Question

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

Explanation:

Step1: Find the measure of ∠BCA

Use the property that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. Also, ∠BCA and the \(100^{\circ}\) angle are supplementary.
\(\angle BCA=180^{\circ}- 100^{\circ}=80^{\circ}\)

Step2: Use the angle - sum property of a triangle

Since \(\angle BAC=\angle BCA = 80^{\circ}\) (given), and in \(\triangle ABC\), the sum of angles is \(180^{\circ}\). Let \(\angle ABC=x\).
By the angle - sum property of a triangle \(\angle BAC+\angle BCA+\angle ABC = 180^{\circ}\)
Substitute \(\angle BAC = 80^{\circ}\), \(\angle BCA = 80^{\circ}\) into the equation:
\(80^{\circ}+80^{\circ}+x=180^{\circ}\)
\(x=180^{\circ}-(80^{\circ}+80^{\circ})\)
\(x = 20^{\circ}\)

Answer:

\(20\)