in the figure below, a || b, and lines c and d intersect at a 42° angle.
what is the measure of x?
select one answer
a 56°
b 72°
c 82°
d 124°

Question

Accepted Answer

# Explanation:
## Step1: Analyze the angle on line a
Since lines $a \parallel b$, and we know one angle on line $b$ with line $c$ is $124^\circ$, the corresponding angle on line $a$ with line $c$ should also be related. But first, let's find the supplementary or related angles. Wait, actually, let's consider the triangle or the angle sum. Wait, the angle between $c$ and $d$ is $42^\circ$, and we have a $124^\circ$ angle on line $b$ with line $c$. Let's find the angle between line $c$ and line $a$ first. Wait, maybe using the fact that $a \parallel b$, so the angle formed by line $c$ and $a$ should be supplementary to $124^\circ$? Wait, no, $180 - 124 = 56^\circ$. Then, in the triangle (or the intersection of lines $c$, $d$, and $a$), we have angles: $56^\circ$, $42^\circ$, and the third angle. Wait, the sum of angles in a triangle is $180^\circ$, so $180 - 56 - 42 = 82^\circ$? No, that's not right. Wait, maybe alternate interior angles. Wait, actually, let's look at the angle $x$. Since $a \parallel b$, the angle $x$ should be equal to the angle formed by line $d$ and $a$ (alternate interior angles). Let's find the angle between line $d$ and $a$. The angle between $c$ and $d$ is $42^\circ$, and the angle between $c$ and $a$ is $180 - 124 = 56^\circ$ (since $a \parallel b$, consecutive interior angles are supplementary). Then, the angle between $d$ and $a$ is $180 - 56 - 42 = 82^\circ$? No, that's not matching. Wait, maybe I made a mistake. Wait, the correct approach: since $a \parallel b$, the angle $x$ is equal to the angle formed by line $d$ and $a$ (alternate interior angles). Let's find the angle at the intersection of $c$, $d$, and $a$. The angle between $c$ and $a$ is $180 - 124 = 56^\circ$ (because $a \parallel b$, so consecutive interior angles are supplementary: $124 + \text{angle on } a = 180$, so angle on $a$ is $56^\circ$). Then, the angle between $d$ and $a$ is $180 - 56 - 42 = 82^\circ$? No, that's not. Wait, no, the angle between $c$ and $d$ is $42^\circ$, the angle between $c$ and $a$ is $56^\circ$, so the angle between $d$ and $a$ is $180 - 56 - 42 = 82^\circ$? But that's not one of the options. Wait, no, maybe I messed up. Wait, the answer options include $124^\circ$? Wait, no, the options are 56, 72, 82, 124. Wait, maybe the angle $x$ is equal to $124^\circ$? No, that can't be. Wait, wait, let's re-examine. The angle on line $b$ with line $c$ is $124^\circ$, so the angle on line $a$ with line $c$ is also $124^\circ$ (corresponding angles, since $a \parallel b$). Then, the angle between line $c$, $d$, and $a$: we have angle $124^\circ$ (between $c$ and $a$), angle $42^\circ$ (between $c$ and $d$), so the angle between $d$ and $a$ is $180 - 124 - 42 = 14^\circ$? No, that's wrong. Wait, I think I confused the diagram. Let's look again. The diagram shows line $b$ with a $124^\circ$ angle with line $c$, line $a$ above $b$, parallel. Lines $c$ and $d$ intersect at $42^\circ$ on line $a$. So, the angle between $c$ and $d$ is $42^\circ$, and the angle between $c$ and $b$ is $124^\circ$. Since $a \parallel b$, the angle between $c$ and $a$ is also $124^\circ$ (corresponding angles). Then, in the triangle formed by lines $c$, $d$, and $a$, the angles are $124^\circ$ (between $c$ and $a$), $42^\circ$ (between $c$ and $d$), so the angle between $d$ and $a$ is $180 - 124 - 42 = 14^\circ$? No, that's not. Wait, maybe the angle $x$ is equal to $124^\circ - 42^\circ$? No. Wait, the correct answer is $82^\circ$? Wait, no, the options have $82^\circ$ as option C. Wait, maybe my initial approach was wrong. Let's use the exterior angle theorem. The angle $124^\circ$ is an exterior angle to the triangle formed by lines $c$, $d$, and $b$. So, the exterior angle is equal to the sum of the two remote interior angles. So, $124 = 42 + x$? No, $124 - 42 = 82$. Ah! That's it. So, since $a \parallel b$, the angle $124^\circ$ is an exterior angle to the triangle (or the angle formed by $d$, $c$, and $b$), so $124 = 42 + x$? Wait, no, exterior angle theorem: the exterior angle is equal to the sum of the two non-adjacent interior angles. So, if we consider the angle at $b$ with line $c$ as $124^\circ$, and the angle between $c$ and $d$ as $42^\circ$, then the angle $x$ (between $d$ and $b$) and $42^\circ$ should add up to $124^\circ$? Wait, $124 - 42 = 82$. Yes! So $x = 124 - 42 = 82^\circ$. Wait, but the option D is $124^\circ$, which is wrong. Wait, no, maybe I mixed up. Wait, let's check again. The angle between $c$ and $b$ is $124^\circ$, the angle between $c$ and $d$ is $42^\circ$, so the angle between $d$ and $b$ (which is $x$) should be $124 - 42 = 82^\circ$. So, the measure of $x$ is $82^\circ$, which is option C. Wait, but the user's diagram shows D selected, but that's wrong. Wait, maybe my diagram analysis is wrong. Wait, no, let's do it properly. Since $a \parallel b$, the alternate interior angles: the angle between $d$ and $a$ is equal to $x$. The angle between $c$ and $a$ is $180 - 124 = 56^\circ$ (consecutive interior angles). Then, in the triangle formed by $c$, $d$, and $a$, the angles are $56^\circ$ (between $c$ and $a$), $42^\circ$ (between $c$ and $d$), so the angle between $d$ and $a$ is $180 - 56 - 42 = 82^\circ$. So, $x = 82^\circ$, which is option C.
## Step2: Confirm the calculation
Using the exterior angle theorem: the exterior angle ($124^\circ$) is equal to the sum of the two remote interior angles ($42^\circ$ and $x$? No, wait, the exterior angle is equal to the sum of the two non-adjacent interior angles. Wait, actually, if we have a triangle with angles $x$, $42^\circ$, and the angle adjacent to $124^\circ$ (which is $180 - 124 = 56^\circ$), then $x + 42 + 56 = 180$, so $x = 180 - 56 - 42 = 82^\circ$. Yes, that's correct.

# Answer:
C. $82^\circ$

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Explanation:

Step1: Analyze the angle on line a

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