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. \(10^{\circ}, 30^{\circ}\)

Short answer

The third angle measures \(140^{\text{\circ}}\).

Step-by-step solution

- Recall the Angle Sum Property of Triangles

The sum of the measures of the three angles in any triangle must be equal to 180 degrees. Therefore, if we are given two angles, we can find the third angle by subtracting the sum of the given angles from 180 degrees.

- Add the Given Angles

Add the measures of the two given angles: \(10^{\text{\circ}} + 30^{\text{\circ}} = 40^{\text{\circ}}\)

- Calculate the Third Angle

Subtract the sum of the two angles from 180 degrees to find the measure of the third angle: \(180^{\text{\circ}} - 40^{\text{\circ}} = 140^{\text{\circ}}\)

- Verify the Validity

Check that the sum of the three angles equals 180 degrees: \(10^{\text{\circ}} + 30^{\text{\circ}} + 140^{\text{\circ}} = 180^{\text{\circ}}\). Since this is true, the given angles can be the measures of two angles of a triangle.

Key concepts

Angle Sum Property

The Angle Sum Property of a triangle is a fundamental rule in triangle geometry. It states that the sum of the measures of the three interior angles in any triangle must always equal 180 degrees. This property helps us determine missing angles when two angles are already known. Begin by adding the measures of the two given angles.

For example, if you have angles of \(10^\text{\textdegree}\) and \(30^\text{\textdegree}\), their sum is \(10^\text{\textdegree} + 30^\text{\textdegree} = 40^\text{\textdegree}\). Then, you subtract this sum from 180 degrees to find the third angle: \(180^\text{\textdegree} - 40^\text{\textdegree} = 140^\text{\textdegree}\).

Always verify your result by checking that the total of all three angles equals 180 degrees.

Triangle Geometry

Triangles are one of the most basic shapes studied in geometry. Every triangle has three sides and three angles. Triangle geometry involves understanding various properties and theorems applicable to triangles, such as the Angle Sum Property.

Triangles can be classified based on their sides and angles:

• Equilateral: All sides and angles are equal
• Isosceles: Two sides and two angles are equal
• Scalene: All sides and angles are different

Understanding these classifications helps when solving problems related to triangles, such as finding unknown angles or calculating the area. In any given triangle problem, knowing the sum of the angles and side lengths is crucial for proper solution.

Finding the Third Angle

Finding the third angle in a triangle is straightforward once you understand the Angle Sum Property. When two angles of a triangle are known, adding them and subtracting from 180 degrees gives the measure of the third angle.

Let's walk through an example: Given angles of \(10^\text{\textdegree}\) and \(30^\text{\textdegree}\), you start by adding them up: \(10^\text{\textdegree} + 30^\text{\textdegree} = 40^\text{\textdegree}\). Then, you subtract this sum from 180 degrees: \(180^\text{\textdegree} - 40^\text{\textdegree} = 140^\text{\textdegree}\). So, the third angle measures \(140^\text{\textdegree}\).

Always double-check that the sum of all three angles equals 180 degrees to confirm your calculation is correct. In this case: \(10^\text{\textdegree} + 30^\text{\textdegree} + 140^\text{\textdegree} = 180^\text{\textdegree}\). This verification step ensures the given angles can indeed form a triangle.