Notice that we are frequently asked two types of questions about parallel lines. If we are told that two lines are parallel, then we may be required to use this information to prove the congruence of two angles (corresponding, alternate interior, etc.). On the other hand, if we are given the fact that two angles are congruent (or perhaps supplementary), we may have to prove that two lines are parallel.
In the figure, , , and . Prove that .
Mathematicians state that if a transversal to two parallel lines is perpendicular to one of the lines, then it is perpendicular to the other. Prove this statement. (Include a labeled drawing with your proof.)
Draw your drawing on paper, take a picture, and upload it using the image upload icon:
If you do not have the ability to upload an image of your work type "Drawing is on paper."
Given a line 𝑙 and a point 𝑃 not on the line, the following directions can be used to draw a line 𝑚 perpendicular to the line 𝑙 through the point 𝑃 based upon a rotation by 180°:
- Pick and label a point 𝐴 on the line 𝑙 so that the circle (center 𝑃, radius 𝐴𝑃) intersects 𝑙 twice.
- Use a protractor to draw a perpendicular line 𝑛 through the point 𝐴 (by constructing a 90° angle).
- Use the directions in Example 2 to construct a parallel line 𝑚 through the point 𝑃.
Do the construction. Why is the line 𝑚 perpendicular to the line 𝑙 in the figure you drew? Why is the line 𝑚 the only perpendicular line to 𝑙 through 𝑃?
Draw your construction on paper, take a picture, and upload it using the image upload icon:
If you do not have the ability to upload an image of your work type "Construction is on paper."