Question

1. in the diagram, ∠cab is a straight angle.

a. describe the relationship among ∠caf, ∠fad, and ∠dab.
b. write an equation for the angle relationship and solve for x.
c. check your solution to the equation.
d. what are the measures of ∠fad, ∠caf, and ∠dac?

Explanation:

Part a

Since \(\angle CAB\) is a straight angle, its measure is \(180^{\circ}\). The three angles \(\angle CAF\), \(\angle FAD\), and \(\angle DAB\) are adjacent and together form \(\angle CAB\), so their measures add up to \(180^{\circ}\).

Part b

Step1: Set up the equation

From the diagram, \(\angle CAF=(2x)^{\circ}\), \(\angle FAD = x^{\circ}\), \(\angle DAB=(2x)^{\circ}\), and \(\angle CAB = 180^{\circ}\). So the equation is \(2x + x+ 2x=180\).

Step2: Combine like terms

Combine the \(x\)-terms: \(5x = 180\).

Step3: Solve for \(x\)

Divide both sides by \(5\): \(x=\frac{180}{5}=36\).

Part c

Step1: Substitute \(x = 36\) into the left - hand side of the equation

Substitute \(x = 36\) into \(2x+x + 2x\): \(2\times36+36 + 2\times36=72 + 36+72\).

Step2: Calculate the sum

\(72+36 + 72=180\), which is equal to the measure of the straight angle \(\angle CAB\). So the solution is correct.

Part d

Answer:

The sum of the measures of \(\angle CAF\), \(\angle FAD\), and \(\angle DAB\) is equal to the measure of the straight angle \(\angle CAB\) (i.e., \(\angle CAF+\angle FAD+\angle DAB = \angle CAB\) and \(\angle CAB = 180^{\circ}\)).