Question

triangle abc has the angle measures shown.
which statement is true about the angles?
$mangle a=(2x)^{circ}$
$mangle b=(3x)^{circ}$
$mangle c=(4x)^{circ}$
$mangle a = 20^{circ}$
$mangle b = 60^{circ}$
$angle a$ and $angle b$ are complementary
$mangle a + mangle c = 100^{circ}$

Explanation:

Step1: Use the triangle - angle - sum theorem

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 \(m\angle A=(2x)^{\circ}\), \(m\angle B=(3x)^{\circ}\), and \(m\angle C=(4x)^{\circ}\) into the equation:
\(2x + 3x+4x=180\)
\(9x = 180\)
\(x=\frac{180}{9}=20\)

Step2: Calculate the measures of each angle

• For \(\angle A\): \(m\angle A=(2x)^{\circ}\), substitute \(x = 20\), then \(m\angle A=2\times20^{\circ}=40^{\circ}\)
• For \(\angle B\): \(m\angle B=(3x)^{\circ}\), substitute \(x = 20\), then \(m\angle B=3\times20^{\circ}=60^{\circ}\)
• For \(\angle C\): \(m\angle C=(4x)^{\circ}\), substitute \(x = 20\), then \(m\angle C=4\times20^{\circ}=80^{\circ}\)

Step3: Check each option

• Option 1: \(m\angle A = 20^{\circ}\) is false, since \(m\angle A=40^{\circ}\)
• Option 2: \(m\angle B = 60^{\circ}\) is true (as calculated above)
• Option 3: Complementary angles sum to \(90^{\circ}\). \(m\angle A+m\angle B=40^{\circ}+60^{\circ}=100^{\circ}

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

• Option 4: \(m\angle A+m\angle C=40^{\circ}+80^{\circ}=120^{\circ}

eq100^{\circ}\)

Answer:

\(m\angle B = 60^{\circ}\) (the second option)