Question

line b intersects line c as shown below. the sum of ∠y and ∠z is 296°. what is the measure of ∠x?

Explanation:

Step1: Identify vertical angles

∠y and ∠z are adjacent to ∠x, and ∠y and ∠z are vertical angles? No, wait, when two lines intersect, vertical angles are equal, and adjacent angles are supplementary (sum to 180°). Wait, actually, ∠y and ∠z: wait, looking at the diagram, when line b and line c intersect, the sum of ∠y and ∠z is 296°, but also, ∠y and ∠z are vertical angles? No, maybe ∠y and ∠z are equal? Wait, no, the problem says sum of ∠y and ∠z is 296°. Wait, actually, when two lines intersect, the sum of all angles around a point is 360°. So ∠x + ∠y + ∠x's vertical angle + ∠z = 360°? Wait, no, let's correct. When two lines intersect, the angles around the intersection point sum to 360°. Also, vertical angles are equal. So ∠y and ∠z: wait, maybe ∠y and ∠z are vertical angles? No, the diagram shows line b (horizontal) and line c (slanted) intersecting. So the four angles: let's label them. Let's say the intersection creates four angles: ∠x, ∠y, ∠z, and the angle opposite to ∠x (let's call it ∠w). So ∠x and ∠w are vertical angles (equal), ∠y and ∠z: wait, maybe ∠y and ∠z are adjacent? Wait, the problem says sum of ∠y and ∠z is 296°. Also, ∠y and ∠z: since they are adjacent to the intersection, maybe ∠y and ∠z are vertical angles? No, vertical angles are equal, so if their sum is 296°, each would be 148°, but then ∠x would be supplementary to ∠y (since they are adjacent, forming a linear pair). Wait, linear pair: adjacent angles on a straight line sum to 180°. So ∠x + ∠y = 180°, and ∠x + ∠z = 180°? No, that can't be unless ∠y = ∠z. Wait, the problem says sum of ∠y and ∠z is 296°, so if ∠y = ∠z, then each is 148°, but maybe ∠y and ∠z are vertical angles? Wait, no, vertical angles are equal, so if their sum is 296°, each is 148°. Then ∠x is supplementary to ∠y (since they form a linear pair), so ∠x = 180° - 148° = 32°? Wait, let's do it properly.

Step2: Sum of angles around a point

The sum of all angles around the intersection point is 360°. So ∠x + ∠y + ∠x's vertical angle + ∠z = 360°. But vertical angles are equal, so ∠x = ∠x's vertical angle. So 2∠x + ∠y + ∠z = 360°. We know ∠y + ∠z = 296°, so 2∠x + 296° = 360°. Then 2∠x = 360° - 296° = 64°, so ∠x = 32°. Wait, that makes sense. Because around the point, sum is 360°, so 2∠x (since ∠x and its vertical angle are equal) plus ∠y and ∠z (sum 296°) equals 360°. So 2∠x = 360 - 296 = 64, so ∠x = 32.

Step1: Sum of angles around a point

When two lines intersect, the sum of all angles around the intersection point is \( 360^\circ \). Let the measure of \( \angle x \) be \( x \). The vertical angle of \( \angle x \) is also \( x \) (vertical angles are equal). The sum of \( \angle y \) and \( \angle z \) is given as \( 296^\circ \).

Step2: Set up the equation

The total sum of angles around the point is:
\( x + x + \angle y + \angle z = 360^\circ \)
Substitute \( \angle y + \angle z = 296^\circ \):
\( 2x + 296^\circ = 360^\circ \)

Step3: Solve for \( x \)

Subtract \( 296^\circ \) from both sides:
\( 2x = 360^\circ - 296^\circ = 64^\circ \)
Divide by 2:
\( x = \frac{64^\circ}{2} = 32^\circ \)

Answer:

\( 32^\circ \)