Question

1. in the diagram shown, all rays share a starting point of a. answer each of the following questions based on the diagram. (a) if the measure of ∠bac is 65° and the measure of ∠dac is 22°, what is the measure of ∠bad? (b) if m∠bac = 65° and the measure of ∠bad is 9° more than the measure of ∠dac, then find the measure of ∠dac. (c) both ∠bac and ∠dac share a ray with ∠bad. but, only one of them is adjacent to ∠bad. which one? explain your choice.

Explanation:

Step1: Identify angle - addition relationship

We know that \(\angle BAD=\angle BAC+\angle DAC\) for part (a).

Step2: Calculate \(\angle BAD\) for part (a)

Given \(\angle BAC = 65^{\circ}\) and \(\angle DAC=22^{\circ}\), then \(\angle BAD=65^{\circ}+ 22^{\circ}=87^{\circ}\).

Step3: Set up an equation for part (b)

Let \(x = \angle DAC\). Then \(\angle BAD=x + 9^{\circ}\). Also, \(\angle BAC=\angle BAD+\angle DAC\), so \(65^{\circ}=(x + 9^{\circ})+x\).

Step4: Solve the equation for part (b)

Combine like - terms: \(65^{\circ}=2x + 9^{\circ}\). Subtract \(9^{\circ}\) from both sides: \(2x=65^{\circ}-9^{\circ}=56^{\circ}\). Divide both sides by 2: \(x = 28^{\circ}\), so \(\angle DAC = 28^{\circ}\).

Step5: Define adjacent angles for part (c)

Adjacent angles share a common side and a common vertex and have no interior points in common.

Step6: Determine the adjacent angle for part (c)

\(\angle DAC\) is adjacent to \(\angle BAD\) because they share ray \(AD\), have a common vertex \(A\), and no interior points in common. \(\angle BAC\) and \(\angle BAD\) do not satisfy the non - overlapping interior points condition as \(\angle BAD\) is inside \(\angle BAC\).

Answer:

(a) \(87^{\circ}\)
(b) \(28^{\circ}\)
(c) \(\angle DAC\) is adjacent to \(\angle BAD\) because they share ray \(AD\), common vertex \(A\) and have no interior points in common.