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. - $m∠jkl = 45°$ - $m∠mkq + m∠pkq = m∠pkm$ - $overrightarrow{pk}$ is an angle bisector. - $∠jkl ≅ ∠qkm$

Explanation:

1. For \(\overrightarrow{KQ}\) being an angle bisector: If \(\overrightarrow{KQ}\) splits \(\angle LKM\) (or relevant angle) into two equal parts, it's an angle bisector. From the diagram and markings, it seems to bisect \(\angle LKQ\) (as \(\angle LKQ\) is marked as bisected), so this is true.
2. For \(\angle LKQ\) being bisected: The marking and the angle bisector \(\overrightarrow{KQ}\) (or related ray) suggest it splits \(\angle LKQ\) into two equal angles, so this is true.
3. For \(m\angle JKL = 45^{\circ}\): There's no indication (like a right angle or bisector for \(\angle JKL\) with known measure) that \(\angle JKL\) is \(45^{\circ}\), so this is false.
4. For \(m\angle MKQ + m\angle PKQ=m\angle PKM\): By the angle addition postulate, the sum of \(\angle MKQ\) and \(\angle PKQ\) should equal \(\angle PKM\), so this is actually true (maybe a mistake in the original unchecked, but based on angle addition, it's true). Wait, but let's re - check: \(\angle PKM\) is composed of \(\angle MKQ\) and \(\angle PKQ\), so \(m\angle MKQ + m\angle PKQ=m\angle PKM\) is true.
5. For \(\overrightarrow{PK}\) being an angle bisector: Since \(\angle JKM\) is a straight angle (\(180^{\circ}\)) and \(\angle JKP\) and \(\angle PKM\) seem to be right angles (from the right - angle marking), \(\overrightarrow{PK}\) bisects \(\angle JKM\), so this is true.
6. For \(\angle JKL\cong\angle QKM\): There's no information (like equal angle measures or bisectors implying equality) to suggest these two angles are congruent, so this is false.

Answer:

• \(\overrightarrow{KQ}\) is an angle bisector: True
• \(\angle LKQ\) is bisected: True
• \(m\angle JKL = 45^{\circ}\): False
• \(m\angle MKQ + m\angle PKQ=m\angle PKM\): True (correction from original unchecked, based on angle addition postulate)
• \(\overrightarrow{PK}\) is an angle bisector: True
• \(\angle JKL\cong\angle QKM\): False

(If we follow the original diagram's markings and standard angle properties, the correct true statements are \(\overrightarrow{KQ}\) is an angle bisector, \(\angle LKQ\) is bisected, \(m\angle MKQ + m\angle PKQ=m\angle PKM\), \(\overrightarrow{PK}\) is an angle bisector. The statement \(m\angle JKL = 45^{\circ}\) and \(\angle JKL\cong\angle QKM\) are false.)