Question

in the figure, ( mangle1=(7x + 8)^{circ} ), ( mangle2=(5x + 11)^{circ} ), and ( mangle4=(13x + 10)^{circ} ). your friend incorrectly says that ( mangle4 = 71^{circ} ). what is ( mangle4 )? what mistake might your friend have made?

Explanation:

Step1: Use the exterior angle theorem

The exterior angle theorem states that \(m\angle4=m\angle1 + m\angle2\).
So, \((13x + 10)=(7x + 8)+(5x + 11)\).

Step2: Solve the equation for \(x\)

Simplify the right - hand side: \((7x + 8)+(5x + 11)=7x+5x + 8 + 11=12x+19\).
The equation becomes \(13x + 10=12x+19\).
Subtract \(12x\) from both sides: \(13x-12x+10=12x - 12x+19\), which gives \(x+10 = 19\).
Subtract \(10\) from both sides: \(x=19 - 10=9\).

Step3: Find \(m\angle4\)

Substitute \(x = 9\) into the formula for \(m\angle4\), \(m\angle4=(13x + 10)^{\circ}\).
\(m\angle4=(13\times9+10)^{\circ}=(117 + 10)^{\circ}=127^{\circ}\).

Answer:

\(127\)

The friend's mistake might be that they set up an incorrect equation (not using the exterior angle theorem correctly, for example, maybe they thought \(m\angle4\) was equal to only one of the non - adjacent interior angles instead of the sum of \(m\angle1\) and \(m\angle2\)).