Question

for # 19-22 use the figure to the right to answer the following question

1. name a pair of vertical angles: ______

if the m∠2 = x + 9
and m∠1 = 4x - 14, then:

1. x = ______
2. m∠1 = ______
3. m∠2 = ______

for problems 23 - 25, use the following lines and angle to do the indicated constructions
diagrams of segments bc, rs, tu, and angle d

1. construct \overline{lm} so that lm = 3bc.
2. construct \angle i so that \angle i \cong \angle d.
3. construct the perpendicular bisector of \overline{rs}.

Explanation:

Problem 19: Name a pair of vertical angles

Vertical angles are formed by two intersecting lines and are opposite each other. From the figure (assuming two intersecting lines), a common pair could be $\angle 1$ and the angle opposite to it, or $\angle 2$ and its opposite. For example, if $\angle 1$ and $\angle 3$ (or other labeled opposite angles) are vertical, but based on typical labeling, a pair like $\angle 1$ and the angle opposite (say $\angle 3$) or $\angle 2$ and its opposite. But since the figure has $\angle 1$ and $\angle 2$ adjacent, likely $\angle 1$ and the angle vertically opposite (e.g., if lines intersect, $\angle 1$ and $\angle 3$, but maybe in the figure, $\angle 1$ and the angle opposite to $\angle 2$'s opposite. Wait, maybe the figure has two intersecting lines, so a pair is $\angle 1$ and the angle opposite (let's assume the other angle is, say, $\angle 3$, but maybe the labels are $\angle 1$, $\angle 2$, and their vertical counterparts. Alternatively, if $\angle 1$ and $\angle 2$ are adjacent, but vertical angles are opposite. So a possible pair is $\angle 1$ and the angle vertically opposite (e.g., if the lines are intersecting, $\angle 1$ and $\angle 3$ where $\angle 3$ is opposite). But maybe in the figure, the two angles given ($\angle 1$ and $\angle 2$) are adjacent, so vertical angles would be $\angle 1$ and the angle opposite to it (let's say $\angle 3$) and $\angle 2$ and its opposite ($\angle 4$). But since the problem says "name a pair", a common example is $\angle 1$ and its vertical angle (e.g., if the lines are intersecting, $\angle 1$ and the angle opposite). But maybe the figure shows $\angle 1$ and $\angle 2$ as adjacent, so vertical angles could be $\angle 1$ and the angle opposite (let's assume the other angle is, say, $\angle 3$). But perhaps the answer is $\angle 1$ and the angle opposite to $\angle 2$'s opposite. Alternatively, maybe the figure has two intersecting lines, so a pair is $\angle 1$ and $\angle 3$ (where $\angle 3$ is opposite), but since the problem is about $\angle 1$ and $\angle 2$, maybe $\angle 1$ and the angle vertically opposite (let's say $\angle 3$) or $\angle 2$ and its opposite. But maybe the answer is $\angle 1$ and the angle opposite (e.g., if the lines are intersecting, $\angle 1$ and $\angle 3$). But without the exact figure, a typical pair is $\angle 1$ and its vertical angle (e.g., $\angle 3$) or $\angle 2$ and its vertical angle. But maybe the figure has $\angle 1$ and $\angle 2$ as adjacent, so vertical angles are $\angle 1$ and the angle opposite (let's say $\angle 3$) and $\angle 2$ and $\angle 4$. So a possible answer is $\boldsymbol{\angle 1}$ and its vertical angle (e.g., $\angle 3$) or $\boldsymbol{\angle 2}$ and its vertical angle. But maybe the figure is two intersecting lines, so a pair is $\angle 1$ and the angle opposite (e.g., $\angle 3$).

Problem 20: Solve for $x$

We assume $\angle 1$ and $\angle 2$ are vertical angles (since vertical angles are equal). So $m\angle 1 = m\angle 2$.

Step 1: Set the measures equal

Since $\angle 1$ and $\angle 2$ are vertical angles, their measures are equal. So:
$4x - 14 = x + 9$

Step 2: Solve for $x$

Subtract $x$ from both sides:
$4x - x - 14 = 9$
$3x - 14 = 9$

Add 14 to both sides:
$3x = 9 + 14$
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Answer:

Step 1: Set the measures equal

Since $\angle 1$ and $\angle 2$ are vertical angles, their measures are equal. So:
$4x - 14 = x + 9$

Step 2: Solve for $x$

Subtract $x$ from both sides:
$4x - x - 14 = 9$
$3x - 14 = 9$

Add 14 to both sides:
$3x = 9 + 14$
$3x = 23$ Wait, no, 9 + 14 is 23? Wait, 9 +14 is 23? Wait, 9 +14 is 23? Wait, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? Wait, 9 +14 is 23? Wait, no, 9 +14 is 23? 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