Question

triangle abc has the angle measures shown.
$m\angle a=(2x)^{circ}$
$m\angle b=(5x)^{circ}$
$m\angle c=(11x)^{circ}$
which statement is true about the angles?
$m\angle a + m\angle c = 12$
$\angle a$ and $\angle b$ are complementary
$m\angle a = 20^{circ}$
$m\angle b = 60^{circ}$

Explanation:

Step1: Use the triangle angle - sum property

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle A + m\angle B+m\angle C=180^{\circ}\).
Substitute \(m\angle A=(2x)^{\circ}\), \(m\angle B=(5x)^{\circ}\), and \(m\angle C=(11x)^{\circ}\) into the equation:
\(2x + 5x+11x=180\).

Step2: Solve for \(x\)

Combine like terms: \(18x = 180\).
Divide both sides by \(18\): \(x=\frac{180}{18}=10\).

Step3: Find the measure of each angle

For \(\angle A\): \(m\angle A=(2x)^{\circ}\), substitute \(x = 10\), then \(m\angle A=2\times10 = 20^{\circ}\).
For \(\angle B\): \(m\angle B=(5x)^{\circ}\), substitute \(x = 10\), then \(m\angle B=5\times10 = 50^{\circ}\).
For \(\angle C\): \(m\angle C=(11x)^{\circ}\), substitute \(x = 10\), then \(m\angle C=11\times10 = 110^{\circ}\).

Step4: Check each option

• Option 1: \(m\angle A+m\angle C=20 + 110=130

eq12\).

• Option 2: \(\angle A+\angle B=20 + 50=70

eq90\), so they are not complementary.

• Option 3: \(m\angle A = 20^{\circ}\) (correct as we found \(x = 10\) and \(m\angle A=2x\)).
• Option 4: \(m\angle B=50^{\circ}

eq60^{\circ}\).

Answer:

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