Question

1. in the given figure, ac extends to point d. if the measure of ∠bac is equal to the measure ∠bca, what is the value of x?

Explanation:

Step1: Find the measure of ∠BCA

Use the property of supplementary angles.
$$\angle BCA + 114^{\circ}=180^{\circ}$$
$$\angle BCA = 180^{\circ}- 114^{\circ}=66^{\circ}$$

Step2: Use the given condition ∠BAC = ∠BCA

Since ∠BAC = ∠BCA = 66°, use the angle - sum property of a triangle.
In \(\triangle ABC\), \(\angle BAC+\angle BCA+\angle ABC = 180^{\circ}\)
Substitute \(\angle BAC = 66^{\circ}\), \(\angle BCA = 66^{\circ}\)
$$66^{\circ}+66^{\circ}+x^{\circ}=180^{\circ}$$

Step3: Solve for \(x\)

Simplify the equation \(66 + 66+x=180\)
$$x=180-(66 + 66)$$
$$x=180 - 132$$
$$x = 48$$

Answer:

\(48\)