Question

1. multiple choice: which of the following must be true? (a) m∠4 = m∠7 (b) m∠6 = m∠7 (c) m∠2 = m∠7 (d) m∠4 = m∠8 if m∠3=(3x + 32)° and m∠6=(4x + 21)°

Explanation:

Step1: Identify angle - relationship

Since $\angle3$ and $\angle6$ are alternate - exterior angles, when two parallel lines are cut by a transversal, alternate - exterior angles are congruent. So, if the lines are parallel, $m\angle3=m\angle6$.
Set up the equation: $3x + 32=4x + 21$.

Step2: Solve the equation for $x$

Subtract $3x$ from both sides: $32=x + 21$.
Subtract 21 from both sides: $x=32 - 21=11$.

Step3: Check angle - equalities

For option (A):
$m\angle4$ and $m\angle7$ are alternate - interior angles. If the lines are parallel, $m\angle4 = m\angle7$.
For option (B):
$m\angle6$ and $m\angle7$ are corresponding angles. If the lines are parallel, $m\angle6 = m\angle7$.
For option (C):
$m\angle2$ and $m\angle7$ are not in a special angle - relationship (like alternate - interior, alternate - exterior, corresponding or vertical) that would guarantee equality without further information.
For option (D):
$m\angle4$ and $m\angle8$ are not in a special angle - relationship that would guarantee equality without further information.
Since when two parallel lines are cut by a transversal, alternate - interior angles are congruent, $m\angle4 = m\angle7$.

Answer:

A. $m\angle4 = m\angle7$