Question

c. explain how you can use the triangle sum theorem to calculate the measure of ∠upm.
d. calculate the measure of ∠upm.
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 - Sum Theorem

The sum of the interior angles of a triangle is 180°. Let the angles of the triangle containing ∠UPM be ∠A, ∠B, and ∠UPM. So, ∠A+∠B + ∠UPM=180°.

Step2: Find known angles

If the measures of ∠A and ∠B are given (not provided in this problem - statement but in a real - scenario), then ∠UPM = 180°-(∠A + ∠B).

For part e:

Step1: Recall angle - side relationship

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

Step2: Identify angle measures

If the measures of the angles of △PMB (∠P, ∠M, ∠B) are known, then order the sides accordingly. For example, if ∠P<∠M<∠B, then the sides in order from shortest to longest are MB < PB < PM.

For part f:

Step1: Recall angle - side relationship

Same as in part e, in a triangle the shortest side is opposite the smallest angle and the longest side is opposite the largest angle.

Step2: Identify angle measures

If the measures of the angles of △PUB (∠P, ∠U, ∠B) are known, then order the sides. For example, if ∠P<∠U<∠B, then the sides in order from shortest to longest are UB < PB < PU.

Since no angle measures are given for calculating ∠UPM or for ordering the sides of the triangles:

Answer:

For part d: Without the measures of the other two angles in the triangle containing ∠UPM, we cannot calculate the measure of ∠UPM. But the formula is ∠UPM = 180°-(sum of the other two angles in the triangle).
For part e: Without angle - measures of △PMB, we cannot list the sides. But the rule is that the side opposite the smallest angle is the shortest and the side opposite the largest angle is the longest.
For part f: Without angle - measures of △PUB, we cannot list the sides. But the rule is that the side opposite the smallest angle is the shortest and the side opposite the largest angle is the longest.