Question

Suppose Q is in the interior of ∠ROS, where S is in the interior of ∠QOP, and P is in the interior of ∠SOT.

Given m∠ROT=127°, m∠SOT=71°, and m∠ROQ=m∠QOS=m∠POT, find m∠QOP.

Answer 1

By using the given angle measurements and the properties of angles, the measure of ∠QOP is found to be 127°.

Step-by-step explanation:

To find m∠QOP, we need to use the given information to solve the angles step by step, keeping in mind that the sum of angles around a point is 360° and the sum of linear pair of angles is 180°. The sum m∠ROQ + m∠QOS + m∠SOT should equal m∠ROT, since Q, S, and T are interior points on adjacent angles leading to point R.

Since m∠ROQ = m∠QOS = m∠POT, let's denote the magnitude of these angles as x. We already know that m∠ROT = 127° and m∠SOT = 71°, so setting up the equation based on the sum of angles gives:

x + x + 71° = 127°

2x + 71° = 127°

2x = 56°

x = 28°

Now that we have x, we can find m∠QOP, which is the sum of m∠QOS, m∠SOT, and m∠POT. So the calculation will be:

m∠QOP = m∠QOS + m∠SOT + m∠POT

m∠QOP = 28° + 71° + 28°

m∠QOP = 127°

Therefore, m∠QOP is 127°.