Question

what is m∠1?
33°
1
80°

m∠1 = °
save answer

Explanation:

Step1: Find the adjacent interior angle

The exterior angle is \(80^\circ\), so the adjacent interior angle is \(180^\circ - 80^\circ = 100^\circ\)? Wait, no, wait. Wait, the exterior angle and the adjacent interior angle are supplementary? Wait, no, in a triangle, the exterior angle is equal to the sum of the two non - adjacent interior angles. Wait, let's correct. The angle adjacent to the \(80^\circ\) exterior angle: the exterior angle and its adjacent interior angle are supplementary, so the adjacent interior angle is \(180 - 80=100^\circ\)? No, that can't be, because the sum of angles in a triangle is \(180^\circ\). Wait, no, the triangle has angles: \(\angle1\), \(33^\circ\), and the angle adjacent to the \(80^\circ\) exterior angle. Wait, the exterior angle theorem: the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. Wait, no, the exterior angle and the adjacent interior angle are supplementary. Wait, let's look at the triangle. The exterior angle is \(80^\circ\), so the interior angle at that vertex is \(180 - 80 = 100^\circ\)? No, that would make the sum of angles in the triangle exceed \(180^\circ\). Wait, I made a mistake. The exterior angle is equal to the sum of the two remote (non - adjacent) interior angles. Wait, no, the exterior angle and the adjacent interior angle are supplementary. Wait, let's start over.

The sum of the interior angles of a triangle is \(180^\circ\). Let the interior angle adjacent to the \(80^\circ\) exterior angle be \(x\). Then \(x + 80^\circ=180^\circ\), so \(x = 100^\circ\)? No, that can't be, because then \(\angle1+33^\circ + 100^\circ=180^\circ\), so \(\angle1=180 - 33 - 100 = 47^\circ\). Wait, no, that's wrong. Wait, no, the exterior angle is equal to the sum of the two non - adjacent interior angles. So the exterior angle (\(80^\circ\)) is equal to \(\angle1+33^\circ\)? Wait, that would mean \(\angle1=80 - 33 = 47^\circ\). Ah, yes! The exterior angle theorem: the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. So if the exterior angle is \(80^\circ\), and one non - adjacent interior angle is \(33^\circ\), then the other non - adjacent interior angle (\(\angle1\)) is \(80^\circ-33^\circ = 47^\circ\). Wait, let's verify with the sum of angles in a triangle. The interior angle adjacent to the \(80^\circ\) exterior angle is \(180 - 80=100^\circ\)? No, that's not right. Wait, no, the exterior angle is formed by extending one side of the triangle. So the two non - adjacent interior angles to the exterior angle are \(\angle1\) and \(33^\circ\), and the exterior angle is equal to their sum. So \(m\angle1+33^\circ = 80^\circ\), so \(m\angle1=80 - 33 = 47^\circ\).

Step1: Recall the exterior angle theorem

The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles.

Step2: Apply the exterior angle theorem

Let \(m\angle1\) be the measure of \(\angle1\). The exterior angle is \(80^\circ\), and one non - adjacent interior angle is \(33^\circ\). According to the exterior angle theorem:
\(m\angle1+33^\circ=80^\circ\)

Step3: Solve for \(m\angle1\)

Subtract \(33^\circ\) from both sides of the equation:
\(m\angle1 = 80^\circ-33^\circ=47^\circ\)

Answer:

\(47\)