PSAT Math Multiple-Choice Question 114: Answer and Explanation

Question: 114

In the figure above, the ratio has the same value as the ratio . Which of the following angle measures must be congruent?

• A. ∠FGH and ∠FHJ
• B. ∠FJG and ∠FJH
• C. ∠GFJ and ∠HFJ
• D. ∠GFH and ∠FJH

Correct Answer: D

Explanation:

D

The question asks for a pair of congruent angles in a figure containing triangles. When given two or more triangles and information about the ratios of the sides, look for similar triangles. Sides FJ and FH are both on the smallest triangle, while FG and GH are both on the largest triangle. Redraw the two triangles side-by-side to better see the similarities. Start with triangle FHJ, redrawing it by itself with the same orientation it currently has, labeling the vertices. Then draw a bigger triangle next to it with the same shape as triangle FHJ, like this:

Since the ratio of is equal to , label the sides of this new triangle correspondingly. Make sure to keep ∠FGH as the smaller angle and ∠GFH as the larger one, like this:

Now it is easier to check corresponding angles. Choice (A) gives the angles on the lower left of the big triangle and the lower right of the small triangle. These are not necessarily congruent, so eliminate (A). Choice (B) gives an angle that does not appear on either triangle and the angle at the top of the small triangle. These are not necessarily congruent, so eliminate (B). Choice (C) gives an angle that does not appear on either triangle and the angle on the lower left of the small triangle. These are not necessarily congruent, so eliminate (C). Choice (D) gives the angles at the tops of both triangles, so these angles are congruent. The correct answer is (D).