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# Which of the following sides are equal if angles RPQ and RQP are similar?A). RP$=$RQB). RP$=$PQC). RQ$=$PQD). None of these

Last updated date: 21st Jul 2024
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Hint: We need to know some properties of a triangle before we solve this problem. An isosceles triangle is one of the types in a triangle. An isosceles triangle is a triangle for which two of the angles are similar and their arms (other than common arm) are equal.

Complete step-by-step solution:
It is given that the angles RPQ and RQP are similar. Hence, it is an isosceles triangle.
From the definition of an isosceles triangle, we know that two of its angles are similar and their arms (other than common arm) are equal.

Consider that the above triangle is isosceles triangle then the marked angles are similar and the marked sides are equal.
Likewise let us see the given diagram and mark its similar angle and equal sides. Since, it is given that angles RPQ and RQP are similar then the sides RP and RQ are equal.

Thus, we got RP $=$RQ.
Let us see the options, option (a) RP$=$RQ, cannot be the correct answer since we got RP $=$RQ.
Option (b) RP$=$PQ, cannot be the correct answer since we got RP $=$RQ.
Option (c) RQ$=$PQ, cannot be the correct answer since we got, RP $=$RQ.
Option (d) None of the above, cannot be the correct answer since we got, RP $=$RQ.
Hence, option (a) RP$=$PQ, is the correct option.

Note: Thus, we need to know the properties of the isosceles triangle to solve this problem. Since, in the question, it is not given directly that the given triangle is an isosceles triangle. It is given that two of its angles are similar, with that we found that it is an isosceles triangle with the help of definition of isosceles triangle.