In the figure below, line AD is parallel to line BE, points
A, C, and D are collinear, and segment BC is congruent
to segment CE. If the measure of angle ABE is 90° and
the measure of angle BCE is 30°, what is the measure of
angle DCE?
Figure
F.
65°
G.
70°
H.
75°
J.
80°
K.
85°
Let's analyze the given information.
We are given that line AD is parallel to line BE, which means angles ABE and BDE are corresponding angles and hence congruent. Since angle ABE is given as 90°, angle BDE is also 90°.
We are also given that segment BC is congruent to segment CE. This implies that angles BCE and CBE are congruent. Since angle BCE is given as 30°, angle CBE is also 30°.
Now, let's consider triangle CDE. Angles BDE and CBE are congruent, so their measures add up to 180° - 90° - 30° = 60°. Since angles BCE and CBE are congruent, angle CBE must be (180° - 60°)/2 = 60°/2 = 30°.
Finally, angle DCE is equal to 180° - angle CDE - angle CBE = 180° - 60° - 30° = 90°.
Therefore, the measure of angle DCE is 90°.
The answer is not provided in the given options.