1. Suppose a car is traveling southwest on Market Street and turns left to go east on 5th Avenue. Draw the angle of turn in the picture. What is the measure of this angle? Explain how you know.

  2. Suppose a car is traveling southwest on Main Street and turns right onto 5th avenue.  Draw the angle of turn in the picture. What is the measure of this angle? Explain using rigid motions. 

  3. A car makes a 35º angle of turn going through the intersection of Main Street and 5th Avenue. Assuming that the car is following these two roads, what can you conclude about the car's route through this intersection? Explain.

 

IM Commentary

The purpose of this task is to apply facts about angles (including congruence of vertical angles and alternate interior angles for parallel lines cut by a transverse) in order to calculate angle measures in the context of a map. This activity helps students visualize vertical, supplementary, and alternate interior angles in terms of a set of street intersections. The teacher may wish to verify that students understand the given information  "If a car traveling northeast on Market Street turns right to go east on 5th Avenue, it turns (clockwise) through a 35 degree angle" before proceeding with work on the task.

It is important for students to understand, in part (b), that the goal is not to apply congruence of alternate interior angles for parallel lines cut by a transverse: rather it is to explain why this is true, in this particular setting. There are many ways to do this. The argument provided in the solution combines a rotation with a translation. One alternative would be to apply a 180 degree rotation about the midpoint between the two street intersections. This map will exchange the given 35 degree angle with the angle given in part (b).

There are many ways the teacher can support students as they work through this task. Students can put themselves into the driver’s seat to help with visualizing. Students might use a small toy car to physically “drive” the routes in the prompts.  It may also be useful to have a compass on hand so students can draw the arcs between the intersections to display how the front end of the car will rotate.  Another suggestion is to create this map on the floor of the classroom using masking tape and have students physically execute the movements in the prompts.  This representation may also help with classroom discussion. It could also be helpful to provide students with patty paper which they can use, along with the supplied map, to execute the rigid motions for part (b). 

Solution

  1. Market Street is a straight line and the angle of turn for the southwest going car is supplementary to the angle of turn for the northeast going car as seen in the picture below:

    We are given that the angle of turn for the northeast traveling car is 35 degrees. We know 35 + 145 = 180 so the car traveling southwest turning left to go east on 5th Avenue turns through a 145 degree angle.

  2. The angle the car turns through, traveling southwest on Main Street and turning right onto 5th avenue, is pictured below:


    To explain why this angle measures 35 degrees, we can apply a 180 degree rotation about the intersection of Market Street and 5th Avenue (shown by the green arrow in the picture below). This maps Market Street to itself and maps the 35 degree angle made by its intersection with 5th avenue to the congruent 35 degree angle marked in the picture below:


    Because Market Street is parallel to Main Street, we can apply a translation (indicated by the arrow in the picture above) to move the intersection of Market Street and 5th Avenue to the intersection of Main Street and 5th Avenue. Since the streets are parallel, this moves Market Street on top of Main Street and maps 5th Avenue to itself. The 180 degree rotation followed by the translation show that the southwest traveling car on Main street makes a 35 degree an angle turn to go west on 5th Avenue.

  3. We have just seen in part (b) that the angle of turn for a car traveling southwest on Main Street turning to go west onto 5th Avenue is 35 degrees. A car traveling west on 5th Avenue turning to go southwest on Main Street also makes a 35 angle of turn (this is the same angle as in the previous case: the car is turning clockwise in the first case and counterclockwise in the second). Similarly a car traveling northeast on Main Street turning to go east on 5th Avenue turns (clockwise) through a 35 degree angle while a car traveling east on 5th avenue and turning to go northeast on Main street turns (counterclockwise) through a 35 degree angle.

Street Intersections

Market Street runs parallel to Main Street and both are intersected by 5th Avenue as shown below: 

 

If a car traveling northeast on Market Street turns right to go east on 5th Avenue, it turns (clockwise) through a 35 degree angle as indicated in the picture.

  1. Suppose a car is traveling southwest on Market Street and turns left to go east on 5th Avenue. Draw the angle of turn in the picture. What is the measure of this angle? Explain how you know.

  2. Suppose a car is traveling southwest on Main Street and turns right onto 5th avenue.  Draw the angle of turn in the picture. What is the measure of this angle? Explain using rigid motions. 

  3. A car makes a 35º angle of turn going through the intersection of Main Street and 5th Avenue. Assuming that the car is following these two roads, what can you conclude about the car's route through this intersection? Explain.

 

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