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: Use angle - sum property of a triangle

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

Step2: Combine like - terms

Combining the \(x\) terms on the left - hand side, \( (2 + 5+11)x=180\), which simplifies to \(18x = 180\).

Step3: Solve for \(x\)

Dividing both sides of the equation \(18x = 180\) by 18, we have \(x=\frac{180}{18}=10\).

Step4: Find the measure of each angle

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

Step5: Check each option

• Option 1: \(m\angle A = 20^{\circ}\), which is correct as we calculated.
• Option 2: \(m\angle B = 50^{\circ}

eq60^{\circ}\), so this option 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:

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