in the figures below, \overleftrightarrow{ps} and \overleftrightarrow{qt} are straight lines. look at the figures and answer the following questions.
1) a) \angle poq and \angle sot are vertical angles. find \angle sot.
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b) name the angles adjacent to 90^\circ.
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c) find \angle qor.
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d) name the linear pair of angles with \overrightarrow{ot} as their common arm.
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2) a) name the angle vertical to \angle sot.
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b) find the complement of 66^\circ.
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c) name the angles adjacent to \angle ros.
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d) \angle pot and \angle poq form a linear pair. find \angle poq.

Question

Accepted Answer

1.
   - a)
     - # Explanation:
       - ## Step1: Use the property of vertical angles
         - Vertical angles are equal. Given that $\angle POQ$ and $\angle SOT$ are vertical angles and $\angle POQ = 70^{\circ}$.
     - # Answer:
       - $\angle SOT=70^{\circ}$
   - b)
     - # Explanation:
       - Adjacent angles share a common side and a common vertex. The angles adjacent to $90^{\circ}$ are $\angle QOP$ (or $70^{\circ}$ angle) and the angle formed by the intersection of $QT$ and $PS$ on the other - side of the $90^{\circ}$ angle (let's call it $\angle SOT$'s adjacent angle to $90^{\circ}$).
     - # Answer:
       - $\angle QOP$ and $\angle SOT$
   - c)
     - # Explanation:
       - ## Step1: Use the sum of angles around a point on a line
         - The sum of angles on a straight line $PS$ at point $O$ for the angles involving $Q$, $O$, and $R$: $\angle QOP+\angle QOR + 90^{\circ}=180^{\circ}$ (since $PS$ is a straight line, the sum of angles on one side of a point on a straight line is $180^{\circ}$). We know $\angle QOP = 70^{\circ}$. Then $\angle QOR=180^{\circ}-90^{\circ}-70^{\circ}$.
       - ## Step2: Calculate the value of $\angle QOR$
         - $\angle QOR = 20^{\circ}$
     - # Answer:
       - $\angle QOR = 20^{\circ}$
   - d)
     - # Explanation:
       - A linear - pair of angles are adjacent angles that form a straight line. If $\overrightarrow{OT}$ is the common arm, then the other arms should form a straight line. The angles are $\angle SOT$ and $\angle POT$.
     - # Answer:
       - $\angle SOT$ and $\angle POT$
2.
   - a)
     - # Explanation:
       - Vertical angles are opposite angles formed by the intersection of two lines. When two lines intersect, the vertical angles are equal. If we consider the intersection of the two lines (one line is $TP$ and the other is the line passing through $O$ and $Q$), the angle vertical to $\angle SOT$ is $\angle POQ$.
     - # Answer:
       - $\angle POQ$
   - b)
     - # Explanation:
       - ## Step1: Use the formula for complementary angles
         - Complementary angles: If two angles $\alpha$ and $\beta$ are complementary, then $\alpha+\beta = 90^{\circ}$. Let the complement of $66^{\circ}$ be $x$. Then $x + 66^{\circ}=90^{\circ}$.
       - ## Step2: Solve for $x$
         - $x=90^{\circ}-66^{\circ}$
     - # Answer:
       - $24^{\circ}$
   - c)
     - # Explanation:
       - Adjacent angles share a common side and a common vertex. The angles adjacent to $\angle ROS$ are $\angle SOP$ (if we assume the lines are arranged in a certain way, considering the intersection of lines at $O$) and $\angle ROQ$.
     - # Answer:
       - $\angle SOP$ and $\angle ROQ$
   - d)
     - # Explanation:
       - ## Step1: Use the property of a linear - pair of angles
         - Since $\angle POT$ and $\angle POQ$ form a linear pair, $\angle POT+\angle POQ = 180^{\circ}$. Given $\angle POT = 66^{\circ}$ (from the figure in part 2).
       - ## Step2: Calculate $\angle POQ$
         - $\angle POQ=180^{\circ}-\angle POT$. Substitute $\angle POT = 66^{\circ}$, then $\angle POQ = 114^{\circ}$
     - # Answer:
       - $\angle POQ = 114^{\circ}$

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