Angles

The answer is C. The measures of the interior angles in a triangle sum to $180°$. The CLT, usually on lower-difficulty questions, will ask you to subtract existing angles from $180$. Answer choice D might be tempting here because it’s true that we don’t have enough information to measure the other two angles of the triangle individually. But that doesn’t mean we can’t find them in combination. There are $180-55=125$ degrees left for the two angles in the triangle not yet known, so the sum of their measure must be $125°$.

The answer is A. We can solve this problem one of two ways. Solution 1: because we know $\angle BEF$ is a right angle (from the right triangle), we know line segment $\overline{BE}$ is perpendicular to $\overrightarrow{DG}$. This means that $\angle CBE$ is also a right angle, because if a certain line (let’s call it $l$) is perpendicular to another line (let’s call it $m$), then line $l$ must be perpendicular to all lines that are parallel to $m$. If $\angle CBE$ is a right angle, then we simply need to subtract $65°$ that $90°$ angle to get $25°$.

Solution 2: because $\overrightarrow{AC}$ and $\overrightarrow{DG}$ are parallel, $\angle CBF$ and $\angle BFE$ are congruent alternate interior angles. This means that the measure of $\angle BFE$ is also $65°$. Knowing that two angles of our triangle are $65°$ and $90°$, we can subtract them both from $180°$ to arrive at $25°$.

The answer is B. The UnCLES method reminds us to read every word carefully; we must not overlook that this an isosceles triangle. This means that not only two of the sides, but also two of the angles, must be congruent. This knowledge eliminates choice $A$ right away, even though it does make the angles sum to $180°$. Choices $B$ and $C$ do guarantee that two of the angles are the same, but choice $C$ is a trap answer tempting to those who mistakenly think that a triangle’s angles sum to $360°$ instead of $180°$.

Answer $B$ looks good, but do we have enough information to be sure this must be the case? Choice $D$ may be appealing if you’re thinking about the fact that we don’t necessarily know which two angles of an isosceles triangle are congruent. But there’s another reality operating here: a triangle can only have one obtuse angle at most, because two obtuse angles would by themselves sum to more than $180°$. So the fact that we’re already given an obtuse angle means that the other two angles must not only be acute but also have equal measures so the triangle is guaranteed to be isosceles. Though tempting, choice $D$ can be eliminated.

The answer is A. We begin, as it often the case, with $180°$ in a triangle. But since none of the three angles is given to us precisely, we’ll do well to set up multiple equations to modeling the situation. The first sentence tells that $A=5B$. (Remember the principle of “help” to confirm that the smaller angle, B, is the one that gets multiplied.) The second equation is more tricky: $C=2A-44$. (Make sure you note that the phrase “$x$ less than” translates to subtracting $x$ from the other term.)

We need to combine the equations to reduce the scenario to just one variable. Since $A$ is present in both equations, it is probably the best candidate. Solving the first equation in terms of A (isolating the other variable) leaves $B=A/5$. We already know what $C$ is in terms of $A$, so we can now use the $180°$ in the triangle to construct the following equation:

$A+A/5+2A-44=180$

Multiplying through by $5$ to get rid of the fraction gives us:

$5A+A+10A-220=900$

So:

$$\begin{array}{rl}16A-220& =900\\ 16A& =1120\\ A& =70\end{array}$$

To get angle $B$, the smallest angle, we can simply divide $70$ by $5$. Angle B is $14°$.

As is often true with challenging problems, there is an alternate strategy here that uses the answers and works backward. Let’s start with one of the middle answers, $24°$, and suppose that’s the measure of angle $B$. Angle $A$, being five times larger, must be $120°$. Angle $C$, which is $44$ less than twice angle $A$, comes out as $196°$. Clearly our numbers are too large for a triangle’s interior angles, which means the answer is $A$! How do we know this? Choice $A$ is the only answer smaller than $B$. Note this strategy and make sure your solutions toolbox includes this backsolving approach, which takes advantage of the fact that CLT answer choices are usually listed in numerical order.

The answer is C. There is a formula for this question, which we will use in a moment, but let’s consider how you might use geometric reasoning to answer this question if you didn’t know the formula. How could you draw diagonals inside a regular octagon to help you divide its angle into recognizable portions? If we keep all our diagonals either vertical or horizontal, we’ll recognize right angles and be able to infer the measures of the other angles. Consider this figure: