In this unit, you proved this theorem: If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides proportionally
For this task, you will first investigate and prove a corollary of the theorem above. (A corollary is a theorem that is proved easily from
another theorem.) Then you will investigate and prove a theorem about angle bisectors.
Question 1
To understand the corollary, consider what happens when three (or more) parallel lines intersect two transversals. You will use
the GeoGebra geometry tool to investigate how multiple parallel lines divide transversals. Open GeoGebra C, and complete
each step below. If you need help, follow these instructions for using GeoGebra.
Part A
Construct three parallel lines. Then construct two arbitrary nonparallel transversals of the parallel lines. Make sure that the
transversals cut the three parallel lines at distinct points. Mark the points of intersection where the transversals cut the parallel
lines. Take a screenshot of your construction, save it, and insert the image below.
Part B
Notice that the two line segments are formed on each transversal between the central parallel line and the outer parallel lines. Measure the lengths of the four line segments
Line SegmentLength
AB/?
BA/?
AB/?
BA/?
Part C
Calculate the ratio of the lengths of the two line segments formed on each transversal. You will have two sets of calculations. Round your answers to the hundredths place. What do you notice about the ratios of the lengths for each transversal. How do they compare?
Part D
Change the orientation of the transversal, and calculate the ratios again. Based on the new ratios, what can you conclude about 3 or more parallel lines that intersect two transversals?
Part E
State your conclusion in the form of a theorem, and then prove the theorem using a two-column proof. When you write the proof, refer to the diagram you created in part A. It will be helpful to use point labels to state what is given and what you have to prove and to use those labels through the proof. As part of the proof, you’ll have to construct a line segment connecting the top intersection point on one of the transversal‘s with the lowest intersection point of the other transversal that’s forming two triangles. Take a screenshot of the construction and insert the image in the space below before writing your proof. Prove:?
Given:?
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