Question

in the triangle below, suppose that ( mangle e=(x + 9)^{circ}, mangle f=(4x - 5)^{circ} ), and ( mangle g=(3x)^{circ} ). find the degree measure of each angle in the triangle.

Explanation:

Step1: Use the triangle angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \((x + 9)+(4x-5)+3x=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \((x+4x + 3x)+(9 - 5)=180\), which gives \(8x+4 = 180\).

Step3: Solve for \(x\)

Subtract \(4\) from both sides: \(8x=180 - 4=176\). Then divide both sides by \(8\): \(x=\frac{176}{8}=22\).

Step4: Find the measure of each angle

• For \(\angle E\): Substitute \(x = 22\) into \(m\angle E=(x + 9)^{\circ}\). So, \(m\angle E=(22 + 9)^{\circ}=31^{\circ}\).
• For \(\angle F\): Substitute \(x = 22\) into \(m\angle F=(4x-5)^{\circ}\). So, \(m\angle F=(4\times22-5)^{\circ}=(88 - 5)^{\circ}=83^{\circ}\).
• For \(\angle G\): Substitute \(x = 22\) into \(m\angle G=(3x)^{\circ}\). So, \(m\angle G=(3\times22)^{\circ}=66^{\circ}\).

Answer:

\(m\angle E = 31^{\circ}\), \(m\angle F = 83^{\circ}\), \(m\angle G = 66^{\circ}\)