Question

ch. 1.6 describing pairs of angles
learning targets: i can identify complementary, supplementary, vertical angles and linear pairs. i can find angle measures in pairs of angles.

1. if the m∠d = 5x and the m∠e = 3(m∠d). find the value of x that makes ∠d and m∠e supplementary angles.
2. if the m∠d = 5x and the m∠e = 3(m∠d). find the value of x that makes ∠d and m∠e complementary angles.

Explanation:

Problem 4

Step1: Recall supplementary angles

Supplementary angles sum to \(180^\circ\). Given \(m\angle D = 5x\) and \(m\angle E = 3(m\angle D)\), substitute \(m\angle D\) into \(m\angle E\): \(m\angle E = 3(5x)=15x\).

Step2: Set up the equation

Since \(\angle D\) and \(\angle E\) are supplementary, \(m\angle D + m\angle E = 180^\circ\). Substitute the expressions: \(5x + 15x = 180\).

Step3: Solve for \(x\)

Combine like terms: \(20x = 180\). Divide both sides by 20: \(x=\frac{180}{20}=9\).

Step1: Recall complementary angles

Complementary angles sum to \(90^\circ\). Given \(m\angle D = 5x\) and \(m\angle E = 3(m\angle D)=15x\) (from earlier substitution).

Step2: Set up the equation

Since \(\angle D\) and \(\angle E\) are complementary, \(m\angle D + m\angle E = 90^\circ\). Substitute: \(5x + 15x = 90\).

Step3: Solve for \(x\)

Combine like terms: \(20x = 90\). Divide both sides by 20: \(x=\frac{90}{20}=4.5\).

Answer:

\(x = 9\)

Problem 5