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Chapter 7: Problem 68

A triangle has a \(45^{\circ}\) angle and a right angle. Find the measure of the third angle.

Short Answer

Expert verified
The measure of the third angle of the triangle is \(45^{\circ}\).

Step by step solution

01

Identify given angles

The problem statement provides two angles of a triangle. The first one is a right angle which is \(90^{\circ}\), and the second one is \(45^{\circ}\).
02

Recall the basic property of a triangle

Remember that the sum of all interior angles in a triangle equals \(180^{\circ}\). So, to calculate the third angle, subtract the sum of the given angles from \(180^{\circ}\).
03

Calculate third angle

Subtract the sum of the given angles (\(90^{\circ} + 45^{\circ} = 135^{\circ}\)) from \(180^{\circ}\). The result is the measure of the third angle, which is \(180^{\circ} - 135^{\circ} = 45^{\circ}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Right Triangle
A right triangle is a type of triangle that has one angle equal to 90 degrees. This 90-degree angle is known as the 'right angle.'
A right triangle has special properties that make it unique among triangles:
  • The side opposite the right angle is called the hypotenuse, and it is the longest side.
  • The other two sides are referred to as the adjacent and opposite sides in relation to a specific angle.
  • Right triangles are often used in geometry and trigonometry because of their predictable angle properties.
For instance, if you know two angles of a right triangle, you can easily determine the third, as the sum of all angles in any triangle, including a right triangle, is always 180 degrees.
Sum of Angles in a Triangle
The sum of angles in any triangle always equals 180 degrees. This is a fundamental rule in geometry known as the Angle Sum Property.
This property applies to all types of triangles, including equilateral, isosceles, scalene, and right triangles.
The sum of angles can be calculated as follows:
  • Consider each angle of the triangle.
  • Add together the angles to check if they equal 180 degrees.
  • If you know two angles, you can find the third angle by subtracting their sum from 180 degrees.
This rule is fundamental in solving various problems related to triangles and is often used to determine unknown angles when the other angles are known.
Angle Calculation
Calculating angles in triangles is straightforward once you understand the basic properties of triangles, specifically the angle sum property.
Consider a triangle with known angles:
  • If two angles are known, the third can be calculated by adding the known angles and subtracting their sum from 180 degrees.
  • For right triangles, one angle is always 90 degrees, leaving the sum of the other two angles to equal 90 degrees as well, since 90 plus 90 equals 180.
For example, in a triangle with a 90-degree angle and a 45-degree angle, the calculation for the third angle would be: 1. Add the two known angles: 90 + 45 = 135. 2. Subtract this sum from 180: 180 - 135 = 45. Therefore, the third angle is 45 degrees. This method can be used universally for all triangles to find missing angle measures.

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Most popular questions from this chapter

You plan to stain the wooden deck attached to your house. The deck measures \(10 \mathrm{ft}\) by \(8 \mathrm{ft}\). If a quart of stain will cover \(50 \mathrm{ft}^{2}\), how many quarts of stain should you buy?

The height of a parallelogram is \(3.4 \mathrm{m},\) and the length of the base is \(5.2 \mathrm{m}\). Find the arga of the parallelogram.

The length of a side of the base of a pyramid is 9 in., and the slant height is 12 in. Find the surface area of the pyramid.

Find the area of a rectangular flower garden that measures \(14 \mathrm{ft}\) by \(9 \mathrm{ft}\).

The height of a triangle is \(5 \mathrm{m}\). The area of the triangle is \(50 \mathrm{m}^{2}\). Find the length of the base of the triangle.
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