Question

Tell whether the given measures could be the angle measures of a triangle. \(45^{\circ}, 55^{\circ}, 80^{\circ}\)

Short answer

No, the sum of the angles is not 180 degrees.

Step-by-step solution

Understand the Problem

To determine if the given measures can be the angles of a triangle, recall that the sum of all interior angles in any triangle must be equal to 180 degrees.

Add the Given Angles

Add the given angle measures together: To do this, sum up the numbers: 51+45+55 = 150. If the sum is 180 degrees, then these measures can form a triangle.

Compare the Sum to 180 Degrees

Check whether the obtained sum is 180 degrees. Conclusion: 155 degrees is not equal to 180 degrees, therefore these measures cannot be the angles of a triangle.

Key concepts

interior angles

Interior angles are the angles found inside a polygon. For a triangle, these are the three angles where each pair of sides meets. These angles are very important to understand because they tell us a lot about the shape and properties of the triangle. Specifically, when dealing with problems about triangles, knowing the interior angles helps to confirm if a set of angles can actually form a triangle.

Remember that the angles inside a triangle are called 'interior angles' because they are inside the boundary of the shape, as opposed to exterior angles which lie outside.

sum of angles

One of the fundamental properties of a triangle is that the sum of its interior angles is always 180 degrees. This is known as the Triangle Angle Sum Theorem. When you have a set of three angles, you can check if they form a triangle by adding them together:

• If the sum is exactly 180 degrees, they can form a triangle.
• If the sum is not 180 degrees, then they cannot form a triangle.

Consider the example given: 45 degrees, 55 degrees, and 80 degrees. Adding these together:

\[45^{\text{°}} + 55^{\text{°}} + 80^{\text{°}} = 180^{\text{°}}. \]

This sum is equal to 180 degrees, meaning these angles can indeed form a triangle.

triangle properties

Triangles have several interesting properties that are useful in geometry. Here are some key properties:

• The sum of the interior angles is always 180 degrees, as detailed above.
• A triangle has three sides and three angles, which can vary in size but will always add up to 180 degrees.
• The type of triangle can be identified based on its angles (acute, right, or obtuse) or its sides (equilateral, isosceles, or scalene).
• In any triangle, the length of one side cannot be greater than the sum of the other two sides (Triangle Inequality Theorem).

Understanding these properties helps in solving many geometric problems, including proving that a set of angle measures cannot form a triangle if their sums do not add up to 180 degrees. When you come across three angles in a triangle exercise, always remember to verify these properties.