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 \((2x)^{\circ}+(5x)^{\circ}+(11x)^{\circ}=180^{\circ}\).

Step2: Combine like terms

\(2x + 5x+11x=180\), which simplifies to \(18x = 180\).

Step3: Solve for \(x\)

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

Step4: Find the measure of each angle

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

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}\).

• 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:

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