Question

towers a lookout tower sits on a network of struts and posts. leslie measured two angles on the tower. if ( mangle1=(7x - 7)^{circ}), ( mangle2=(4x + 2)^{circ}), and ( mangle3=(2x + 6)^{circ}), what is ( mangle1)?

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, \(m\angle1=m\angle2 + m\angle3\).
Substitute the given angle expressions: \((7x - 7)=(4x + 2)+(2x + 6)\).

Step2: Simplify the right - hand side

Simplify \((4x + 2)+(2x + 6)\):
\((4x+2x)+(2 + 6)=6x+8\).
The equation becomes \(7x-7 = 6x+8\).

Step3: Solve for \(x\)

Subtract \(6x\) from both sides: \(7x-6x-7=6x - 6x+8\), which gives \(x-7 = 8\).
Add \(7\) to both sides: \(x-7 + 7=8 + 7\), so \(x = 15\).

Step4: Find \(m\angle1\)

Substitute \(x = 15\) into the expression for \(m\angle1\): \(m\angle1=(7x-7)\).
\(m\angle1=7\times15-7\).
First, calculate \(7\times15 = 105\), then \(105-7=98\).

Answer:

\(m\angle1 = 98^{\circ}\)