Question

in the triangle, suppose that ( mangle e = (6x - 8)^circ ), ( mangle f = (5x - 4)^circ ), and ( mangle g = x^circ ).
(a) write an equation to find ( x ). make sure you use an \=\ sign in your answer.
equation: (square)
(b) find the degree measure of each angle.
( mangle e = square^circ )
( mangle f = square^circ )
( mangle g = square^circ )

Explanation:

Part (a)

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\). So, \(m\angle E + m\angle F + m\angle G = 180\).

Step2: Substitute angle expressions

Substitute \(m\angle E=(6x - 8)^\circ\), \(m\angle F=(5x - 4)^\circ\), \(m\angle G = x^\circ\) into the equation: \((6x - 8)+(5x - 4)+x = 180\).

Step1: Solve the equation for \(x\)

Simplify the left - hand side of the equation \((6x - 8)+(5x - 4)+x = 180\):
Combine like terms: \(6x+5x + x-8 - 4=180\), which is \(12x-12 = 180\).
Add 12 to both sides: \(12x-12 + 12=180 + 12\), so \(12x=192\).
Divide both sides by 12: \(x=\frac{192}{12}=16\).

Step2: Find \(m\angle E\)

Substitute \(x = 16\) into \(m\angle E=(6x - 8)^\circ\): \(m\angle E=6\times16-8=96 - 8 = 88^\circ\).

Step3: Find \(m\angle F\)

Substitute \(x = 16\) into \(m\angle F=(5x - 4)^\circ\): \(m\angle F=5\times16-4=80 - 4 = 76^\circ\).

Step4: Find \(m\angle G\)

Substitute \(x = 16\) into \(m\angle G=x^\circ\): \(m\angle G = 16^\circ\).

Answer:

\((6x - 8)+(5x - 4)+x = 180\)

Part (b)