Question

the measure of the other non - right angle is
and the lengths of the other sides are:
(and there is a right - triangle image with one angle 30°, one leg length 14, right angle, sides labeled b and c)

Explanation:

Step1: Find the other non - right angle

In a right - triangle, the sum of the interior angles is \(180^{\circ}\), and one angle is \(90^{\circ}\), one angle is \(30^{\circ}\). Let the other non - right angle be \(x\). Then we use the angle - sum property of a triangle: \(90^{\circ}+30^{\circ}+x = 180^{\circ}\).
Simplify the left - hand side: \(120^{\circ}+x=180^{\circ}\).
Subtract \(120^{\circ}\) from both sides: \(x = 180^{\circ}- 120^{\circ}=60^{\circ}\).

Step2: Analyze the sides of the 30 - 60 - 90 triangle

In a 30 - 60 - 90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\), where the side opposite the \(30^{\circ}\) angle is the shortest side (let's call it \(a\)), the side opposite the \(60^{\circ}\) angle is \(a\sqrt{3}\), and the hypotenuse is \(2a\).
From the diagram, the side adjacent to the \(30^{\circ}\) angle (the leg with length 14) is the side opposite the \(60^{\circ}\) angle? Wait, no. Wait, in the right - triangle, the side with length 14: let's see the angles. The angle of \(30^{\circ}\), the right angle (\(90^{\circ}\)), and the other angle \(60^{\circ}\).
If the side adjacent to the \(30^{\circ}\) angle (the leg) has length 14, and we know that in a 30 - 60 - 90 triangle, the side adjacent to the \(30^{\circ}\) angle (the leg) is \(a\sqrt{3}\) (where \(a\) is the side opposite the \(30^{\circ}\) angle), and the hypotenuse is \(2a\), and the side opposite the \(60^{\circ}\) angle is \(a\sqrt{3}\). Wait, maybe I got it reversed. Let's correct:
In a 30 - 60 - 90 triangle, the side opposite the \(30^{\circ}\) angle (\(a\)) is the shortest leg, the side opposite the \(60^{\circ}\) angle (\(b\)) is \(a\sqrt{3}\), and the hypotenuse (\(c\)) is \(2a\).
Looking at the triangle, the leg with length 14: let's assume that the leg of length 14 is the side opposite the \(60^{\circ}\) angle. So if \(b\) (opposite \(60^{\circ}\)) is 14, then \(a\sqrt{3}=14\), so \(a=\frac{14}{\sqrt{3}}=\frac{14\sqrt{3}}{3}\)? No, wait, maybe the leg of length 14 is the adjacent side to the \(30^{\circ}\) angle, which is the side opposite the \(60^{\circ}\) angle. Wait, no, let's re - express.
Wait, the angle of \(30^{\circ}\): the side opposite the \(30^{\circ}\) angle is the shorter leg (let's call it \(b\)? No, the diagram shows the leg with length 14, and the other leg \(b\), and hypotenuse \(c\).
Wait, maybe the side with length 14 is the leg adjacent to the \(30^{\circ}\) angle, so it is the side opposite the \(60^{\circ}\) angle. So in the ratio, the side opposite \(60^{\circ}\) is \(a\sqrt{3}\), the side opposite \(30^{\circ}\) is \(a\), and hypotenuse \(2a\).
If \(a\sqrt{3}=14\), then \(a = \frac{14}{\sqrt{3}}=\frac{14\sqrt{3}}{3}\) (the side opposite \(30^{\circ}\), which is the shorter leg), and the hypotenuse \(c = 2a=\frac{28\sqrt{3}}{3}\)? No, that can't be right. Wait, maybe I made a mistake. Wait, maybe the side with length 14 is the hypotenuse? No, because the hypotenuse is the longest side. Wait, no, the side with length 14 is a leg. Wait, let's start over.
In a right - triangle, the sum of angles: \(30^{\circ}+90^{\circ}+\theta=180^{\circ}\), so \(\theta = 60^{\circ}\), that's the other non - right angle.
Now, for the sides:
Let's denote:
- Let the side opposite \(30^{\circ}\) be \(b\) (the shorter leg).
- The side opposite \(60^{\circ}\) be 14 (the longer leg).
- The hypotenuse be \(c\).

In a 30 - 60 - 90 triangle, \(\frac{\text{shorter leg}}{\text{longer leg}}=\frac{1}{\sqrt{3}}\), so \(\frac{b}{14}=\frac{1}{\sqrt{3}}\), so \(b=\frac{14}{\sqrt{3}}=\frac{14\sqrt{3}}{3}\). And the hypotenuse \(c = 2b=\frac{28\sqrt{3}}{3}\)? No, that seems complicated. Wait, maybe the side with length 14 is the hypotenuse? No, because the hypotenuse should be longer than the legs. Wait, maybe the side with length 14 is the side opposite the \(30^{\circ}\) angle. Then:
If the side opposite \(30^{\circ}\) (shorter leg) is 14, then the longer leg (opposite \(60^{\circ}\)) is \(14\sqrt{3}\), and the hypotenuse is \(2\times14 = 28\).

Ah, that makes more sense. So let's check the angle: the side opposite \(30^{\circ}\) is the shorter leg. So if the leg with length 14 is opposite \(30^{\circ}\), then:
- Shorter leg (\(b\) opposite \(30^{\circ}\)): 14
- Longer leg (opposite \(60^{\circ}\)): \(14\sqrt{3}\)
- Hypotenuse (\(c\)): \(2\times14 = 28\)

Yes, that fits the 30 - 60 - 90 triangle ratio (\(1:\sqrt{3}:2\)), where the shorter leg (opposite \(30^{\circ}\)) is \(a = 14\), the longer leg (opposite \(60^{\circ}\)) is \(a\sqrt{3}=14\sqrt{3}\), and the hypotenuse is \(2a = 28\).

Answer:

The measure of the other non - right angle is \(60^{\circ}\). The length of the shorter leg (opposite \(30^{\circ}\)) is 14, the length of the longer leg (opposite \(60^{\circ}\)) is \(14\sqrt{3}\), and the length of the hypotenuse is 28. (If we consider the side with length 14 as the shorter leg opposite the \(30^{\circ}\) angle)