Question

in the triangle, suppose that ( mangle q=(x + 8)^{circ}, mangle r=(7x + 4)^{circ} ), and ( mangle s=(4x)^{circ} ).
(a) write an equation to find ( x ). make sure you use an \=\ sign in your answer.
equation:
(b) find the degree measure of each angle.
( mangle q=) (circ)
( mangle r=) (circ)
( mangle s=) (circ)

Explanation:

Step1: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle Q+m\angle R + m\angle S=180^{\circ}\).
Substitute \(m\angle Q=(x + 8)^{\circ}\), \(m\angle R=(7x + 4)^{\circ}\), and \(m\angle S=(4x)^{\circ}\) into the equation: \((x + 8)+(7x + 4)+4x=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \((x+7x + 4x)+(8 + 4)=180\), which gives \(12x+12 = 180\).

Step3: Solve for \(x\)

Subtract 12 from both sides: \(12x=180 - 12\), so \(12x=168\).
Divide both sides by 12: \(x=\frac{168}{12}=14\).

Step4: Find \(m\angle Q\)

Substitute \(x = 14\) into \(m\angle Q=(x + 8)^{\circ}\): \(m\angle Q=(14 + 8)^{\circ}=22^{\circ}\).

Step5: Find \(m\angle R\)

Substitute \(x = 14\) into \(m\angle R=(7x + 4)^{\circ}\): \(m\angle R=(7\times14 + 4)^{\circ}=(98+4)^{\circ}=102^{\circ}\).

Step6: Find \(m\angle S\)

Substitute \(x = 14\) into \(m\angle S=(4x)^{\circ}\): \(m\angle S=(4\times14)^{\circ}=56^{\circ}\).

Answer:

(a) Equation: \((x + 8)+(7x + 4)+4x=180\)
(b) \(m\angle Q = 22^{\circ}\), \(m\angle R=102^{\circ}\), \(m\angle S = 56^{\circ}\)