Question

If the given measures could be the measures of two angles of a triangle, give the measure of the third angle. If not, explain why. \(45^{\circ}, 45^{\circ}\)

Short answer

The measure of the third angle is \( 90^{\text{circ}} \).

Step-by-step solution

Understand the sum of angles in a triangle

In any triangle, the sum of the three interior angles is always 180 degrees. Let's denote the angles by \(A\), \(B\), and \(C\). Then, we have \(A + B + C = 180^{\text{circ}}\).

Identify the given angles

The given measures of the two angles are \(45^{\text{circ}}\) and \(45^{\text{circ}}\). Denote these by \(A = 45^{\text{circ}}\) and \(B = 45^{\text{circ}}\). Now, we need to find the measure of the third angle \( C \).

Use the sum of angles to find the third angle

Using the equation from Step 1: \( A + B + C = 180^{\text{circ}} \), substitute \( A = 45^{\text{circ}} \) and \( B = 45^{\text{circ}} \). This gives: \( 45^{\text{circ}} + 45^{\text{circ}} + C = 180^{\text{circ}} \)

Solve for the third angle

Combine the given angles: \( 45^{\text{circ}} + 45^{\text{circ}} = 90^{\text{circ}} \). Therefore, the equation becomes: \( 90^{\text{circ}} + C = 180^{\text{circ}} \). Solving for \( C \): \( C = 180^{\text{circ}} - 90^{\text{circ}} = 90^{\text{circ}} \).

Key concepts

Understanding Interior Angles

In any triangle, the sum of the three interior angles is always 180 degrees. These angles are the ones inside the triangle. Imagine cutting out a triangle from a piece of paper and measuring each corner with a protractor. The three angles added together will give you exactly 180 degrees. This is true for all triangles, regardless of their shape or size. It’s a fundamental property of triangles that helps us solve problems involving angle measures.

Key Triangle Properties

Triangles have several essential properties that are useful for solving geometry problems. One of the key properties is that the sum of the interior angles is always 180 degrees. Here are some other important properties to remember:

• Triangles can be classified based on their angles: acute, right, or obtuse.
• The sum of any two sides of a triangle is always greater than the length of the third side.
• If a triangle has two equal angles, the sides opposite those angles are also equal, making it an isosceles triangle.
• A triangle with all three equal sides and angles is called an equilateral triangle, with each angle measuring 60 degrees.
• In a right triangle, the right angle is 90 degrees, and the other two angles must sum up to 90 degrees.

Understanding the properties of triangles is crucial for working through geometry problems efficiently.

Angle Calculation in Triangles

Now let's focus on angle calculation within triangles. If we know the measures of two angles, we can easily find the third angle using the fact that the sum of the interior angles is 180 degrees. This is done by subtracting the sum of the known angles from 180 degrees. Here’s a step-by-step process:

• Identify the known angles and denote them, for example, as A and B.
• Recall that the sum of the angles in any triangle is given by the equation: \( A + B + C = 180^{\circ} \).
• Substitute the values of A and B for the given angles.
• Solve for the third angle, C, by rearranging the equation to \( C = 180^{\circ} - (A + B) \).

Applying this method makes it easy to find the missing angle. For example, if you know two angles are 45 degrees each, substitute them into the formula: \( 45^{\circ} + 45^{\circ} + C = 180^{\circ} \). Simplifying this gives \( 90^{\circ} + C = 180^{\circ} \), so \( C = 90^{\circ} \). Therefore, the third angle is 90 degrees, making this a right triangle.