14. in the diagram shown, $\overline{qw}$ and $\overline{sv}$ intersect at point $r$ and $\overline{rt}$ is drawn such that $\angle srt \cong \angle wrt$. \
(a) name a linear pair of angles from the diagram. \
(b) name a pair of vertical angles from the diagram. \
(c) if $m\angle srt = 70^\circ$, then find the measure of $\angle qrv$ and the measure of $\angle qrs$. \
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15. in a linear angle pair, the measure of the larger angle is 15 degrees less than twice the measure of the smaller angle. find the measures of both angles. show the work that leads to your answers. \
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16. lines $r$ and $s$ intersect as shown in the diagram below. find the value of $x$ algebraically. justify your approach. \
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17. two lines, $j$ and $k$, intersect to form four angles, one of which is a right angle as marked. explain why each of the other four angles must also be right angles.

Question

14. in the diagram shown, $\overline{qw}$ and $\overline{sv}$ intersect at point $r$ and $\overline{rt}$ is drawn such that $\angle srt \cong \angle wrt$.  \
(a) name a linear pair of angles from the diagram.  \
(b) name a pair of vertical angles from the diagram.  \
(c) if $m\angle srt = 70^\circ$, then find the measure of $\angle qrv$ and the measure of $\angle qrs$.  \
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15. in a linear angle pair, the measure of the larger angle is 15 degrees less than twice the measure of the smaller angle. find the measures of both angles. show the work that leads to your answers.  \
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16. lines $r$ and $s$ intersect as shown in the diagram below. find the value of $x$ algebraically. justify your approach.  \
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17. two lines, $j$ and $k$, intersect to form four angles, one of which is a right angle as marked. explain why each of the other four angles must also be right angles.

Accepted Answer

### Question 14 (a)
# Brief Explanations:
A linear pair of angles are adjacent angles that form a straight line (sum to $180^\circ$). From the diagram, $\angle QRS$ and $\angle SRW$ (or other pairs like $\angle SRT$ and $\angle WRT$ but wait, no—wait, $\angle QRS$ and $\angle SRW$ share a common side $RS$ and their non - common sides form a straight line $QW$. Alternatively, $\angle QRS$ and $\angle SRW$ (but actually, looking at the diagram, $\angle QRS$ and $\angle SRW$ is a linear pair? Wait, no, $\angle QRS$ and $\angle SRW$ – wait, $QW$ is a straight line, so $\angle QRS$ and $\angle SRW$ are adjacent and form a straight line. Or $\angle SRT$ and $\angle WRT$ – no, the problem says $\angle SRT\cong\angle WRT$, but a linear pair sums to $180^\circ$, so if $\angle SRT\cong\angle WRT$, then each is $90^\circ$, but for a linear pair, we can take $\angle QRS$ and $\angle SRW$ (but actually, let's see: $QW$ is a straight line, $SV$ intersects at $R$, and $RT$ is drawn. So a linear pair could be $\angle QRS$ and $\angle SRW$ (since they are adjacent and on a straight line $QW$). Or $\angle QRV$ and $\angle VRW$, etc. But a common one is $\angle QRS$ and $\angle SRW$ (or $\angle SRT$ and $\angle WRT$ if we consider, but since $\angle SRT\cong\angle WRT$, they are a linear pair only if they sum to $180^\circ$, so each is $90^\circ$, but the question just asks to name a linear pair. So one example is $\angle QRS$ and $\angle SRW$ (or $\angle SRT$ and $\angle WRT$ as they are adjacent and form a straight line? Wait, no, $\angle SRT$ and $\angle WRT$ share the side $RT$ and their non - common sides $SR$ and $WR$ form a straight line $SW$? Wait, no, $QW$ is a straight line. Wait, maybe $\angle QRS$ and $\angle SRW$ is a linear pair.
# Answer:
$\angle QRS$ and $\angle SRW$ (or other valid linear pairs like $\angle SRT$ and $\angle WRT$ if they are adjacent and form a straight line, but $\angle QRS$ and $\angle SRW$ is a more straightforward example as they are on the straight line $QW$)

### Question 14 (b)
# Brief Explanations:
Vertical angles are opposite angles formed by the intersection of two lines. When $QW$ and $SV$ intersect at $R$, the vertical angles are $\angle QRS$ and $\angle VRW$ (or $\angle QRV$ and $\angle SRW$). Because vertical angles are equal and are opposite each other when two lines cross.
# Answer:
$\angle QRS$ and $\angle VRW$ (or $\angle QRV$ and $\angle SRW$)

### Question 14 (c)
# Explanation:
## Step 1: Find $m\angle WRT$
Given that $\angle SRT\cong\angle WRT$ and $m\angle SRT = 70^\circ$, so $m\angle WRT=70^\circ$.

## Step 2: Find $m\angle QRV$
$\angle QRV$ and $\angle SRW$ are vertical angles? Wait, no. Wait, $\angle SRT$ and $\angle WRT$ are equal, and $QW$ is a straight line, so $\angle SRT+\angle WRT+\angle QRS+\angle VRW = 360^\circ$? No, wait, $QW$ is a straight line, so the sum of angles on a straight line is $180^\circ$. Wait, $\angle SRT$ and $\angle WRT$ are adjacent to $\angle QRS$ and $\angle VRW$? Wait, no, let's re - examine. Since $\angle SRT\cong\angle WRT = 70^\circ$, then $\angle SRW=\angle SRT+\angle WRT = 70^\circ + 70^\circ=140^\circ$. And $\angle QRV$ is vertical to $\angle SRW$, so $m\angle QRV = 140^\circ$.

## Step 3: Find $m\angle QRS$
Since $QW$ is a straight line, $\angle QRS+\angle SRW = 180^\circ$ (linear pair). We know $m\angle SRW = 140^\circ$, so $m\angle QRS=180^\circ - 140^\circ = 40^\circ$? Wait, no, wait. Wait, $\angle SRT = 70^\circ$, and $\angle QRS$ and $\angle SRT$: Wait, maybe I made a mistake. Let's start over.

Given $\angle SRT\cong\angle WRT$, so $RT$ is the angle bisector of $\angle SRW$. So $\angle SRW=\angle SRT+\angle WRT = 70^\circ+70^\circ = 140^\circ$. Now, $\angle QRS$ and $\angle SRW$ are a linear pair? No, $QW$ is a straight line, so $\angle QRS+\angle SRW = 180^\circ$? Wait, no, $QW$ is a straight line, so the sum of angles on one side of $QW$ at point $R$ is $180^\circ$. So $\angle QRS+\angle SRT+\angle WRT=180^\circ$? Wait, no, the straight line is $QW$, so the angles along $QW$ at $R$ are $\angle QRS$, $\angle SRT$, $\angle WRT$? No, the lines are $QW$ (horizontal) and $SV$ (diagonal) intersecting at $R$, and $RT$ is a ray from $R$ such that $\angle SRT=\angle WRT$. So the straight line is $QW$, so $\angle QRS+\angle SRW = 180^\circ$, and $\angle SRW=\angle SRT+\angle WRT = 140^\circ$, so $\angle QRS = 180 - 140=40^\circ$. And $\angle QRV$ is vertical to $\angle SRW$, so $\angle QRV=\angle SRW = 140^\circ$. Wait, no, vertical angles: when two lines intersect, vertical angles are equal. The lines $QW$ and $SV$ intersect at $R$, so $\angle QRS$ and $\angle VRW$ are vertical angles, and $\angle QRV$ and $\angle SRW$ are vertical angles. So if $\angle SRW = 140^\circ$, then $\angle QRV = 140^\circ$, and $\angle QRS=180 - 140 = 40^\circ$.

Wait, another way: Since $\angle SRT=\angle WRT = 70^\circ$, then $\angle SRW = 140^\circ$. $\angle QRV$ and $\angle SRW$ are vertical angles (because $QW$ and $SV$ intersect at $R$), so $m\angle QRV=m\angle SRW = 140^\circ$. And $\angle QRS$ and $\angle SRW$ are a linear pair (they form a straight line $QW$), so $m\angle QRS + m\angle SRW=180^\circ$, so $m\angle QRS=180 - 140 = 40^\circ$.

# Answer:
$m\angle QRV = 140^\circ$, $m\angle QRS = 40^\circ$

### Question 15
# Explanation:
## Step 1: Define variables
Let the measure of the smaller angle be $x$ degrees. Then the measure of the larger angle is $2x - 15$ degrees (since it is 15 degrees less than twice the smaller angle).

## Step 2: Use the property of linear pair
Since they are a linear pair, their sum is $180^\circ$. So we set up the equation: $x+(2x - 15)=180$.

## Step 3: Solve the equation
Combine like terms: $3x-15 = 180$.

Add 15 to both sides: $3x=180 + 15=195$.

Divide both sides by 3: $x=\frac{195}{3}=65$.

## Step 4: Find the larger angle
The larger angle is $2x - 15=2\times65-15 = 130 - 15 = 115$ degrees.

# Answer:
The smaller angle is $65^\circ$ and the larger angle is $115^\circ$

### Question 16
# Explanation:
## Step 1: Identify the relationship
When two lines intersect, vertical angles are equal. The angles $10x - 26$ and $8(x + 1)$ are vertical angles, so they are equal.

## Step 2: Set up the equation
$10x-26=8(x + 1)$

## Step 3: Solve the equation
Expand the right - hand side: $10x-26 = 8x+8$.

Subtract $8x$ from both sides: $10x-8x-26=8x - 8x+8$, which simplifies to $2x-26 = 8$.

Add 26 to both sides: $2x-26 + 26=8 + 26$, so $2x=34$.

Divide both sides by 2: $x=\frac{34}{2}=17$.

# Answer:
$x = 17$

### Question 17
# Brief Explanations:
When two lines $j$ and $k$ intersect, if one angle is a right angle ($90^\circ$):
1. **Linear Pair Property**: Angles that form a linear pair sum to $180^\circ$. If one angle (say $\angle1$) is $90^\circ$, and $\angle1$ and $\angle2$ form a linear pair, then $\angle1+\angle2 = 180^\circ$, so $\angle2=180^\circ - 90^\circ = 90^\circ$. Similarly, $\angle1$ and $\angle3$ form a linear pair, so $\angle3 = 90^\circ$.
2. **Vertical Angles Property**: Vertical angles are equal. $\angle1$ and $\angle$ (the angle opposite to $\angle1$) are vertical angles, so it is also $90^\circ$. $\angle2$ and its vertical angle are equal, so the vertical angle of $\angle2$ is $90^\circ$, and $\angle3$ and its vertical angle are equal, so the vertical angle of $\angle3$ is $90^\circ$. So all four angles are right angles.

# Answer:
When two lines intersect, (1) Linear pairs of angles sum to $180^\circ$. If one angle is $90^\circ$, its linear - pair angle is $180^\circ - 90^\circ=90^\circ$. (2) Vertical angles are equal, so the angle opposite to a $90^\circ$ angle is also $90^\circ$. By applying these two properties (linear pair and vertical angles) to each of the four angles formed by the intersection of lines $j$ and $k$, we conclude that all four angles must be right angles.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 14 (a)

Step 1: Find \(m\angle WRT\)

Step 2: Find \(m\angle QRV\)

Step 3: Find \(m\angle QRS\)

Answer:

Question 14 (b)