Question of the Week No.5 (ACT Math)

Consider the following figure:

In this figure, the lengths of segments: AD, BD, and BC are the same. In addition to this, the measure of angle ABD is 25 degrees. What is the degree measure of angle BCE?

A) 50 degrees

B) 100 degrees

C) 105 degrees

D) 130 degrees

E) 160 degrees

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Explanation: It might seem that a solution to this question is improbable given the small amount of information the question gives us. Most ACT questions are usually presented in this way, but, the key to solving questions like these, is approaching them like the building of a puzzle.

Our objective is to find the measure of angle BCE. So to do that, lets go ahead and analyze in further detail the triangle setup we have here. If you look closely enough, triangle ABC, is composed of two triangles, triangle ABD and triangle DBC. Herein, lies the key to answering this question.

Lets go ahead and consider triangle ABD. In triangle ABD, we know that the measurement of angle ABD is 25 degrees. BUT, additionally, we know that this triangle is also an ISCOCELES TRIANGLE, thus the measurement of angle BAD (lol) is also 25 degrees.

Remember that for ALL TRIANGLES, the sum of all the interior angles have to add up to 180 degrees. Thus, the angle ADB has a measurement of 130 degrees.

So now that we’ve fully analyzed triangle ABD, lets go ahead and bridge the gaps between this triangle and triangle DBC and we’ll get closer to calculating our mystery angle.

From here on out, it’s just easy sailing. The other angle that we can easily figure out from the red triangle is angle BDC. It might seem that we don’t have enough information but we actually do. If you think about the straight angle ADC, you know that it has a measurement of 180 degrees (shown in blue).

Thus, we know that 130 degrees + angle BDC has to equal 180 degrees. Therefore, angle BDC has a measurement of 50 degrees.

Additionally, we know that this triangle (triangle DBC), is an ISOSCELES TRIANGLE, and therefore, angle DCB is also 50 degrees.

Finally, when we consider the straight angle ACE (shown in green), we know that it has a measurement of 180 degrees. So, if you want to find the mystery angle BCE, all you have to solve is the following equation:

50 degrees + Angle BCE (mystery angle) = 180 degrees.

Therefore, the measurement of angle BCE is 130 degrees and the answer is Option D.