Question

c. explain how you can use the triangle - sum theorem to calculate the measure of ∠upm.
subtract the sum of the other two interior angles of the triangle from 180° to find the measure of ∠upm.
calculate the measure of ∠upm.
∠upm = 80°
e. list the sides of △pmb in order from shortest to longest. explain how you determined your answer.
f. list the sides of △pub in order from shortest to longest. explain how you determined your answer.

Explanation:

Step1: Recall triangle - angle - side relationship

In a triangle, the shortest side is opposite the smallest angle and the longest side is opposite the largest angle.

Step2: Analyze \(\triangle PMB\)

We need to know the measures of the angles in \(\triangle PMB\). If we assume we have angle - measure information (not given in full here, but in a real - problem scenario), we order the sides based on the angle measures. For example, if \(\angle P<\angle M<\angle B\), then the sides in order from shortest to longest are \(MB < PB< PM\).

Step3: Analyze \(\triangle PUB\)

Similarly, for \(\triangle PUB\), we use the angle - measure information. If \(\angle P<\angle U<\angle B\), then the sides in order from shortest to longest are \(UB < PB < PU\).

Answer:

For \(\triangle PMB\): Without angle - measure values, we can't give a specific order. But the method is to order sides based on opposite - angle measures.
For \(\triangle PUB\): Without angle - measure values, we can't give a specific order. But the method is to order sides based on opposite - angle measures.