Question

in the figure, ( mangle1=(4x + 8)^{circ},mangle2=(2x + 18)^{circ} ), and ( mangle3=(7x + 5)^{circ} ). what is ( mangle3 )?

Explanation:

Step1: Use the exterior angle theorem

The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. So, \(m\angle3=m\angle1 + m\angle2\).
Substitute the given angle expressions: \((7x + 5)=(4x + 8)+(2x + 18)\).

Step2: Solve the equation for \(x\)

Simplify the right - hand side: \((4x + 8)+(2x + 18)=4x+2x + 8 + 18=6x+26\).
The equation becomes \(7x + 5=6x+26\).
Subtract \(6x\) from both sides: \(7x-6x+5=6x - 6x+26\), which gives \(x+5 = 26\).
Subtract \(5\) from both sides: \(x=26 - 5=21\).

Step3: Find \(m\angle3\)

Substitute \(x = 21\) into the expression for \(m\angle3\): \(m\angle3=(7x + 5)^{\circ}\).
\(m\angle3=7\times21+5\).
First, calculate \(7\times21 = 147\). Then \(147+5=152\).

Answer:

\(152\)