Question

the measure of angle 1 is (3x + 10)° and the measure of angle 4 is (4x - 15)°. what is the measure of angle 7?°

Explanation:

Step1: Find the value of \(x\)

Since \(\angle1\) and \(\angle4\) are supplementary (they form a linear pair), we have \((3x + 10)+(4x-15)=180\).
Simplify the left - hand side: \(3x+10 + 4x-15=7x - 5\).
So, \(7x-5 = 180\).
Add \(5\) to both sides: \(7x=180 + 5=185\).
Divide both sides by \(7\): \(x=\frac{185}{7}\) is incorrect. Wait, no, actually \(\angle1\) and \(\angle4\) are vertical angles (opposite angles formed by two intersecting lines). 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 \(\angle1\)

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

Step3: Find the measure of \(\angle7\)

\(\angle1\) and \(\angle5\) are corresponding angles (parallel lines \(c\) and \(b\) cut by transversal \(a\)), so \(\angle1=\angle5 = 85^{\circ}\).
\(\angle5\) and \(\angle7\) are vertical angles. So \(\angle7=\angle5\).

Answer:

\(85\)