Question

determining an unknown angle measure
the measure of angle 1 is (3x + 10)° and the measure of angle 4 is (4x − 15)°. what is the measure of angle 7?
diagram: horizontal line ( a ) intersects two parallel vertical lines ( c ) (left) and ( b ) (right). angles at ( a )-( c ) intersection: ( 1 ) (bottom-left), ( 2 ) (top-left), ( 3 ) (bottom-right), ( 4 ) (top-right). angles at ( a )-( b ) intersection: ( 5 ) (bottom-left), ( 6 ) (top-left), ( 7 ) (bottom-right), ( 8 ) (top-right).

Explanation:

Step1: Identify relationship between angle 1 and angle 4

Angle 1 and angle 4 are vertical angles? No, wait, looking at the diagram, lines c and b are parallel? Wait, no, angle 1 and angle 4: actually, angle 1 and angle 4 are adjacent and form a linear pair? Wait, no, the diagram shows two parallel lines (c and b) cut by a transversal (line a). Wait, angle 1 and angle 4: actually, angle 1 and angle 4 are equal? Wait, no, maybe angle 1 and angle 4 are alternate interior or something? Wait, no, the problem: angle 1 is (3x + 10)° and angle 4 is (4x - 15)°. Wait, maybe angle 1 and angle 4 are equal? Wait, no, looking at the diagram, angle 1 and angle 4: if lines c and b are parallel, but actually, the two vertical lines (c and b) are parallel, and the horizontal line is the transversal. Wait, angle 1 and angle 4: maybe they are equal? Wait, no, angle 1 and angle 4: let's see, angle 1 and angle 3 are vertical, angle 4 and angle 2 are vertical. Wait, maybe angle 1 and angle 4 are equal? Wait, no, maybe they are supplementary? Wait, no, the problem: maybe angle 1 and angle 4 are equal because they are corresponding angles? Wait, no, let's think again. Wait, the two vertical lines (c and b) are parallel, so the transversal (the horizontal line) creates corresponding angles. Wait, angle 1 and angle 5? No, angle 4 and angle 6? Wait, maybe angle 1 and angle 4 are equal? Wait, the problem says "the measure of angle 1 is (3x + 10)° and the measure of angle 4 is (4x - 15)°". Wait, maybe angle 1 and angle 4 are equal (vertical angles? No, vertical angles are opposite. Wait, angle 1 and angle 3 are vertical, angle 4 and angle 2 are vertical. Wait, angle 1 and angle 4: maybe they are equal because lines c and b are parallel, so angle 1 and angle 5 are equal, angle 4 and angle 6 are equal, but angle 5 and angle 7 are vertical, angle 6 and angle 8 are vertical. Wait, maybe angle 1 and angle 4 are equal? Wait, let's solve for x first. If angle 1 and angle 4 are equal (maybe they are alternate interior angles? Wait, no, angle 1 is at line c, angle 4 is also at line c. Wait, maybe angle 1 and angle 4 are supplementary? No, angle 1 and angle 2 are supplementary, angle 2 and angle 4 are vertical. Wait, maybe angle 1 and angle 4 are equal. Let's assume angle 1 = angle 4 (maybe they are corresponding angles or something). So set 3x + 10 = 4x - 15.

Step2: Solve for x

3x + 10 = 4x - 15
Subtract 3x from both sides: 10 = x - 15
Add 15 to both sides: x = 25

Step3: Find measure of angle 1

Substitute x = 25 into angle 1: 3(25) + 10 = 75 + 10 = 85°

Step4: Find measure of angle 7

Angle 1 and angle 7: angle 1 and angle 5 are equal (corresponding angles, since lines c and b are parallel, transversal a), and angle 5 and angle 7 are vertical angles, so angle 5 = angle 7. Therefore, angle 1 = angle 7. So angle 7 is 85°.

Answer:

85