Question

1. if m∠2 = 7x + 7, m∠3 = 4y, and m∠4 = 112, find the values of x and y.

options:
○ x=17, y=15
○ x=15, y=17
○ x=112, y=68
○ x=68, y=112
(image: two intersecting lines forming four angles labeled 1, 2, 3, 4)

Explanation:

Step1: Find y using vertical angles

∠3 and ∠4 are adjacent supplementary angles? Wait, no, ∠2 and ∠4 are vertical? Wait, looking at the intersecting lines, ∠2 and ∠4: Wait, actually, ∠3 and ∠4 are adjacent, forming a linear pair? Wait, no, when two lines intersect, vertical angles are equal, and linear pairs are supplementary (sum to 180°). Wait, ∠2 and ∠4: Wait, ∠3 and ∠4: Wait, let's see the diagram. Angles 2 and 4: Wait, angle 2 and angle 4 are vertical? No, angle 2 and angle 4: Wait, angle 3 and angle 1 are vertical, angle 2 and angle 4 are vertical? Wait, no, when two lines intersect, opposite angles are vertical (equal). So angle 2 and angle 4: Wait, m∠4 is 112, so angle 2 should be equal to angle 4? Wait, no, angle 2 and angle 4: Wait, maybe angle 3 and angle 4 are supplementary? Wait, no, let's re-examine.

Wait, the problem: m∠2 = 7x +7, m∠3 =4y, m∠4=112.

First, angle 2 and angle 4: are they vertical angles? Wait, when two lines intersect, vertical angles are equal. So angle 2 and angle 4: Wait, no, angle 2 and angle 3 are adjacent, forming a linear pair (sum to 180°), angle 3 and angle 4 are adjacent, forming a linear pair (sum to 180°), angle 4 and angle 1, etc. Wait, no, angle 2 and angle 4: Let's see, angle 2 and angle 4: if the lines are intersecting, then angle 2 and angle 4 are vertical angles? Wait, no, angle 2 and angle 4: Wait, maybe angle 2 and angle 4 are equal? Wait, m∠4 is 112, so angle 2 should be equal to angle 4? Wait, no, angle 2 and angle 3 are supplementary (sum to 180°), angle 3 and angle 4 are supplementary (sum to 180°), so angle 2 = angle 4 (since both are supplementary to angle 3). So m∠2 = m∠4. So 7x +7 = 112.

Also, angle 3 and angle 4 are supplementary? Wait, no, angle 3 and angle 4: if angle 4 is 112, then angle 3 + angle 4 = 180? Wait, no, angle 3 and angle 4: let's see, angle 3 and angle 4 are adjacent, forming a linear pair, so they should sum to 180. Wait, no, angle 3 and angle 4: if angle 4 is 112, then angle 3 = 180 - 112 = 68? Wait, no, wait, angle 3 and angle 2: angle 2 + angle 3 = 180 (linear pair). So angle 3 = 180 - angle 2. But angle 2 is equal to angle 4 (vertical angles), so angle 2 = 112, so angle 3 = 180 - 112 = 68. But m∠3 =4y, so 4y = 68? Wait, no, that would make y=17. Wait, let's do step by step.

Step1: Find x using vertical angles (∠2 and ∠4)

∠2 and ∠4 are vertical angles, so they are equal. So m∠2 = m∠4.
Given m∠2 = 7x +7, m∠4 =112.
So 7x +7 = 112.
Subtract 7 from both sides: 7x = 112 -7 = 105.
Divide by 7: x = 105 /7 =15.

Step2: Find y using linear pair (∠3 and ∠4)

∠3 and ∠4 form a linear pair, so they are supplementary (sum to 180°).
So m∠3 + m∠4 = 180.
Given m∠3 =4y, m∠4=112.
So 4y +112 = 180.
Subtract 112: 4y = 180 -112 =68.
Divide by 4: y=68/4=17.

Answer:

x = 15, y = 17 (Option: x=15, y=17)