Question

vertical and adjacent angles • practice
example 1
refer to the figure.

1. name two adjacent angles
2. name two vertical angles
3. find ( mangle sut ).

Explanation:

Sub - question 1: Name two adjacent angles

Adjacent angles are two angles that share a common side and a common vertex, and their non - common sides are on opposite sides of the common side. Looking at the figure, $\angle MSR$ and $\angle RSP$ share the common side $SR$ and the common vertex $S$. Another pair could be $\angle SUT$ and $\angle TUP$ (if we consider the relevant lines), or $\angle MSU$ and $\angle SUT$ (since they share the common vertex $S$ and a common side, and their non - common sides are in a linear or adjacent configuration). For simplicity, let's take $\angle MSR$ and $\angle RSP$.

Vertical angles are formed by two intersecting lines and are opposite each other. When two lines intersect, the vertical angles are equal in measure. In the figure, if we consider the intersection of lines (for example, the intersection of the lines that form angles at a point), $\angle MSR$ and $\angle PTR$ (assuming the relevant intersection) or more clearly, if we look at the intersection of the lines at some point, say the intersection of the line containing $M, G, N$ and the line containing $T, R, S$ (or other intersecting lines), a common pair of vertical angles could be $\angle MGR$ and $\angle TRN$ (but more accurately, looking at the right angle at $S$, if we consider the intersection of lines, $\angle MSN$ and $\angle TSP$ (assuming appropriate intersections). A more straightforward pair: if we consider the angles formed at the intersection of two lines, for example, when two lines cross, the vertical angles are opposite. Let's assume the lines $MN$ and $TP$ intersect at $R$, then $\angle MRP$ and $\angle NRT$ are vertical angles. Another example: if we look at the angle at $U$, but actually, a better pair is $\angle MSR$ and $\angle PTR$ (but maybe a more obvious one is $\angle SGU$ and $\angle RTP$ - however, a standard pair could be $\angle MGR$ and $\angle TRN$ or $\angle MSN$ and $\angle TSP$. Alternatively, considering the right angle at $S$, if we have two intersecting lines, say line $MS$ and line $UP$ intersecting with other lines, but perhaps the most clear is $\angle MSR$ and $\angle TSP$ (vertical angles formed by the intersection of two lines).

Step 1: Identify the right angle and the given angle

We know that $\angle SUV$ is a straight angle (since $V - U - T$? Wait, no, looking at the figure, $\angle MSV$ is a right angle (the square at $S$), and the angle given is $58^{\circ}$ at $U$ between $UT$ and the vertical line. Wait, actually, $\angle SUT$ and the $58^{\circ}$ angle and the right angle? Wait, no, the line $SU$ is vertical, and $\angle SUV$ is a straight line? Wait, no, the square at $S$ indicates that $\angle MSU$ is a right angle? Wait, no, the figure has a right angle at $S$ (the square), so $SU$ is perpendicular to $SP$, so $\angle SUP = 90^{\circ}$? Wait, no, the angle at $U$ is $58^{\circ}$ between $UT$ and $UV$? Wait, no, let's re - examine. The line $UV$ and $UT$: $\angle VUT$ is a straight line? No, $V - U - T$? Wait, no, the angle at $U$: we have a right angle at $S$ (so $SU\perp SP$), and we need to find $\angle SUT$. We know that $\angle SUT$ and the $58^{\circ}$ angle are complementary? Wait, no, if $SU$ is vertical and there is a right angle at $S$, then $\angle SUT + 58^{\circ}=90^{\circ}$? Wait, no, the right angle at $S$ means that $SU$ is perpendicular to $SP$, so the angle between $SU$ and $SP$ is $90^{\circ}$. The angle between $UT$ and $UV$ is $58^{\circ}$, and since $UV$ is a straight line (or $SU$ is vertical), $\angle SUT + 58^{\circ}=90^{\circ}$? Wait, no, let's think again. The square at $S$ indicates that $\angle MSP$ is a right angle? No, the square is at $S$ between $MS$ and $SU$? Wait, the figure shows a square at $S$, so $MS\perp SU$, so $\angle MSU = 90^{\circ}$. Then, the line $UT$ makes a $58^{\circ}$ angle with $UV$ (where $V - U - T$ is a straight line? No, $V$ and $T$ are on a line through $U$? Wait, the angle at $U$ is $58^{\circ}$ between $UT$ and $UV$, and we need to find $\angle SUT$. Since $SU$ is perpendicular to $MS$, but actually, $SU$ is vertical, and $SP$ is horizontal (because of the right angle at $S$). So the angle between $SU$ (vertical) and $UT$: we know that the angle between $UT$ and $UV$ (which is a straight line opposite to $SU$? No, $UV$ is a line going left from $U$, and $UT$ is going to $T$. The right angle at $S$ means that $SU\perp SP$, so $SU$ is vertical, $SP$ is horizontal. Then, the angle between $UT$ and $UV$ is $58^{\circ}$, and since $SU$ is vertical, $\angle SUT=90^{\circ}- 58^{\circ}=32^{\circ}$? Wait, no, maybe $\angle SUT$ and the $58^{\circ}$ angle are complementary because $SU$ is perpendicular to $SP$, and $UT$ is a line such that the angle between $UT$ and $UV$ is $58^{\circ}$, and $UV$ is a straight line with respect to $SU$? Wait, let's do the calculation. If we have a right angle (90 degrees) at $S$ related to $SU$ and another line, and the angle between $UT$ and $UV$ is $58^{\circ}$, then $\angle SUT = 90^{\circ}-58^{\circ}=32^{\circ}$? Wait, no, maybe the angle we need is such that $\angle SUT$ and the $58^{\circ}$ angle are complementary. So:

Step 1: Recall the right angle

The square at $S$ indicates that $\angle SUP = 90^{\circ}$ (assuming $SP$ is horizontal and $SU$ is vertical).

Step 2: Calculate $\angle SUT$

We know that $\angle SUT + 58^{\circ}=\angle SUP = 90^{\circ}$ (since they are adjacent angles forming the right angle). So, $m\angle SUT=90^{\circ}- 58^{\circ}$.
$m\angle SUT = 32^{\circ}$

Answer:

$\angle MSR$ and $\angle RSP$ (other valid pairs are also possible, e.g., $\angle MSU$ and $\angle SUT$, $\angle SUT$ and $\angle TUP$ etc.)

Sub - question 2: Name two vertical angles