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What is 𝑚∠𝑅𝑀𝑆 in the following figure?

What is the measure of angle 𝑅𝑀𝑆 in the following figure?

In this question, we need to calculate the measure of angle 𝑅𝑀𝑆 as shown in the diagram. To do this, we will recall a number of angle properties and relationships. Firstly, we note that 𝑄𝑇 is a straight line, and we recall that angles on a straight line sum to 180 degrees. This means that the measure of angle 𝑄𝑀𝑃 plus the measure of angle 𝑇𝑀𝑃 must equal 180 degrees. From the diagram, we see that angle 𝑇𝑀𝑃 measures 146 degrees. And subtracting this from both sides of our equation, we have the measure of angle 𝑄𝑀𝑃 is 34 degrees.

Next, we recall that vertically opposite angles are equal. This means that the measure of angle 𝑆𝑀𝑇 must be equal to the measure of angle 𝑄𝑀𝑃. And we already know this is equal to 34 degrees. We now have the measures of four of the five angles on the diagram. We will therefore use one final property. Angles at a point sum to 360 degrees. The four known angles sum to 304 degrees. And this means that the measure of angle 𝑅𝑀𝑆 plus 304 degrees is equal to 360 degrees. Subtracting 304 degrees from both sides, we have the measure of angle 𝑅𝑀𝑆 is equal to 56 degrees. And this is the final answer to this question.

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