Question

which statement regarding the interior and exterior angles of a triangle is always true?
an exterior angle is supplementary to the adjacent interior angle.
an adjacent interior angle is supplementary to a remote interior angle.
a remote interior angle is congruent to the exterior angle.
an exterior angle is supplementary to a remote interior angle.

Explanation:

• By the definition of supplementary angles (two angles whose sum is \(180^{\circ}\)) and the relationship between an exterior angle and its adjacent interior angle in a triangle.
• An exterior angle and its adjacent interior angle form a linear pair. A linear pair of angles is supplementary (sum to \(180^{\circ}\)).
• For the second option: An adjacent interior angle and a remote interior angle are not supplementary in general. For example, in an equilateral triangle, each interior angle is \(60^{\circ}\), and adjacent - remote pairs do not add to \(180^{\circ}\).
• For the third option: A remote interior angle is not congruent to the exterior angle. By the exterior - angle theorem, the exterior angle is equal to the sum of the two remote interior angles.
• For the fourth option: An exterior angle is not supplementary to a remote interior angle. If the exterior angle is \(E\) and the remote interior angles are \(x\) and \(y\) (\(E=x + y\) by the exterior - angle theorem), and \(E+y=(x + y)+y=x + 2y

eq180^{\circ}\) in general.

Answer:

An exterior angle is supplementary to the adjacent interior angle.