Question

color guard shannon is designing a new rectangular flag for the schools color guard and is determining the angles at which to cut the fabric. she wants the measure of ∠2 to be three times as great as the measure of ∠1. she thinks the measures of ∠3 and ∠4 should be equal. finally, she wants the measure of ∠6 to be half that of ∠5. determine the measures of the angles.

m∠1 =

m∠2 =

m∠3 =

m∠4 =

m∠5 =

m∠6 =

Explanation:

Step1: Find \(m\angle1\) and \(m\angle2\)

Since \(\angle1\) and \(\angle2\) are adjacent angles forming a right - angle (\(90^{\circ}\)) in a rectangle. Let \(m\angle1 = x\), then \(m\angle2=3x\). Using the equation \(x + 3x=90^{\circ}\), we have \(4x = 90^{\circ}\), so \(x=\frac{90^{\circ}}{4}=22.5^{\circ}\). Thus \(m\angle1 = 22.5^{\circ}\) and \(m\angle2=3\times22.5^{\circ}=67.5^{\circ}\)

Step2: Find \(m\angle3\) and \(m\angle4\)

In a rectangle, the sum of angles at a vertex is \(90^{\circ}\). Let \(m\angle3 = m\angle4=y\). We know that \(y + y+22.5^{\circ}=90^{\circ}\), \(2y=90^{\circ}- 22.5^{\circ}=67.5^{\circ}\), \(y = 33.75^{\circ}\). So \(m\angle3 = 33.75^{\circ}\) and \(m\angle4 = 33.75^{\circ}\)

Step3: Find \(m\angle5\) and \(m\angle6\)

Let \(m\angle6 = z\), then \(m\angle5 = 2z\). Since \(z+2z + 33.75^{\circ}+33.75^{\circ}=180^{\circ}\) (sum of angles on a straight line is \(180^{\circ}\)), \(3z=180^{\circ}-67.5^{\circ}=112.5^{\circ}\), \(z = 30^{\circ}\). So \(m\angle6 = 30^{\circ}\) and \(m\angle5=60^{\circ}\)

Answer:

\(m\angle1 = 22.5^{\circ}\), \(m\angle2 = 67.5^{\circ}\), \(m\angle3 = 33.75^{\circ}\), \(m\angle4 = 33.75^{\circ}\), \(m\angle5 = 60^{\circ}\), \(m\angle6 = 30^{\circ}\)