Question

in the triangle, suppose that ( m angle a=(5 x-4)^{circ}, m angle b=(4 x-6)^{circ} ), and ( m angle c=x^{circ} ).
(a) write an equation to find ( x ). make sure you use an \= \ sign in your answer.

(b) find the degree measure of each angle.
( m angle a=)( m angle b=)( m angle c=)

Explanation:

Step1: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle A+m\angle B + m\angle C=180^{\circ}\).
Substitute \(m\angle A=(5x - 4)^{\circ}\), \(m\angle B=(4x - 6)^{\circ}\), and \(m\angle C = x^{\circ}\) into the equation:
\((5x-4)+(4x - 6)+x=180\)
Simplify the left - hand side:
\(5x-4 + 4x-6+x=(5x + 4x+x)-(4 + 6)=10x-10\)
So the equation is \(10x-10 = 180\)

Step2: Solve the equation \(10x-10 = 180\) for \(x\)

Add \(10\) to both sides of the equation:
\(10x-10+10=180 + 10\)
\(10x=190\)
Divide both sides by \(10\):
\(x=\frac{190}{10}=19\)

Step3: Find \(m\angle A\)

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

Step4: Find \(m\angle B\)

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

Step5: Find \(m\angle C\)

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

Answer:

(a) Equation: \(10x-10 = 180\)
(b) \(m\angle A = 91^{\circ}\), \(m\angle B=70^{\circ}\), \(m\angle C = 19^{\circ}\)