Activity 1: proof
a busy, two-lane split highway crosses another busy split highway. the northbound lanes are divided from the southbound lanes by a borrow pit. to maintain a direct line of traffic flow, the lines of traffic must be parallel. to verify this, you need to prove that in the following diagram the lines of traffic are parallel.
given:
prove:
is parallel to
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activity 2: precision
in construction, the precision used to build a structure directly influences the strength of the structure. in the rocky mountains, many houses’ roofs are built using trusses, which are strong enough to support the heavy load from snow, but light enough to build in a stockyard and transport to the construction site. chris works in the stockyard. he has been told he needs to build a series of trusses for a new house using the pattern below. he has also been told the pattern may be inaccurate—so he needs to check and make sure the following features are all parallel, and adjust the pattern before starting.
describe the angles you would need to know to verify that , , and are all parallel to each other, and , , and are also parallel.
what could be an appropriate measure for ? be prepared to explain your choice for that angle. use that measure to find all other angles involved.
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come to the exploration meeting prepared to discuss the following:
what method(s) can chris use to make sure the segments in the pattern are all parallel and that his new trusses are built precisely and strongly?
what method(s) can chris’s boss use to verify the precision of the trusses chris builds?
if you lived in a mountainous area with a lot of snow, how much leeway would you feel comfortable giving your home’s builders as they design and build your trusses? why?
identify any alternate interior angles, corresponding angles, and/or alternate exterior angles.
exercise 3: railroads
railroad tracks’ rails must be a precise distance from each other. if the distance between the rails is too narrow or too wide, the train will derail. come prepared to discuss the following questions:
imagine traveling back to 1865 when railroads were being built all across the american frontier. how do you think they built the tracks to be the right width apart from each other? what could have gone wrong? could you teach the railroad workers about parallelism (alternate interior and exterior angles, etc.) so they could always build tracks the right width apart?
do you think it would be practical to use principles of parallelism for building railroads during that time period? why or why not?
what other methods could be used to ensure parallel rails (any method, not just the geometry kinds)?