Question

in the diagram, ∠jkm is a straight angle. which statements about the diagram are true? check all that apply. ☐ $overrightarrow{kq}$ is an angle bisector. ☐ ∠lkq is bisected. ☐ $mangle jkl = 45^circ$ ☐ $mangle mkq + mangle pkq = mangle pkm$ ☐ $overrightarrow{pk}$ is an angle bisector. ☐ $angle jkl cong angle qkm$

Explanation:

1. For $m\angle MKQ + m\angle PKQ = m\angle PKM$: The angle addition postulate states that the sum of two adjacent angles (sharing a common side and vertex) equals the measure of the larger angle they form. Here, $\angle MKQ$ and $\angle PKQ$ are adjacent and form $\angle PKM$, so this holds.
2. For $\overrightarrow{PK}$ being an angle bisector: $\angle JKM$ is a straight angle ($180^\circ$). If $\overrightarrow{PK}$ creates two equal angles (e.g., right angles) from $\angle JKM$, it bisects $\angle JKM$.
3. For $\angle JKL \cong \angle QKM$: If $\overrightarrow{PK}$ bisects $\angle JKM$ and other angle congruences (from the diagram’s symmetry or given right angles) hold, these two angles will have equal measures, so they are congruent.

The other statements lack sufficient information (or are inconsistent with the diagram’s implied structure) to be true.

Answer:

• $\overrightarrow{KQ}$ is an angle bisector: False (no info to confirm it bisects an angle)
• $\angle LKQ$ is bisected: False (no info about bisecting $\angle LKQ$)
• $m\angle JKL = 45^\circ$: False (no info to confirm this measure)
• $m\angle MKQ + m\angle PKQ = m\angle PKM$: True (by angle addition postulate, as $\angle PKM$ is composed of $\angle MKQ$ and $\angle PKQ$)
• $\overrightarrow{PK}$ is an angle bisector: True (since $\angle JKM$ is a straight angle ($180^\circ$) and $\angle JKP$ and $\angle PKM$ appear to be right angles or equal, so $\overrightarrow{PK}$ bisects $\angle JKM$)
• $\angle JKL \cong \angle QKM$: True (if $\overrightarrow{PK}$ bisects $\angle JKM$ and other angle relationships hold, these angles would be congruent)