Question

in the triangle below, suppose that ( mangle a=(4x - 6)^{circ}, mangle b=(5x - 4)^{circ} ), and ( mangle c=x^{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, \((4x - 6)+(5x - 4)+x=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \((4x+5x + x)+(-6 - 4)=180\), which gives \(10x-10 = 180\).

Step3: Solve for \(x\)

Add \(10\) to both sides of the equation: \(10x-10 + 10=180 + 10\), so \(10x=190\). Then divide both sides by \(10\): \(x=\frac{190}{10}=19\).

Step4: Find the measure of \(\angle A\)

Substitute \(x = 19\) into \(m\angle A=(4x - 6)^{\circ}\). Then \(m\angle A=(4\times19 - 6)^{\circ}=(76 - 6)^{\circ}=70^{\circ}\).

Step5: Find the measure of \(\angle B\)

Substitute \(x = 19\) into \(m\angle B=(5x - 4)^{\circ}\). Then \(m\angle B=(5\times19 - 4)^{\circ}=(95 - 4)^{\circ}=91^{\circ}\).

Step6: Find the measure of \(\angle C\)

Substitute \(x = 19\) into \(m\angle C=x^{\circ}\). Then \(m\angle C = 19^{\circ}\).

Answer:

\(m\angle A = 70^{\circ}\), \(m\angle B=91^{\circ}\), \(m\angle C = 19^{\circ}\)