Question

1. what is the measure of the missing angle in the figure below? 47° ? 93° 46° 87° 133° 140°

Explanation:

Step1: Find the adjacent interior angle

The exterior angle is \(93^\circ\), so the adjacent interior angle is \(180^\circ - 93^\circ = 87^\circ\) (linear pair).

Step2: Calculate the missing angle

In a triangle, the sum of interior angles is \(180^\circ\). Let the missing angle be \(x\). Then \(x + 47^\circ + 87^\circ = 180^\circ\). Solving for \(x\): \(x = 180^\circ - 47^\circ - 87^\circ = 46^\circ\). Wait, no, wait, correction: Wait, the exterior angle theorem: the exterior angle is equal to the sum of the two non - adjacent interior angles. So \(93^\circ=47^\circ + x\), then \(x = 93^\circ - 47^\circ=46^\circ\)? Wait, no, that's wrong. Wait, the exterior angle is supplementary to the adjacent interior angle. So adjacent interior angle is \(180 - 93 = 87\). Then the sum of the three interior angles is \(180\). So \(47+87 + x=180\). \(134 + x = 180\), so \(x = 180 - 134 = 46\)? Wait, but let's re - check. The exterior angle is equal to the sum of the two remote (non - adjacent) interior angles. So the exterior angle \(93^\circ\) should be equal to \(47^\circ\) plus the missing angle? No, wait, no. Wait, the adjacent angle to the exterior angle is \(180 - 93 = 87\). Then the three angles: \(47\), \(87\), and the missing angle. So \(47+87 + \text{missing}=180\). \(134+\text{missing}=180\), so missing is \(46\). Wait, but the options have \(46^\circ\) as an option. Wait, maybe I made a mistake in the exterior angle theorem. Wait, the exterior angle is equal to the sum of the two non - adjacent interior angles. So the exterior angle \(93^\circ\) is equal to \(47^\circ\) plus the missing angle? No, that would be \(93 = 47 + \text{missing}\), so missing is \(46\). Yes, that's correct. So the missing angle is \(46^\circ\).

Answer:

\(46^\circ\) (the option with \(46^\circ\))