Question

part 2 - free response questions: all work must be shown correctly to receive full credit. 9. given triangle abc. if m∠a = 2x, m∠b = x + 12 and m∠c = 3x + 48, determine the measure of the largest angle. 10. in the diagram below, line ℓ is parallel to line m, and line w is a transversal. if m∠2 = 3x + 17 and m∠3 = 5x - 21, what is m∠2?

Explanation:

Step1: Use angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. So, \(m\angle A + m\angle B+m\angle C = 180^{\circ}\). Substitute \(m\angle A = 2x\), \(m\angle B=x + 12\), and \(m\angle C=3x + 48\) into the equation: \(2x+(x + 12)+(3x + 48)=180\).

Step2: Simplify the left - hand side

Combine like terms: \((2x+x + 3x)+(12 + 48)=180\), which gives \(6x+60 = 180\).

Step3: Solve for \(x\)

Subtract 60 from both sides: \(6x=180 - 60=120\). Then divide both sides by 6: \(x=\frac{120}{6}=20\).

Step4: Find the measure of each angle

\(m\angle A=2x=2\times20 = 40^{\circ}\), \(m\angle B=x + 12=20+12 = 32^{\circ}\), \(m\angle C=3x + 48=3\times20+48=60 + 48=108^{\circ}\).

for second part:

Step1: Use the property of corresponding angles

Since line \(t\) is parallel to line \(m\) and \(w\) is a transversal, \(\angle2\) and \(\angle3\) are corresponding angles, so \(m\angle2=m\angle3\). Set up the equation \(3x + 17=5x-21\).

Step2: Solve for \(x\)

Subtract \(3x\) from both sides: \(17=5x-3x - 21\), which simplifies to \(17 = 2x-21\). Add 21 to both sides: \(17+21=2x\), so \(38 = 2x\). Divide both sides by 2: \(x = 19\).

Step3: Find \(m\angle2\)

Substitute \(x = 19\) into the expression for \(m\angle2\): \(m\angle2=3x + 17=3\times19+17=57 + 17=74^{\circ}\).

Answer:

The measure of the largest angle is \(108^{\circ}\)