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

Question

Accepted Answer

# Brief Explanations:
1. For $\overrightarrow{KQ}$: There's no info it bisects an angle, so false.
2. For $\angle LKQ$ being bisected: No indication of bisecting $\angle LKQ$, so false.
3. For $m\angle JKL = 45^\circ$: No info to confirm this, so false.
4. For $m\angle MKQ + m\angle PKQ = m\angle PKM$: By angle addition postulate, this is true.
5. For $\overline{PK}$ being an angle bisector: $\angle JKM$ is straight ($180^\circ$), $\angle JKP$ and $\angle PKM$ are both $90^\circ$ (since $\angle PKM$ is right), so $\overline{PK}$ bisects $\angle JKM$, true.
6. For $\angle JKL\cong\angle QKM$: $\angle JKL + \angle LKP = 90^\circ$ and $\angle QKM + \angle PKQ=\theta$? Wait, no—$\angle JKP = 90^\circ$ (since $\angle PKM$ is right, so $\angle JKP = 90^\circ$) and $\angle PKM = 90^\circ$. If $\overline{PK}$ bisects, and $\angle JKL$ and $\angle QKM$: Wait, actually, $\angle JKL$ and $\angle QKM$: Let's see, $\angle JKP = 90^\circ$, so $\angle JKL + \angle LKP = 90^\circ$, and $\angle PKM = 90^\circ$, so $\angle QKM + \angle PKQ = 90^\circ$. But also, if $\overrightarrow{KQ}$ and $\overrightarrow{KL}$ are symmetric? Wait, no, the diagram shows $\angle JKL$ and $\angle QKM$: Wait, actually, $\angle JKL$ and $\angle QKM$: Let's check the right angles. $\angle JKP = 90^\circ$ (since $\angle PKM$ is right, so $\angle JKP = 180^\circ - 90^\circ = 90^\circ$). So $\angle JKL + \angle LKP = 90^\circ$, and $\angle QKM + \angle PKQ = 90^\circ$. But also, if $\overline{PK}$ is perpendicular to $JM$, then $\angle JKP = \angle PKM = 90^\circ$. Now, $\angle JKL$ and $\angle QKM$: Let's see, $\angle JKL$ and $\angle QKM$: If $\overrightarrow{KQ}$ and $\overrightarrow{KL}$ are such that $\angle LKP = \angle PKQ$ (if $\overrightarrow{KQ}$ bisects? No, earlier we saw $\overline{PK}$ bisects. Wait, actually, $\angle JKL$ and $\angle QKM$: Since $\angle JKP = 90^\circ$ and $\angle PKM = 90^\circ$, and if $\angle JKL$ and $\angle QKM$ are equal? Wait, maybe I made a mistake. Wait, the correct ones: Let's re - evaluate.

Wait, the correct true statements:

- $m\angle MKQ + m\angle PKQ = m\angle PKM$: Angle addition, true.

- $\overline{PK}$ is an angle bisector: $\angle JKM = 180^\circ$, $\angle JKP=\angle PKM = 90^\circ$, so it bisects $\angle JKM$, true.

- $\angle JKL\cong\angle QKM$: Wait, $\angle JKL + \angle LKP = 90^\circ$ and $\angle QKM + \angle PKQ = 90^\circ$. But also, if $\overrightarrow{KQ}$ and $\overrightarrow{KL}$ are symmetric with respect to $PK$, then $\angle LKP=\angle PKQ$, so $\angle JKL=\angle QKM$, true? Wait, maybe. Wait, the diagram: $\angle JKL$ and $\angle QKM$: Let's see, $\angle JKL$ and $\angle QKM$: Since $\angle JKP = 90^\circ$ and $\angle PKM = 90^\circ$, and if $\overline{PK}$ is the bisector, then $\angle JKL$ and $\angle QKM$ should be equal. So that's true.

Wait, earlier mistake: Let's list again:

1. $\overrightarrow{KQ}$ is an angle bisector: No, no info it bisects $\angle LKM$ or something.

2. $\angle LKQ$ is bisected: No, no line bisecting it.

3. $m\angle JKL = 45^\circ$: No info, could be any angle.

4. $m\angle MKQ + m\angle PKQ = m\angle PKM$: True (angle addition).

5. $\overline{PK}$ is an angle bisector: True (bisects $\angle JKM$ into two $90^\circ$ angles).

6. $\angle JKL\cong\angle QKM$: True (since $\angle JKL + \angle LKP = 90^\circ$, $\angle QKM + \angle PKQ = 90^\circ$, and if $\angle LKP=\angle PKQ$ (from symmetry or $\overrightarrow{KQ}$ and $\overrightarrow{KL}$ being symmetric with respect to $PK$), then $\angle JKL=\angle QKM$).

Wait, maybe I was wrong about some. Let's check standard angle bisector and angle addition:

- Angle addition: $m\angle MKQ + m\angle PKQ = m\angle PKM$ is angle addition, so true.

- $\overline{PK}$ is angle bisector: $\angle JKM = 180^\circ$, $\angle JKP = \angle PKM = 90^\circ$, so yes, bisects $\angle JKM$, true.

- $\angle JKL\cong\angle QKM$: Let's see, $\angle JKL + \angle LKP = 90^\circ$ (since $\angle JKP = 90^\circ$) and $\angle QKM + \angle PKQ = 90^\circ$ (since $\angle PKM = 90^\circ$). If $\angle LKP=\angle PKQ$ (which would be the case if $\overrightarrow{KQ}$ and $\overrightarrow{KL}$ are symmetric with respect to $PK$), then $\angle JKL=\angle QKM$, so they are congruent, true.

Wait, but the first option: $\overrightarrow{KQ}$ is angle bisector: Is there an angle it bisects? $\angle LKM$? No, we don't know. $\angle LKQ$ is bisected? No, no line. $m\angle JKL = 45^\circ$: No, we can't assume that.

So the true statements are:

- $m\angle MKQ + m\angle PKQ = m\angle PKM$

- $\overline{PK}$ is an angle bisector

- $\angle JKL\cong\angle QKM$

Wait, but let's check the diagram again. The right angle at $PKM$ (so $\angle PKM = 90^\circ$), so $\angle JKP = 90^\circ$ (since $JKM$ is straight). So $\overline{PK}$ is perpendicular to $JM$, so it bisects $\angle JKM$ (180° into two 90° angles), so $\overline{PK}$ is an angle bisector (true).

For $m\angle MKQ + m\angle PKQ = m\angle PKM$: By angle addition postulate, the sum of two adjacent angles ($\angle MKQ$ and $\angle PKQ$) is equal to the larger angle ($\angle PKM$) they form, so true.

For $\angle JKL\cong\angle QKM$: Since $\angle JKL + \angle LKP = 90^\circ$ (because $\angle JKP = 90^\circ$) and $\angle QKM + \angle PKQ = 90^\circ$ (because $\angle PKM = 90^\circ$). Also, if we assume that $\overrightarrow{KL}$ and $\overrightarrow{KQ}$ are symmetric with respect to $\overline{PK}$ (which is a common diagram setup for such angle problems), then $\angle LKP=\angle PKQ$, so by the subtraction property of equality, $\angle JKL=\angle QKM$, so they are congruent, true.

# Answer:
- $m\angle MKQ + m\angle PKQ = m\angle PKM$ (the option with this statement)
- $\overline{PK}$ is an angle bisector (the option with this statement)
- $\angle JKL\cong\angle QKM$ (the option with this statement)

(In boxed form for each correct option, but since it's multiple checkboxes, list the correct ones:

- $\boldsymbol{m\angle MKQ + m\angle PKQ = m\angle PKM}$

- $\boldsymbol{\overline{PK}}$ is an angle bisector

- $\boldsymbol{\angle JKL \cong \angle QKM}$

And also, wait, earlier I thought $\overline{PK}$ is angle bisector, angle addition, and $\angle JKL\cong\angle QKM$ are true. Let's confirm with the diagram:

- $\overline{PK}$ is perpendicular to $JM$, so it splits $\angle JKM$ (180°) into two 90° angles, so it's an angle bisector: true.

- Angle addition: $m\angle MKQ + m\angle PKQ = m\angle PKM$ is basic angle addition, true.

- $\angle JKL$ and $\angle QKM$: Since $\angle JKP = \angle PKM = 90^\circ$, and if we consider the symmetry (since the diagram looks symmetric with respect to $PK$), then $\angle JKL$ and $\angle QKM$ are congruent: true.

So the correct options are:

- $\overrightarrow{KQ}$ is an angle bisector: False

- $\angle LKQ$ is bisected: False

- $m\angle JKL = 45^\circ$: False

- $m\angle MKQ + m\angle PKQ = m\angle PKM$: True

- $\overline{PK}$ is an angle bisector: True

- $\angle JKL \cong \angle QKM$: True)

So the answer is the options:

- $m\angle MKQ + m\angle PKQ = m\angle PKM$

- $\overline{PK}$ is an angle bisector

- $\angle JKL \cong \angle QKM$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Answer: