Question

when the measure of angle c decreases, what happens to the measures of angles a and b on line p?
both sections increase.
both sections decrease.
one section increases, and the other section decreases.
what will happen to the individual sections cannot be determined, but the size of the two sections together increases

Explanation:

Step1: Recall Triangle Angle Sum

The sum of angles in a triangle is $180^\circ$, so $m\angle A + m\angle B + m\angle C = 180^\circ$.

Step2: Analyze Change in $\angle C$

If $m\angle C$ decreases, let the new measure be $m\angle C' < m\angle C$. Then $m\angle A' + m\angle B' + m\angle C' = 180^\circ$. Substituting, $m\angle A' + m\angle B' = 180^\circ - m\angle C'$. Since $m\angle C' < m\angle C$, $180^\circ - m\angle C' > 180^\circ - m\angle C = m\angle A + m\angle B$. But for the semicircle (sum of angles on line $p$? Wait, no—wait, the triangle's angles: when $\angle C$ decreases, to keep the sum $180^\circ$, the sum of $\angle A$ and $\angle B$ must increase. But looking at the options, the key is that in a triangle, if one angle decreases, the sum of the other two increases. But the options: wait, the diagram shows the angles on a line? Wait, no, the triangle angles. Wait, the options: "Both sections increase"—wait, maybe the semicircle represents the sum? Wait, no, the triangle's angles: $A + B + C = 180$. If $C$ decreases, $A + B$ must increase. But the options: the first option is "Both sections increase"—so if $C$ decreases, $A$ and $B$ (their measures) must increase (since their sum increases, and assuming they adjust such that both increase? Wait, maybe in the context, the angles are on a straight line (semicircle, $180^\circ$), but no, the triangle. Wait, maybe the problem is that when angle $C$ (in the triangle) decreases, the angles $A$ and $B$ (also in the triangle) must increase to keep the sum $180$. So both $A$ and $B$ increase.

Answer:

Both sections increase. (Corresponding to the first option: "Both sections increase.")