vertical and adjacent angles · practice example 1 refer to the figure. 1. name two adjacent angles 2. name two vertical angles 3. find ( mangle suv ).

Question

Accepted Answer

### 1. Naming two adjacent angles
# Brief Explanations:
Adjacent angles share a common side and vertex. For example, $\angle MSQ$ and $\angle QSP$ share side $SQ$ and vertex $S$. Another pair could be $\angle SQU$ and $\angle UST$ (depending on the figure's details, but a common valid pair is $\angle MSQ$ and $\angle QSP$).
# Answer:
$\angle MSQ$ and $\angle QSP$ (or other valid adjacent angle pair like $\angle SUT$ and $\angle TUP$ etc., depending on the figure's angle connections)

### 2. Naming two vertical angles
# Brief Explanations:
Vertical angles are opposite angles formed by intersecting lines. For example, if lines intersect at $Q$, $\angle MQS$ and $\angle PQN$ (or if lines intersect at $R$, $\angle SRN$ and $\angle PRM$). A common pair from intersecting lines at $Q$: $\angle MQS$ and $\angle PQN$.
# Answer:
$\angle MQS$ and $\angle PQN$ (or other valid vertical angle pair from intersecting lines in the figure)

### 3. Finding $m\angle SUV$
# Explanation:
## Step1: Identify the right angle and given angle
We see $S$ has a right angle (marked $\perp$), so the line $SU$ is part of a straight line? Wait, no, $\angle SUV$: looking at the figure, $SV$ is a straight line? Wait, the angle at $U$: the given angle is $58^\circ$, and $\angle SUV$: since $SU$ is perpendicular? Wait, no, the right angle is at $S$ (between $MS$ and $SP$). Wait, the angle at $U$: the angle between $UV$ and $UT$ is $58^\circ$, and $\angle SUV$: since $UV$ and $SU$ form a straight line? Wait, no, $SU$ is vertical, $UV$ is a straight line with $SU$? Wait, no, the right angle is at $S$ (so $MS \perp SP$). Wait, for $\angle SUV$: the angle between $SU$ and $UV$. Wait, the line $SU$ is vertical, and $UV$ is a straight line? Wait, no, the angle at $U$: the given angle is $58^\circ$, and $\angle SUV$: since $SU$ and $UV$ form a linear pair with the $58^\circ$ angle? Wait, no, the right angle is at $S$, but $\angle SUV$: wait, the figure has $SU$ vertical, $UV$ going left, and $UT$ at $58^\circ$ from $SU$. Wait, actually, $\angle SUV$: since $SU$ is a straight line? No, $SU$ and $UV$: wait, the angle between $SU$ (vertical) and $UV$ (horizontal left? No, the diagram: $U$ is on a straight line $V - U - T$? Wait, no, the key is that $\angle SUV$: if $SU$ is perpendicular? No, the right angle is at $S$ (between $MS$ and $SP$). Wait, maybe $\angle SUV$ is a straight angle? No, wait, the given angle is $58^\circ$ at $U$ between $SU$ and $UT$, so $\angle SUV$: since $UV$ and $UT$ are a straight line? Wait, no, $V - U - T$ is a straight line? Then $\angle SUV + 58^\circ = 180^\circ$? No, wait, $SU$ is vertical, $UV$ is a straight line with $SU$? Wait, no, the right angle is at $S$ (so $MS \perp SP$), but $SU$ is vertical, so $SP$ is horizontal. Then $UT$ is a line from $U$ to $T$ (on $SP$). So $\angle SUV$: the angle between $SU$ (vertical) and $UV$ (leftward). Wait, actually, $SU$ and $UV$ form a straight line? No, $V - U - S$? Wait, the diagram: $V$ is left of $U$, $U$ is below $S$, $S$ is connected to $M$ and $P$. So $SU$ is a vertical segment (from $S$ down to $U$), $UV$ is a horizontal segment left from $U$, and $UT$ is a segment from $U$ to $T$ (on $SP$) at $58^\circ$ from $SU$. Wait, no, the angle at $U$ between $SU$ and $UT$ is $58^\circ$, so $\angle SUV$: since $UV$ is a straight line with $UT$? No, $V - U - T$ is a straight line, so $\angle SUV + 58^\circ = 180^\circ$? No, that can't be. Wait, no, $SU$ is perpendicular to $SP$ (right angle at $S$), so $SP$ is horizontal, $SU$ is vertical. Then $UT$ is a line from $U$ to $T$ (on $SP$), so triangle $SUT$? Wait, no, the problem is to find $m\angle SUV$. Wait, maybe $SU$ and $UV$ form a straight line, and the angle between $SU$ and $UT$ is $58^\circ$, so $\angle SUV$ is $180^\circ - 58^\circ$? No, that would be if $UT$ and $UV$ are a straight line. Wait, the diagram shows $V - U - T$ as a straight line? Then $\angle SUV + \angle SUT = 180^\circ$. But $\angle SUT$ is $58^\circ$? Wait, no, the given angle is $58^\circ$ at $U$ between $SU$ and $UT$, so $\angle SUT = 58^\circ$, so $\angle SUV = 180^\circ - 58^\circ = 122^\circ$? Wait, no, maybe $SU$ is perpendicular to $SP$, so $SP$ is horizontal, $SU$ is vertical, so $\angle S$ is $90^\circ$, but $\angle SUV$: wait, maybe I misread. Wait, the right angle is at $S$ (between $MS$ and $SP$), so $MS \perp SP$. Then $SU$ is a vertical line (same as $MS$? No, $S$ is connected to $U$ below, so $SU$ is vertical, $SP$ is horizontal. Then $UT$ is a line from $U$ to $T$ (on $SP$), so $\angle SUT$ is $58^\circ$, so $\angle SUV$: since $UV$ is a straight line with $UT$? No, $V$ is left of $U$, $U$ is below $S$, $T$ is on $SP$ (right of $U$). So $V - U - T$ is a straight line, so $\angle SUV + \angle SUT = 180^\circ$. So $\angle SUV = 180^\circ - 58^\circ = 122^\circ$? Wait, no, maybe the right angle is at $S$ (between $SU$ and $SP$), so $SU \perp SP$, so $\angle USP = 90^\circ$. Then $UT$ is a line from $U$ to $T$ (on $SP$), so $\angle SUT = 58^\circ$, so $\angle SUV$: since $UV$ is a straight line with $UT$? No, $V$ is left of $U$, $U$ is below $S$, $T$ is on $SP$ (right of $U$). So $V - U - T$ is a straight line, so $\angle SUV + \angle SUT = 180^\circ$. So $\angle SUV = 180 - 58 = 122$? Wait, but maybe $SU$ is vertical, $UV$ is horizontal, so $\angle SUV$ is $90^\circ$? No, the diagram has a $58^\circ$ angle at $U$ between $SU$ and $UT$. Wait, maybe the right angle is at $S$ (between $MS$ and $SP$), so $MS \perp SP$, and $SU$ is part of $MS$? No, $S$ is connected to $M$ above and $U$ below, so $MSU$ is a vertical line. Then $SP$ is horizontal from $S$ to $P$, $UT$ is a line from $U$ to $T$ (on $SP$), making a $58^\circ$ angle with $SU$. So $\angle SUV$: $UV$ is a straight line from $U$ to $V$ (left), so $SU$ (vertical) and $UV$ (horizontal left) form a right angle? No, that would be $90^\circ$, but the $58^\circ$ is there. Wait, I think I made a mistake. Let's re-express:

The key is that $\angle SUV$: looking at the figure, $SU$ is vertical, $UV$ is a straight line, and the angle between $SU$ and $UT$ is $58^\circ$. Since $UV$ and $UT$ are a straight line (linear pair), $\angle SUV + 58^\circ = 180^\circ$? No, that would be if $UV$ and $UT$ are opposite rays. Wait, $V - U - T$: so $UV$ and $UT$ are opposite rays, so they form a straight line (180 degrees). Then $\angle SUV$ and $\angle SUT$ are adjacent angles forming a linear pair, so their sum is 180 degrees. Given $\angle SUT = 58^\circ$, then $\angle SUV = 180^\circ - 58^\circ = 122^\circ$? Wait, but maybe $SU$ is perpendicular to $SP$, so $SP$ is horizontal, $SU$ is vertical, so $\angle USP = 90^\circ$, but that's at $S$. At $U$, the angle between $SU$ (vertical) and $UT$ (going to $T$ on $SP$) is $58^\circ$, so $UT$ makes $58^\circ$ with $SU$, so $UV$ is a straight line from $U$ to $V$ (left), so $UV$ is opposite to $UT$? No, $V$ is left, $T$ is right, so $UV$ and $UT$ are opposite rays, so linear pair. Thus, $\angle SUV = 180 - 58 = 122$ degrees.

## Step1: Identify linear pair
$\angle SUV$ and $\angle SUT$ form a linear pair (they are adjacent and their non - common sides form a straight line), so $m\angle SUV + m\angle SUT=180^{\circ}$.

## Step2: Substitute known angle
We know that $m\angle SUT = 58^{\circ}$. Substitute into the equation: $m\angle SUV+58^{\circ}=180^{\circ}$.

## Step3: Solve for $m\angle SUV$
$m\angle SUV=180^{\circ}- 58^{\circ}=122^{\circ}$.

# Answer:
$122^{\circ}$

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QUESTION IMAGE

Question

Explanation:

1. Naming two adjacent angles

Step1: Identify the right angle and given angle

Answer:

2. Naming two vertical angles