Question

the measure of angle 1 is $(3x + 10)^{circ}$ and the measure of angle 4 is $(4x - 15)^{circ}$. what is the measure of angle 7?

Explanation:

Step1: Find the value of \(x\)

Since angle \(1\) and angle \(4\) are supplementary (they form a linear - pair), \((3x + 10)+(4x-15)=180\).
Combine like terms: \(3x+4x+10 - 15=180\), \(7x-5 = 180\).
Add \(5\) to both sides: \(7x=180 + 5=185\).
Divide both sides by \(7\): \(x=\frac{185}{7}\) is wrong. Wait, no, actually, angle \(1\) and angle \(4\) are vertical angles (assuming lines \(a\) and \(b\) are parallel and \(c\) is a transversal, angle \(1\) and angle \(4\) are not supplementary. Angle \(1\) and angle \(2\) are supplementary, angle \(4\) and angle \(2\) are supplementary. Wait, no, angle \(1\) and angle \(4\) are vertical angles. So \(3x + 10=4x-15\).
Subtract \(3x\) from both sides: \(10=x - 15\).
Add \(15\) to both sides: \(x=25\).

Step2: Find the measure of angle \(1\)

Substitute \(x = 25\) into the expression for angle \(1\): \(\angle1=(3x + 10)^{\circ}=(3\times25+10)^{\circ}=(75 + 10)^{\circ}=85^{\circ}\).

Step3: Use the property of parallel lines and transversals

Since lines \(c\) and \(b\) are parallel and \(a\) is a transversal, angle \(1\) and angle \(5\) are congruent (alternate interior angles). Angle \(5\) and angle \(7\) are vertical angles. So \(\angle7=\angle5=\angle1\).

Answer:

\(85\)