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Chapter 11: Problem 3

Solve for the remaining side(s) and angle(s) if possible. As in the text, \((\alpha, a)\), \((\beta, b)\) and \((\gamma, c)\) are angle-side opposite pairs. $$ \alpha=95^{\circ}, \beta=85^{\circ}, a=33.33 $$

Short Answer

Expert verified
Given angles are invalid; they add up to the total internal angle of a triangle, leading to no valid solution.

Step by step solution

01

Understand the given information

We are given angles \(\alpha = 95^{\circ}\) and \(\beta = 85^{\circ}\), and the side \(a = 33.33\) opposite \(\alpha\). We need to find the remaining angle \(\gamma\) and the side \(b\) opposite \(\beta\).
02

Calculate the remaining angle \(\gamma\)

Since the sum of angles in a triangle is \(180^{\circ}\), we can find \(\gamma\) using:\[\gamma = 180^{\circ} - \alpha - \beta = 180^{\circ} - 95^{\circ} - 85^{\circ} = 0^{\circ}.\] This is an unexpected result, indicating a potentially incorrect problem setup because a triangle cannot have an angle of \(0^{\circ}\).
03

Verify the angles' correctness

Given that the computed angle \(\gamma = 0^{\circ}\), which is not possible in a valid triangle, we need to verify whether the angles \(\alpha\) and \(\beta\) provided are possible. The sum given exceeds \(180^{\circ}\), invalidating the triangle.
04

Conclude from the findings

Since \(\alpha\) and \(\beta\) sum up to \(180^{\circ}\), an error exists in the problem because the angles should sum up to less than \(180^{\circ}\) for a valid triangle. Thus, there are no valid remaining sides or angles to find.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angle Sum Property
In every triangle, the sum of the interior angles must be exactly \(180^{\circ}\). This is known as the Angle Sum Property of triangles. To visualize this, imagine spreading the angles out and they should perfectly fill a straight line, which measures \(180^{\circ}\).

When you have two angles of a triangle, finding the third one becomes straightforward because you subtract the sum of the two known angles from \(180^{\circ}\). For instance, if you know angles \(\alpha\) and \(\beta\), the third angle \(\gamma\) is calculated as follows:
  • \(\gamma = 180^{\circ} - \alpha - \beta\)
Hence, having angles \(\alpha = 95^{\circ}\) and \(\beta = 85^{\circ}\) should immediately indicate an issue: their total is already \(180^{\circ}\), meaning no room is left for \(\gamma\). It's important to check your given angles to ensure they can form a valid triangle.
Triangle Validation
Validating a triangle involves ensuring that the sum of its angles is exactly \(180^{\circ}\). This requirement is pivotal because it speaks to the physical properties of triangles. If your angles add up to more than \(180^{\circ}\), like in our exercise, or even less, it suggests an error.

Using the Angle Sum Property, you can easily validate a given triangle by simply adding up the provided angles. Ask yourself:
  • Do these angles sum to \(180^{\circ}\)?
  • Are all components鈥攁ngles and sides鈥攑hysically reasonable?
If you answer "no" to the first question, as we do here due to \(\alpha = 95^{\circ}\) and \(\beta = 85^{\circ}\), this means something is off in your setup. Beyond that, each individual angle must be a positive number, and all three sides must have lengths greater than zero.
Incorrect Triangle Setup
Sometimes, problems can occur, leading to an incorrect triangle setup. This is what happened in the exercise where we're given \(\alpha = 95^{\circ}\) and \(\beta = 85^{\circ}\), leading to \(\gamma = 0^{\circ}\).

Unlike possible triangles where all sides and angles are positive and logically arranged, this setup fails the basic Angle Sum Property, indicating an error or misinterpretation.

To identify such issues, always verify:
  • That all angles provided add up to \(180^{\circ}\).
  • Each angle is less than \(180^{\circ}\) and more than \(0^{\circ}\).
  • Each side length is positive.
In our example, having any angle as \(0^{\circ}\) or any pair summing to \(180^{\circ}\) shows we're not working with a valid triangle, hence no valid solution can be obtained.

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