Question

A rower is designed to simulate rowing. In the diagram shown, notice that the base of the rower, along with the lower portion of the oar, and the hydraulic resistance bar form a triangle. Suppose the measures of two of the angles are 15 and 100 . What is the measure of the third angle?

Short answer

65 degrees

Step-by-step solution

Understanding Triangle Angle Sum

In a triangle, the sum of the interior angles is always 180 degrees. This is a fundamental property of triangles that will help us determine the unknown angle in the given triangle.

Identifying Known Angles

We are given two angles in the triangle: 15 degrees and 100 degrees. We will use these angles to find the third angle.

Calculating the Third Angle

Subtract the sum of the known angles from 180 degrees to find the measure of the third angle. The calculation is: \( 180 - 15 - 100 = 65 \).

Conclusion

The measure of the third angle in the triangle is 65 degrees. This completes the solution.

Key concepts

Interior Angles

In the realm of geometry, the concept of interior angles is vital. Each polygon, including triangles, is composed of several angles inside it, which are called interior angles. For triangles, the principle is simple but crucial: the sum of all three interior angles is always 180 degrees. This principle holds true for any triangle, regardless of its shape, whether it is an equilateral, isosceles, or scalene triangle.
To illustrate this further, let's explore the problem given. We had two angles, 15 degrees and 100 degrees. To find the missing angle, we used the triangle angle sum property. By subtracting the sum of the known angles from 180, we found the third angle: 65 degrees. This fundamental property allows us to solve many problems related to triangles seamlessly.

Geometry Concepts

Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. One of the significant concepts in geometry is understanding and working with different angles. Angles are an essential part of geometric shapes, and knowing how to calculate them can solve many geometric problems.
When working with triangles, specifically, the angles are crucial as they determine the type of triangle. Here's why angle measurement is important:

• The sum of angles helps confirm the shape as a triangle if it totals 180 degrees.
• It helps in recognizing the triangle type (e.g., acute, right, obtuse) based on the angles.
• Provides the basis for further geometric calculations, such as determining side lengths using trigonometry.

This problem, with given angles, is a part of these essential geometry concepts, allowing for precise conclusions about the triangle formed.

Triangle Properties

Triangles are one of the simplest polygons with distinct properties that make them fascinating and widely studied geometries. Understanding triangle properties lays the groundwork for more complex geometric studies. Here are some key properties:

• Triangles, regardless of their type, always have interior angles that sum up to 180 degrees.
• They can be categorized into three types—equilateral, isosceles, and scalene—based on side lengths and angles.
• Each type provides unique attributes, like equal angles in equilateral triangles, or two equal angles in isosceles triangles.

In our particular exercise involving the rowing equipment, recognizing these properties helped solve the angle problem effectively. By applying the property of angle sums, we determined the missing angle was 65 degrees, reinforcing the consistent nature of triangles in geometry. Knowing these properties enables a deeper understanding and ability to solve different geometric problems efficiently.