use the figure and given information to answer questions 1-6. determine if each statement is true (t) or false (f). vertical angles: ∠1 and ∠2 linear pairs: ∠1 and ∠3, ∠1 and ∠4 (hint: add angle numbers to the figure) 1. if m∠3 = 30°, then m∠4 = 150° t f 2. if m∠1 = 140°, then m∠4 = 40° t f 3. ∠2 and ∠4 are congruent t f 4. m∠3 + m∠1 = m∠4 + m∠2 t f 5. ∠3 ≅ ∠4 t f 6. m∠3 = 180° − m∠2 t f

Question

Accepted Answer

### Question 1:
# Explanation:
## Step1: Recall linear pair property
Linear pair angles sum to $180^\circ$. $\angle3$ and $\angle4$ form a linear pair.
## Step2: Calculate $m\angle4$
If $m\angle3 = 30^\circ$, then $m\angle4=180^\circ - 30^\circ = 150^\circ$.
# Answer: T

### Question 2:
# Explanation:
## Step1: Recall linear pair property
$\angle1$ and $\angle4$ are a linear pair, so $m\angle1 + m\angle4 = 180^\circ$.
## Step2: Calculate $m\angle4$
If $m\angle1 = 140^\circ$, then $m\angle4 = 180^\circ - 140^\circ = 40^\circ$.
# Answer: T

### Question 3:
# Explanation:
## Step1: Recall vertical angles
$\angle1$ and $\angle2$ are vertical angles (congruent). $\angle1$ and $\angle4$ are linear pair, $\angle2$ and $\angle3$ are linear pair.
## Step2: Relate $\angle2$ and $\angle4$
Since $\angle1=\angle2$ and $m\angle1 + m\angle4 = 180^\circ$, $m\angle2 + m\angle4 = 180^\circ$ only if $\angle2$ and $\angle4$ are supplementary, not necessarily congruent. Wait, correction: $\angle2$ and $\angle4$: $\angle1$ and $\angle2$ are vertical (so $m\angle1 = m\angle2$), $\angle1$ and $\angle4$ are linear pair ($m\angle1 + m\angle4 = 180^\circ$), so $m\angle2 + m\angle4 = 180^\circ$ – no, wait, actually $\angle2$ and $\angle4$: let's see, $\angle2$ and $\angle3$ are linear pair, $\angle3$ and $\angle4$ are linear pair. So $\angle2$ and $\angle4$: since $\angle2 + \angle3 = 180^\circ$ and $\angle3 + \angle4 = 180^\circ$, then $\angle2=\angle4$ (subtracting $\angle3$ from both equations). Oh right! So $\angle2$ and $\angle4$ are congruent.
# Answer: T

### Question 4:
# Explanation:
## Step1: Recall linear pair and vertical angles
$\angle1$ and $\angle3$ are linear pair ($m\angle1 + m\angle3 = 180^\circ$), $\angle2$ and $\angle4$ are congruent (from Q3), and $\angle1$ and $\angle2$ are vertical (so $m\angle1 = m\angle2$), $\angle3$ and $\angle4$ are supplementary? Wait no, $\angle3$ and $\angle4$ are linear pair ($m\angle3 + m\angle4 = 180^\circ$). Also, $m\angle1 = m\angle2$ (vertical angles), $m\angle3 + m\angle1 = 180^\circ$, $m\angle4 + m\angle2 = m\angle4 + m\angle1$ (since $m\angle2 = m\angle1$). But $m\angle3 + m\angle4 = 180^\circ$, and $m\angle1 = m\angle2$. Wait, let's substitute: $m\angle3 + m\angle1$ and $m\angle4 + m\angle2$. Since $m\angle2 = m\angle1$, this is $m\angle3 + m\angle1$ vs $m\angle4 + m\angle1$. But $m\angle3 + m\angle4 = 180^\circ$, and $m\angle1 + m\angle3 = 180^\circ$ (linear pair), so $m\angle1 = 180^\circ - m\angle3$, and $m\angle2 = 180^\circ - m\angle3$. Then $m\angle4 + m\angle2 = m\angle4 + (180^\circ - m\angle3)$. But $m\angle4 = 180^\circ - m\angle3$ (linear pair of $\angle3$), so substituting: $(180^\circ - m\angle3) + (180^\circ - m\angle3) = 360^\circ - 2m\angle3$. Wait, no, I messed up. Wait, $\angle1$ and $\angle3$ are linear pair: $m\angle1 + m\angle3 = 180^\circ$. $\angle2$ and $\angle4$: from Q3, $\angle2 = \angle4$, and $\angle1 = \angle2$ (vertical angles), so $\angle1 = \angle2 = \angle4$. Then $m\angle4 + m\angle2 = 2m\angle1$. And $m\angle3 + m\angle1 = 180^\circ$. Wait, no, let's take actual values. Let $m\angle3 = 30^\circ$, then $m\angle1 = 150^\circ$ (linear pair), $m\angle2 = 150^\circ$ (vertical to $\angle1$), $m\angle4 = 150^\circ$ (from Q3). Then $m\angle3 + m\angle1 = 30 + 150 = 180$, $m\angle4 + m\angle2 = 150 + 150 = 300$. Wait, that's not equal. Wait, no, I made a mistake in Q3. Wait, $\angle2$ and $\angle4$: $\angle1$ and $\angle2$ are vertical (so $m\angle1 = m\angle2$), $\angle1$ and $\angle4$ are linear pair ($m\angle1 + m\angle4 = 180^\circ$), so $m\angle2 + m\angle4 = 180^\circ$, not congruent. Oh! I messed up Q3 earlier. So $\angle2$ and $\angle4$ are supplementary, not congruent. So let's redo Q3: $\angle1$ and $\angle2$ are vertical (congruent), $\angle1$ and $\angle4$ are linear pair (supplementary), so $\angle2$ and $\angle4$ are supplementary (since $\angle2 = \angle1$, so $\angle2 + \angle4 = \angle1 + \angle4 = 180^\circ$). So they are not congruent unless each is $90^\circ$. So Q3 is F. Then Q4: $m\angle3 + m\angle1 = 180^\circ$ (linear pair), $m\angle4 + m\angle2$: $\angle2 = \angle1$ (vertical), $\angle4 = 180^\circ - \angle1$ (linear pair with $\angle1$), so $m\angle4 + m\angle2 = (180^\circ - \angle1) + \angle1 = 180^\circ$. So $m\angle3 + m\angle1 = 180^\circ$ and $m\angle4 + m\angle2 = 180^\circ$, so they are equal. Ah, right! Because $m\angle3 + m\angle1 = 180^\circ$ (linear pair), and $m\angle4 + m\angle2$: $\angle2 = \angle1$ (vertical), $\angle4 = 180^\circ - \angle1$ (linear pair with $\angle1$), so $m\angle4 + m\angle2 = (180^\circ - \angle1) + \angle1 = 180^\circ$. So both sums are $180^\circ$, so equal.
# Answer: T

### Question 5:
# Explanation:
## Step1: Recall linear pair
$\angle3$ and $\angle4$ are a linear pair, so $m\angle3 + m\angle4 = 180^\circ$. They are congruent only if each is $90^\circ$, which isn't given. So $\angle3
ot\cong \angle4$ (unless right angles, but no info).
# Answer: F

### Question 6:
# Explanation:
## Step1: Recall vertical and linear pairs
$\angle1$ and $\angle2$ are vertical (so $m\angle1 = m\angle2$), $\angle1$ and $\angle3$ are linear pair ($m\angle1 + m\angle3 = 180^\circ$). Substitute $m\angle1$ with $m\angle2$: $m\angle2 + m\angle3 = 180^\circ$, so $m\angle3 = 180^\circ - m\angle2$.
# Answer: T

### Final Answers for Each Question:
1. T
2. T
3. F (corrected from earlier mistake: $\angle2$ and $\angle4$ are supplementary, not congruent)
4. T
5. F
6. T

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QUESTION IMAGE

Question

Explanation:

Question 1:

Step1: Recall linear pair property

Step2: Calculate \(m\angle4\)

Step1: Recall linear pair property

Step2: Calculate \(m\angle4\)

Step1: Recall vertical angles

Step2: Relate \(\angle2\) and \(\angle4\)

Answer:

Question 2: