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 \\( \angle 2 \\) to be three times as great as the measure of \\( \angle 1 \\). she thinks the measures of \\( \angle 3 \\) and \\( \angle 4 \\) should be equal. finally, she wants the measure of \\( \angle 6 \\) to be half that of \\( \angle 5 \\). determine the measures of the angles. \\( m \angle 1= \\) \\( m \angle 2= \\) \\( m \angle 3= \\) \\( m \angle 4= \\) \\( m \angle 5= \\) \\( m \angle 6= \\)

Explanation:

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

Since \(\angle1\) and \(\angle2\) are complementary (\(\angle1+\angle2 = 90^{\circ}\)), and \(m\angle2 = 3m\angle1\).
Substitute \(m\angle2\) into the complementary - angle equation: \(m\angle1+3m\angle1=90^{\circ}\).
Combine like terms: \(4m\angle1 = 90^{\circ}\), so \(m\angle1=\frac{90^{\circ}}{4}=22.5^{\circ}\).
Then \(m\angle2 = 3\times22.5^{\circ}=67.5^{\circ}\).

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

The sum of angles in a triangle is \(180^{\circ}\). In the right - triangle (the triangle with \(\angle1,\angle3,\angle4\)), \(\angle1+\angle3+\angle4 = 90^{\circ}\) (because one angle is \(90^{\circ}\) in the rectangle). Since \(m\angle3 = m\angle4\), and \(m\angle1 = 22.5^{\circ}\), then \(22.5^{\circ}+2m\angle3=90^{\circ}\).
Subtract \(22.5^{\circ}\) from both sides: \(2m\angle3=90^{\circ}-22.5^{\circ}=67.5^{\circ}\).
Divide by 2: \(m\angle3=m\angle4 = 33.75^{\circ}\).

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

Since \(\angle5\) and \(\angle6\) are complementary (\(\angle5+\angle6 = 90^{\circ}\)), and \(m\angle6=\frac{1}{2}m\angle5\).
Substitute \(m\angle6\) into the complementary - angle equation: \(m\angle5+\frac{1}{2}m\angle5=90^{\circ}\).
Combine like terms: \(\frac{3}{2}m\angle5 = 90^{\circ}\).
Solve for \(m\angle5\): \(m\angle5=90^{\circ}\times\frac{2}{3}=60^{\circ}\).
Then \(m\angle6=\frac{1}{2}\times60^{\circ}=30^{\circ}\).

Answer:

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