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. \(60^{\circ}, 60^{\circ}\)

Short answer

The measure of the third angle is \(60^{\circ}\).

Step-by-step solution

- Understand the Sum of Angles in a Triangle

In any triangle, the sum of all three interior angles is always equal to 180 degrees. This is a fundamental property of triangles.

- Add the Given Angles

Add the measures of the two given angles: \( 60^{\circ} + 60^{\circ} \ = 120^{\circ} \)

Subtract from 180 Degrees

Subtract the sum of the given angles from 180 degrees to find the third angle: \( 180^{\circ} - 120^{\circ} = 60^{\circ} \)

Key concepts

Sum of angles in a triangle

In a triangle, the sum of all three interior angles is always 180 degrees. This is true for every type of triangle, whether it's an acute, obtuse, or right triangle. This fundamental property allows us to solve for any missing angle if we know the other two angles.

For instance, consider the given angles, both measuring 60 degrees each. To find the third angle, you sum the given angles and then subtract this sum from 180 degrees. Remember, this property simplifies many geometric problems, making it very useful.

Triangle properties

Triangles are defined by their sides and angles. Here are some basic properties:

• Three sides.
• Three angles.
• The sum of the interior angles is always 180 degrees.

Let's consider the types of triangles based on angles:

• Acute Triangle: All angles are less than 90 degrees.
• Obtuse Triangle: One angle is more than 90 degrees.
• Right Triangle: One angle is exactly 90 degrees.

Understanding these properties helps in identifying and solving problems related to triangles.

Interior angles of triangles

Interior angles are the angles inside a triangle formed by its sides. Knowing that these angles always add up to 180 degrees is key. For example, if two angles are given as 60 degrees each, you can find the third angle by subtracting the sum of these angles from 180 degrees:
\[ 180^\text{°} - (60^\text{°} + 60^\text{°}) = 60^\text{°} \] This property ensures you can always find a missing angle if the other two are known, making geometric calculations straightforward and manageable.
Remember, each angle plays a crucial role in determining the type and properties of the triangle.