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: Use vertical angles property

Vertical angles are equal. Angle 1 and angle 4 are supplementary (since \(c\parallel b\) and \(a\) is a transversal, \(\angle1+\angle4 = 180^{\circ}\)). So, \((3x + 10)+(4x-15)=180\).

Step2: Solve the equation for \(x\)

Combine like - terms: \(3x+4x+10 - 15=180\), \(7x-5 = 180\). Add 5 to both sides: \(7x=180 + 5=185\). Then \(x=\frac{185}{7}=25\).

Step3: Find \(\angle1\)

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

Step4: Use corresponding angles property

Since \(c\parallel b\) and \(a\) is a transversal, \(\angle1\) and \(\angle7\) are corresponding angles. Corresponding angles are equal when two parallel lines are cut by a transversal.

Answer:

\(85\)