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:

• For the first option:
• By the definition of adjacent angles forming a linear pair. An exterior angle and its adjacent interior angle form a linear pair. The sum of angles in a linear pair is \(180^{\circ}\). So, an exterior angle is supplementary to the adjacent interior angle.
• 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 \(60^{\circ}\). If we consider an adjacent interior angle (\(60^{\circ}\)) and a remote interior angle (\(60^{\circ}\)), their sum is \(120^{\circ}

eq180^{\circ}\).

• For the third option:
• By the exterior - angle theorem, an exterior angle of a triangle is equal to the sum of the two non - adjacent (remote) interior angles. So, a remote interior angle is not congruent to the exterior angle. For example, if two remote interior angles are \(30^{\circ}\) and \(40^{\circ}\), the exterior angle is \(30^{\circ}+ 40^{\circ}=70^{\circ}\), and \(30^{\circ}

eq70^{\circ}\) and \(40^{\circ}
eq70^{\circ}\).

• For the fourth option:
• An exterior angle is equal to the sum of the two remote interior angles (by the exterior - angle theorem: \(\angle ACD=\angle A+\angle B\) for \(\triangle ABC\) with exterior angle \(\angle ACD\)). They are not supplementary. If \(\angle A = 30^{\circ}\) and \(\angle B=40^{\circ}\), \(\angle ACD = 70^{\circ}\), and \(30^{\circ}+70^{\circ}=100^{\circ}

eq180^{\circ}\), \(40^{\circ}+70^{\circ}=110^{\circ}
eq180^{\circ}\)

Answer:

An exterior angle is supplementary to the adjacent interior angle.