In triangles MNO and PQR, angles M and P each have measure 60 ∘ , MN=15, and PQ=45. Which additional piece of information is sufficient to prove that triangle MNO is similar to triangle PQR?AMO=9 and PR=27BMO=9 and PR=36CThe measure of angles N and Q are 120 ∘ and 120 ∘ , respectively.DThe measure of angles O and R are 30 ∘ and 60 ∘ , respectively.
Question
In triangles MNO and PQR, angles M and P each have measure 60 ∘ , MN=15, and PQ=45. Which additional piece of information is sufficient to prove that triangle MNO is similar to triangle PQR?AMO=9 and PR=27BMO=9 and PR=36CThe measure of angles N and Q are 120 ∘ and 120 ∘ , respectively.DThe measure of angles O and R are 30 ∘ and 60 ∘ , respectively.
Solution
To prove that two triangles are similar, we need to show that their corresponding angles are equal and their corresponding sides are proportional.
We already know that angles M and P are equal (60 degrees each).
Let's examine each option:
A) MO=9 and PR=27. This would mean that the sides MO and PR are in the same ratio as MN and PQ (15:45 = 1:3 and 9:27 = 1:3). This would be sufficient to prove similarity.
B) MO=9 and PR=36. This would mean that the sides MO and PR are not in the same ratio as MN and PQ (15:45 = 1:3 and 9:36 = 1:4). This would not be sufficient to prove similarity.
C) The measure of angles N and Q are 120 degrees each. This would mean that another pair of corresponding angles are equal, which is a good start, but we still need to show that the sides are proportional. Without this information, we cannot prove similarity.
D) The measure of angles O and R are 30 degrees and 60 degrees, respectively. This would mean that these angles are not equal, so the triangles cannot be similar.
Therefore, the additional piece of information that is sufficient to prove that triangle MNO is similar to triangle PQR is option A: MO=9 and PR=27.
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