select all of the statements that must be true if m∠4 = m∠6.
image of angle diagram with lines a, b, c and angles 1-8
a. a ⊥ b
b. m∠4 = 90
c. m∠2 ≠ m∠8
d. m∠3 ≠ m∠4
e. m∠3 = m∠7

Question

Accepted Answer

# Explanation:
## Step1: Analyze ∠4 and ∠6 relationship
∠4 and ∠6 are same - side interior angles. If $m\angle4 = m\angle6$, and for same - side interior angles, when $m\angle4=m\angle6$, since same - side interior angles are supplementary (i.e., $m\angle4 + m\angle6=180^{\circ}$) in general, if $m\angle4 = m\angle6$, then $m\angle4=m\angle6 = 90^{\circ}$. So lines $b$ and $c$ are parallel? Wait, no, ∠4 and ∠6 are formed by transversal $a$ cutting $b$ and $c$. Wait, actually, ∠4 and ∠6: ∠4 and ∠5 are supplementary (linear pair), ∠5 and ∠6 are vertical angles? No, ∠4 and ∠6: let's look at the lines. Line $a$ is a transversal, lines $b$ and $c$ are two lines. ∠4 and ∠6: if $m\angle4 = m\angle6$, and ∠4 + ∠6 = 180° (same - side interior angles), then $2m\angle4=180^{\circ}$, so $m\angle4 = 90^{\circ}$, so line $a$ is perpendicular to lines $b$ and $c$? Wait, no, ∠4 and ∠6: if $m\angle4=m\angle6$ and they are same - side interior angles, then they must be right angles. So $a\perp b$ and $a\perp c$, so $b\parallel c$ (if two lines are perpendicular to the same transversal, they are parallel). Now let's check each option:

- Option A: $a\perp b$. Since $m\angle4 = 90^{\circ}$ (because $m\angle4=m\angle6$ and $m\angle4 + m\angle6 = 180^{\circ}$), so the angle between $a$ and $b$ is $90^{\circ}$, so $a\perp b$. This is true.
- Option B: $m\angle4 = 90^{\circ}$. As we saw, since $m\angle4=m\angle6$ and $m\angle4 + m\angle6=180^{\circ}$, solving $2m\angle4 = 180^{\circ}$ gives $m\angle4 = 90^{\circ}$. This is true.
- Option C: $m\angle2
eq m\angle8$. ∠2 and ∠8: ∠2 and ∠4 are vertical angles? No, ∠2 and ∠3 are vertical angles, ∠4 and ∠1 are vertical angles. Wait, ∠2 and ∠8: ∠2 and ∠4 are supplementary (linear pair), ∠4 = 90°, so ∠2 = 90°. ∠8 and ∠6 are vertical angles, ∠6 = 90°, so ∠8 = 90°. So $m\angle2=m\angle8 = 90^{\circ}$, so this statement is false.
- Option D: $m\angle3
eq m\angle4$. ∠3 and ∠4: they are adjacent angles forming a linear pair? Wait, ∠3 and ∠4: if $a\perp b$, then ∠3 and ∠4 are both 90°? Wait, no, ∠3 and ∠4: if $a\perp b$, then the angle between $a$ and $b$ is 90°, so ∠3 and ∠4 are complementary? Wait, no, ∠3 + ∠4=180°? No, ∠3 and ∠4: if $a\perp b$, then ∠3 and ∠4 are both 90°? Wait, no, in the diagram, ∠3 and ∠4 are adjacent angles at the intersection of $a$ and $b$. If $a\perp b$, then ∠3 = ∠4 = 90°, so $m\angle3=m\angle4$, so this statement is false.
- Option E: $m\angle3 = m\angle7$. ∠3 and ∠7: ∠3 and ∠4 are 90° (if $a\perp b$), ∠7 and ∠6 are vertical angles, ∠6 = 90°, so ∠7 = 90°. ∠3 = 90°, so $m\angle3=m\angle7$. This is true. Wait, but let's re - examine. If $a\perp b$ and $a\perp c$, then $b\parallel c$ (corresponding angles are equal, since both are 90° with transversal $a$). Then ∠3 and ∠7: ∠3 and ∠5 are equal (alternate interior angles, since $b\parallel c$), and ∠5 and ∠7 are vertical angles, so ∠3 = ∠5 = ∠7. So $m\angle3 = m\angle7$ is true. Wait, but earlier when we thought about option D, we made a mistake. Wait, ∠3 and ∠4: if $a\perp b$, then ∠3 and ∠4 are both 90°? Wait, no, ∠3 and ∠4: in the intersection of $a$ and $b$, ∠3 and ∠4 are adjacent angles. If $a\perp b$, then the four angles formed are all 90°, so ∠3 = ∠4 = 90°, so option D is false.

Wait, let's start over. Given $m\angle4=m\angle6$. ∠4 and ∠6 are same - side interior angles (formed by transversal $a$ cutting lines $b$ and $c$). The same - side interior angles theorem states that if two lines are parallel, then same - side interior angles are supplementary. But here $m\angle4 = m\angle6$, so $m\angle4+m\angle6 = 2m\angle4=180^{\circ}$, so $m\angle4 = 90^{\circ}$. So:

- A: $a\perp b$: since the angle between $a$ and $b$ at their intersection is ∠4 (or ∠1, ∠2, ∠3), and $m\angle4 = 90^{\circ}$, so $a\perp b$. True.
- B: $m\angle4 = 90^{\circ}$: as shown, since $m\angle4=m\angle6$ and $m\angle4 + m\angle6=180^{\circ}$, so $m\angle4 = 90^{\circ}$. True.
- C: $m\angle2
eq m\angle8$: ∠2 and ∠4 are supplementary (linear pair), so $m\angle2 = 180 - 90=90^{\circ}$. ∠8 and ∠6 are vertical angles, $m\angle6 = 90^{\circ}$, so $m\angle8 = 90^{\circ}$. So $m\angle2=m\angle8$, so this is false.
- D: $m\angle3
eq m\angle4$: ∠3 and ∠4 are adjacent angles forming a linear pair? Wait, no, ∠3 and ∠4: if $a\perp b$, then ∠3 and ∠4 are both 90° (since the intersection of two perpendicular lines forms four right angles). So $m\angle3=m\angle4 = 90^{\circ}$, so this is false.
- E: $m\angle3 = m\angle7$: ∠3 = 90° (from $a\perp b$), ∠7 and ∠6 are vertical angles, $m\angle6 = 90^{\circ}$, so $m\angle7 = 90^{\circ}$. So $m\angle3=m\angle7$. True. Wait, but let's check again. If $a\perp b$ and $a\perp c$, then $b\parallel c$ (converse of the corresponding angles theorem, since corresponding angles are 90°). Then ∠3 and ∠7: ∠3 and ∠5 are alternate interior angles (since $b\parallel c$), so $m\angle3=m\angle5$. ∠5 and ∠7 are vertical angles, so $m\angle5=m\angle7$. Thus $m\angle3=m\angle7$. So E is true.

Wait, but the problem says "select all that must be true". So A, B, E? Wait, no, let's re - check A: $a\perp b$. Since $m\angle4 = 90^{\circ}$, which is the angle between $a$ and $b$, so $a\perp b$ is true. B: $m\angle4 = 90^{\circ}$ is true. E: $m\angle3 = m\angle7$ is true. But wait, maybe I made a mistake with E. Let's see, ∠3 and ∠7: if $b\parallel c$ (because $a\perp b$ and $a\perp c$), then ∠3 and ∠7 are corresponding angles? Wait, ∠3 and ∠7: ∠3 is at the intersection of $a$ and $b$, ∠7 is at the intersection of $a$ and $c$. Since $b\parallel c$, corresponding angles are equal. ∠3 and ∠7: are they corresponding angles? Let's see the positions. ∠3 and ∠7: ∠3 is above line $b$, left of transversal $a$; ∠7 is below line $c$, left of transversal $a$. Wait, maybe alternate exterior angles? No, ∠3 and ∠7: if $b\parallel c$, then ∠3 (which is equal to ∠1, vertical angles) and ∠7 (which is equal to ∠5, vertical angles). And since $b\parallel c$, ∠1 and ∠5 are corresponding angles? Wait, maybe my initial analysis of E is correct.

But let's check the options again. The options are A, B, C, D, E.

Wait, maybe the correct answers are A, B, E? But let's check with the diagram again.

Wait, ∠4 and ∠6: same - side interior angles. If $m\angle4 = m\angle6$, then $m\angle4 + m\angle6=2m\angle4 = 180^{\circ}$, so $m\angle4 = 90^{\circ}$. So:

- A: $a\perp b$: yes, because the angle between $a$ and $b$ is $90^{\circ}$ (∠4 is 90°).
- B: $m\angle4 = 90^{\circ}$: yes, as shown.
- C: $m\angle2
eq m\angle8$: ∠2 and ∠4 are supplementary, so ∠2 = 90°, ∠8 and ∠6 are vertical angles, ∠6 = 90°, so ∠8 = 90°, so $m\angle2=m\angle8$, so C is false.
- D: $m\angle3
eq m\angle4$: ∠3 and ∠4 are adjacent angles forming a linear pair? No, ∠3 and ∠4 are adjacent angles at the intersection of $a$ and $b$, and since $a\perp b$, they are both 90°, so $m\angle3=m\angle4$, so D is false.
- E: $m\angle3 = m\angle7$: since $b\parallel c$ (because $a\perp b$ and $a\perp c$), ∠3 and ∠7 are equal (corresponding angles or alternate interior angles). So E is true.

So the correct answers are A, B, E? Wait, but maybe the problem is designed so that A, B, E are correct. But let's check with the initial equations.

Alternatively, maybe I made a mistake with E. Let's see, ∠3 and ∠7: ∠3 is 90°, ∠7 is 90°, so they are equal. So E is true.

So the statements that must be true are A, B, E.

# Answer:
A. $a \perp b$, B. $m\angle4 = 90$, E. $m\angle3 = m\angle7$

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QUESTION IMAGE

Question

Explanation:

Step1: Analyze ∠4 and ∠6 relationship

Answer: