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Magazine Chapter 1: Problem 47
Two angles in a triangle cannot have measures \(82^{\circ}\) and \(67^{\circ}\).
Short Answer
Expert verified
The statement is incorrect; the angles can form a triangle.
Step by step solution
01
Understand the Problem
We have a triangle, and it's given that two angles have measures of \(82^{\circ}\) and \(67^{\circ}\). We need to determine if these angle measures are possible in a triangle, based on the properties of triangles.
02
Recall Triangle Sum Theorem
The Triangle Sum Theorem states that the sum of the angles in a triangle must always equal \(180^{\circ}\). We will use this theorem to check if the given angles can be part of a triangle.
03
Calculate Sum of Given Angles
Add the two given angles: \(82^{\circ} + 67^{\circ} = 149^{\circ}\).
04
Calculate the Third Angle
Subtract the sum of the two given angles from \(180^{\circ}\):\[180^{\circ} - 149^{\circ} = 31^{\circ}.\]This result is the measure of the third angle.
05
Examine Feasibility of Angle Measures
To determine if the angles can exist within a triangle, check if each angle is greater than 0, which they are. Additionally, check that the angle measures do not sum to more than \(180^{\circ}\), which they do not since \(82^{\circ} + 67^{\circ} + 31^{\circ} = 180^{\circ}\). Thus, these angles can form a triangle.
06
Check Constraint of the Problem
Re-evaluate the statement: 'Two angles in a triangle cannot have measures \(82^{\circ}\) and \(67^{\circ}\)'. This statement is incorrect. These two angles, along with a third angle of \(31^{\circ}\), can sum up to \(180^{\circ}\), satisfying the condition of a triangle.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Triangle Sum Theorem
In the world of geometry, the Triangle Sum Theorem plays a crucial role in understanding triangle properties. This theorem states that the sum of the interior angles in any triangle is always \(180^{\circ}\).
No matter the type or size of the triangle, this rule remains steadfast, helping us analyze and solve many geometric problems efficiently.
Whenever you are dealing with triangles, remember:
No matter the type or size of the triangle, this rule remains steadfast, helping us analyze and solve many geometric problems efficiently.
Whenever you are dealing with triangles, remember:
- Add up all three angles, and the result should be \(180^{\circ}\).
- If given two angles, subtract their sum from \(180^{\circ}\) to find the third angle.
Angle Sum Property
The Angle Sum Property is a key component when discussing triangles. This property distinguishes triangles by asserting that the three angles' contributions to the triangle add up precisely to \(180^{\circ}\).
In our exercise, we started by noting two angles with measures \(82^{\circ}\) and \(67^{\circ}\).
Let's break it down mathematically:
In our exercise, we started by noting two angles with measures \(82^{\circ}\) and \(67^{\circ}\).
Let's break it down mathematically:
- Add \(82^{\circ}\) and \(67^{\circ}\): \(82^{\circ} + 67^{\circ} = 149^{\circ}\)
- Use the Angle Sum Property to find the third angle: \(180^{\circ} - 149^{\circ} = 31^{\circ}\)
Feasibility of Angle Measures
To determine if a given set of angles can form a triangle, understanding the feasibility of their measures is essential.
The major requirements are:
The major requirements are:
- Each angle must be greater than \(0^{\circ}\).
- The sum of all angles must consistently equal \(180^{\circ}\).
- Sum of given angles: \(82^{\circ} + 67^{\circ} = 149^{\circ}\)
- Calculating the third angle using \(180^{\circ} - 149^{\circ} = 31^{\circ}\)
- Ensuring each angle (\(82^{\circ}, 67^{\circ}, 31^{\circ}\)) is positive and collectively sums to exactly \(180^{\circ}\)
