Question

in the triangle below, suppose that ( mangle q=(x + 6)^{circ}, mangle r=(3x + 9)^{circ} ), and ( mangle s=(7x)^{circ} ).
find the degree measure of each angle in the triangle.
your answer
( 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, \((x + 6)+(3x + 9)+7x=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(x+3x + 7x+6 + 9=180\), which gives \(11x+15 = 180\).

Step3: Solve for \(x\)

Subtract 15 from both sides: \(11x=180 - 15=165\). Then divide both sides by 11: \(x=\frac{165}{11}=15\).

Step4: Find \(m\angle Q\)

Substitute \(x = 15\) into \(m\angle Q=(x + 6)^{\circ}\). So, \(m\angle Q=(15 + 6)^{\circ}=21^{\circ}\).

Step5: Find \(m\angle R\)

Substitute \(x = 15\) into \(m\angle R=(3x + 9)^{\circ}\). So, \(m\angle R=(3\times15 + 9)^{\circ}=(45+9)^{\circ}=54^{\circ}\).

Step6: Find \(m\angle S\)

Substitute \(x = 15\) into \(m\angle S=(7x)^{\circ}\). So, \(m\angle S=(7\times15)^{\circ}=105^{\circ}\).

Answer:

\(m\angle Q = 21^{\circ}\), \(m\angle R = 54^{\circ}\), \(m\angle S = 105^{\circ}\)