Question

in the triangle, suppose that ( mangle q=(x + 3)^{circ}, mangle r=(2x + 7)^{circ} ), and ( mangle s=(7x)^{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=square^{circ} )
( mangle r=square^{circ} )
( mangle s=square^{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 \(m\angle Q=(x + 3)^{\circ}\), \(m\angle R=(2x + 7)^{\circ}\), and \(m\angle S=(7x)^{\circ}\) into the equation: \((x + 3)+(2x + 7)+7x=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \((x+2x + 7x)+(3 + 7)=180\), which is \(10x+10 = 180\).

Step3: Solve for \(x\)

Subtract \(10\) from both sides: \(10x=180 - 10=170\).
Divide both sides by \(10\): \(x=\frac{170}{10}=17\).

Step4: Find \(m\angle Q\)

Substitute \(x = 17\) into \(m\angle Q=(x + 3)^{\circ}\). Then \(m\angle Q=(17+3)^{\circ}=20^{\circ}\).

Step5: Find \(m\angle R\)

Substitute \(x = 17\) into \(m\angle R=(2x + 7)^{\circ}\). Then \(m\angle R=(2\times17 + 7)^{\circ}=(34 + 7)^{\circ}=41^{\circ}\).

Step6: Find \(m\angle S\)

Substitute \(x = 17\) into \(m\angle S=(7x)^{\circ}\). Then \(m\angle S=(7\times17)^{\circ}=119^{\circ}\).

Answer:

(a) Equation: \((x + 3)+(2x + 7)+7x=180\)
(b) \(m\angle Q = 20^{\circ}\), \(m\angle R = 41^{\circ}\), \(m\angle S = 119^{\circ}\)