Line CD intersects with segment AB at E so that the measure of AE equals the measure of EB and angle CEB is a right angle.

Question

Kyle is creating a frame for a model car. He begins by piecing two rods together, as shown in the diagram. Justify why ∠AEC is a right angle.
Definition of parallel lines
Definition of perpendicular bisector
Definition of complementary angles
Definition of adjacent angles

Answer 1

To justify why ∠AEC is a right angle, we can use the information provided about the intersection of line CD with segment AB at point E.

Given that AE equals EB, we deduce that point E is the midpoint of segment AB. According to the definition of a perpendicular bisector, line CD not only bisects AB at E but is also perpendicular to it, since angle CEB is a right angle.

From this, we can make the following conclusions:

1. Since angle CEB is a right angle, it measures 90 degrees.
2. The line segment CE extends perpendicularly from line segment AB (because of the right angle at E), which implies that line CD must act as a perpendicular bisector of segment AB.

Now, by the definition of perpendicular lines, if a line is perpendicular to another line at a point, then it forms right angles with that line at that point.

Therefore, ∠AEC, which is formed by line AE and line CD (which is perpendicular to AB at E), must also be a right angle.

In conclusion, the justification for why ∠AEC is a right angle relies on the understanding that CD is the perpendicular bisector of AB at E, establishing that ∠AEC = 90 degrees.