Question

triangle abc has the angle measures shown. m∠a=(2x)°, m∠b=(5x)°, m∠c=(11x)°. which statement is true about the angles? m∠a = 20°, m∠b = 60°, ∠a and ∠b are complementary, m∠a + m∠c = 120°

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. So, \(m\angle A + m\angle B+m\angle C=180^{\circ}\). Substituting the given angle - measures \(m\angle A=(2x)^{\circ}\), \(m\angle B=(5x)^{\circ}\), and \(m\angle C=(11x)^{\circ}\), we get the equation \(2x + 5x+11x=180\).

Step2: Solve the equation for \(x\)

Combining like - terms on the left - hand side of the equation \(2x + 5x+11x=180\), we have \(18x = 180\). Dividing both sides by 18, we get \(x=\frac{180}{18}=10\).

Step3: Find the measure of each angle

• \(m\angle A=(2x)^{\circ}=2\times10 = 20^{\circ}\).
• \(m\angle B=(5x)^{\circ}=5\times10 = 50^{\circ}\).
• \(m\angle C=(11x)^{\circ}=11\times10 = 110^{\circ}\).

Step4: Check each option

• Option 1: \(m\angle A = 20^{\circ}\), which is correct as we found \(m\angle A=2\times10 = 20^{\circ}\).
• Option 2: \(m\angle B = 60^{\circ}\), but we found \(m\angle B = 50^{\circ}\), so this is incorrect.
• Option 3: Complementary angles add up to 90°. \(m\angle A+m\angle B=20^{\circ}+50^{\circ}=70^{\circ}

eq90^{\circ}\), so \(\angle A\) and \(\angle B\) are not complementary.

• Option 4: \(m\angle A + m\angle C=20^{\circ}+110^{\circ}=130^{\circ}

eq120^{\circ}\).

Answer:

\(m\angle A = 20^{\circ}\)