Question

Geometry The measure of two angles of a triangle are given. Find the measure of the third angle. \(23^{\circ}\) and \(47^{\circ}\)

Short answer

The measure of the third angle is 110 degrees.

Step-by-step solution

- Understand the sum of angles in a triangle

The sum of the interior angles in any triangle is always 180 degrees. This is a fundamental property of triangles.

- Add the given angles

Add the measures of the two given angles: \[23^{\text{\circ}} + 47^{\text{\circ}} = 70^{\text{\circ}}\]

- Subtract the sum from 180 degrees

Subtract the sum of the given angles from 180 degrees to find the measure of the third angle: \[180^{\text{\circ}} - 70^{\text{\circ}} = 110^{\text{\circ}}\]

Key concepts

Geometry

Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. One of its fundamental elements is the study of triangles, which are simple polygons with three edges and three vertices. Understanding the basic properties of triangles helps us solve various problems related to shapes. In the context of our exercise, we focus on the angles inside a triangle, known as interior angles.

Interior Angles

The interior angles of a triangle are the angles formed between two sides of the triangle that are inside the shape. A fundamental property of these angles is that their sum is always 180 degrees. This means that if you know two angles in a triangle, you can always find the third angle by subtracting the sum of the known angles from 180 degrees. In our example, the given angles are 23 degrees and 47 degrees. Adding them, we get:

\(23^{\text{\circ}} + 47^{\text{\circ}} = 70^{\text{\circ}}\)

To find the third angle, we subtract this sum from 180 degrees:

\(180^{\text{\circ}} - 70^{\text{\circ}} = 110^{\text{\circ}}\).

Thus, the third angle measures 110 degrees.

Triangle Properties

Triangles exhibit several key properties that are useful when solving geometric problems.

• Types of Triangles: Triangles can be classified based on their side lengths (equilateral, isosceles, and scalene) or their angles (acute, obtuse, and right triangle).
• Sum of Angles: As already mentioned, the interior angles of any triangle always add up to 180 degrees.
• Exterior Angles: An exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.
• Angle-Side Relationship: The side opposite the largest angle is the longest, while the side opposite the smallest angle is the shortest.

These properties are essential when analyzing and solving problems involving triangles, as they provide a framework to understand and calculate unknown values.