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, 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 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.

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 transversals, and calculate the ratios again. Based on the new ratios, what can you conclude about three 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 throughout the proof.

As part of the proof, you’ll have to construct a line segment connecting the top intersection point on one of the transversals with the lowest intersection point of the other transversal, thus forming two triangles. Take a screenshot of the construction, save it, and insert the image in the space below before you begin your written proof.

Question 2
Now you’ll look at how an angle bisector of a triangle divides the side opposite to the angle. Reopen GeoGebra, and complete each step below.

Part A
Construct a random triangle, , and construct the angle bisector of . The angle bisector divides the opposite side, , into two line segments. Mark the point where the angle bisector intersects , and label it D. To verify that you have bisected the angle, measure and and display the angle measures. Take a screenshot of your construction, save it, and insert the image below.

Part B
Record the measurements for line segments of AB, BC, AD, and DC .

Part C
Calculate the ratio of AB to BC and the ratio of AD to DC. Round your answers to the hundredths place. What do you notice about the ratios?

Part D
Based on the ratios in part C, what can you conclude about the angle bisector?

Part E
Now write your conclusion in part D as a theorem, and prove the theorem using a two-column proof. When you write the proof, refer to the diagram you created in part A. As part of the proof, you will have to construct a line through point A parallel to . Mark the point where the new line intersects the angle bisector, , and label the point E. Take a screenshot of the construction, save it, and insert the image in the space below before you begin your written proof.