Question

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

Explanation:

• Exterior and adjacent interior angles:
• By the definition of a linear pair of angles (two adjacent angles that form a straight line), an exterior angle of a triangle and its adjacent interior angle form a linear pair.
• The sum of the measures of angles in a linear pair is \(180^{\circ}\). So, if we let the measure of the exterior angle be \(E\) and the measure of the adjacent interior angle be \(A\), then \(E + A=180^{\circ}\).
• Analysis of other options:
• Option 1 (Exterior - remote interior):
• The exterior angle theorem states that \(E=a + b\) (where \(a\) and \(b\) are the remote interior angles). If \(E + a=180^{\circ}\), then \(a + b+a = 180^{\circ}\), which is not always True. For example, in an equilateral triangle, each interior angle is \(60^{\circ}\) and each exterior angle is \(120^{\circ}\). If \(a = 60^{\circ}\) (remote interior) and \(E = 120^{\circ}\), \(E+a=180^{\circ}\) (works here), but consider a triangle with interior angles \(30^{\circ},40^{\circ},110^{\circ}\). The exterior angle adjacent to \(110^{\circ}\) is \(70^{\circ}\). If \(a = 30^{\circ}\) (remote interior), \(E + a=70+30 = 100

eq180\).

• Option 2 (Remote interior - exterior):
• From the exterior angle theorem \(E=a + b\) (where \(a\) and \(b\) are non - adjacent or remote interior angles). So \(E

eq a\) (or \(E
eq b\)) in general. For example, in a right - angled triangle with angles \(30^{\circ},60^{\circ},90^{\circ}\), the exterior angle adjacent to \(90^{\circ}\) is \(90^{\circ}\), and the remote interior angles are \(30^{\circ}\) and \(60^{\circ}\). \(30
eq90\) and \(60
eq90\).

• Option 4 (Adjacent - remote interior):
• Let the angles of the triangle be \(x\) (adjacent interior), \(y\) (remote interior), and \(z\) (remote interior). We know that \(x + y+z = 180^{\circ}\). If \(x + y=180^{\circ}\), then \(z = 0^{\circ}\), which is not possible for a triangle.

Answer:

An exterior angle is supplementary to the adjacent interior angle.