Question

triangle abc has the angle measures shown. m∠a=(2x)°, m∠b=(3x)°, m∠c=(4x)°. which statement is true about the angles? m∠a = 20°, m∠b = 60°, ∠a and ∠b are complementary, m∠a + m∠c = 100°

Explanation:

1. First, use the angle - sum property of a triangle:

• The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle A + m\angle B+m\angle C = 180^{\circ}\).
• Given \(m\angle A=(2x)^{\circ}\), \(m\angle B=(3x)^{\circ}\), and \(m\angle C=(4x)^{\circ}\), we substitute these into the angle - sum formula: \(2x + 3x+4x=180\).
• Combine like - terms: \(9x = 180\).
• Solve for \(x\): \(x=\frac{180}{9}=20\).

1. Then, find the measure of each angle:

• \(m\angle A=(2x)^{\circ}=2\times20 = 40^{\circ}\).
• \(m\angle B=(3x)^{\circ}=3\times20 = 60^{\circ}\).
• \(m\angle C=(4x)^{\circ}=4\times20 = 80^{\circ}\).

1. Now, 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.
• Option 3: Complementary angles add up to \(90^{\circ}\). \(m\angle A+m\angle B=40 + 60=100

eq90\), so \(\angle A\) and \(\angle B\) are not complementary.

• Option 4: \(m\angle A+m\angle C=40 + 80 = 120

eq100\).

Answer:

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