Question

Which of the following could be the degree measures of two angles in a right triangle? Indicate all such angles. a. \(20^{\circ}\) and \(70^{\circ}\) b. \(30^{\circ}\) and \(60^{\circ}\) c. \(45^{\circ}\) and \(45^{\circ}\) d. \(55^{\circ}\) and \(55^{\circ}\) e. \(75^{\circ}\) and \(75^{\circ}\)

Short answer

The pairs of angle measures that could belong to a right triangle are: a. \(20^{\circ}\) and \(70^{\circ}\), b. \(30^{\circ}\) and \(60^{\circ}\), and c. \(45^{\circ}\) and \(45^{\circ}\). This is based on the rule for a right triangle that one angle measures \(90^{\circ}\) and the other two angles are acute with their sum equaling \(90^{\circ}\). The pairs d. \(55^{\circ}\) and \(55^{\circ}\) and e. \(75^{\circ}\) and \(75^{\circ}\) could not be in a right triangle as their sums are more than \(90^{\circ}\).

Step-by-step solution

Analyze the first pair of angles

The sum of \(20^{\circ}\) and \(70^{\circ}\) is \(90^{\circ}\). Since the sum of these angles is \(90^{\circ}\), and a right triangle also has a \(90^{\circ}\) angle, these angles could be in a right triangle. So, answer 'a' is possible.

Analyze the second pair of angles

The sum of \(30^{\circ}\) and \(60^{\circ}\) is \(90^{\circ}\). Similar to step one, as the sum equals \(90^{\circ}\), these angles could be in a right triangle. So, answer 'b' is also possible.

Analyze the third pair of angles

Similar to steps one and two, the sum of \(45^{\circ}\) and \(45^{\circ}\) is \(90^{\circ}\). Therefore, this pair of angles could be in a right triangle as well. Therefore, answer 'c' is another possible solution.

Analyze the fourth pair of angles

The sum of \(55^{\circ}\) and \(55^{\circ}\) is \(110^{\circ}\). As this sum is more than \(90^{\circ}\), these angles cannot be part of a right triangle. Thus, answer 'd' is not possible.

Analyze the fifth pair of angles

The sum of \(75^{\circ}\) and \(75^{\circ}\) is \(150^{\circ}\). This sum is more than \(90^{\circ}\), which means these angles cannot be part of a right triangle. Therefore, answer 'e' is not possible.

Key concepts

Right Triangle

A right triangle is a special type of triangle in geometry known for having one angle that measures exactly 90 degrees. This 90-degree angle is called a right angle. A key property of right triangles is that the sum of the other two angles must always be 90 degrees as well, since the total sum of angles inside any triangle is 180 degrees. For example, if one angle is 90 degrees (the right angle), then the sum of the remaining two angles must be 90 degrees for the triangle's total to reach 180 degrees.

Angle Measures

Angle measures are crucial in determining the type of triangle. In the context of right triangles, the two non-right angles must always add up to 90 degrees. Here’s a quick look at some common angle pairings:

• 30° and 60°
• 45° and 45°
• 20° and 70°

All these pairs of angles sum up to 90 degrees. This makes them suitable angle measures for the non-right angles in a right triangle.

Triangle Properties

Understanding triangle properties helps us classify different types of triangles. A key property is that the sum of all internal angles in any triangle is always 180 degrees.

• In an equilateral triangle, all three angles are equal, each being 60 degrees.
• In an isosceles triangle, two angles are equal.
• In a scalene triangle, all angles are different.

With right triangles, this property specifies that one of the angles must be 90 degrees, leaving the other two to sum to 90 degrees. This constraint shapes the kinds of problems you'll often solve in geometry.

Mathematical Reasoning

Mathematical reasoning is the logical process used to solve problems. In considering right triangles, you apply mathematical reasoning by evaluating whether given angles make sense within the triangle's properties. For the problem at hand, recognizing that two angle measures must add up to 90 degrees is crucial. This reasoning guided the solution and elimination process:

• Pairs like 30° and 60° or 45° and 45° satisfy the requirement, thus they are valid.
• Pairs like 55° and 55° or 75° and 75° do not, as they exceed 90 degrees, making them invalid.

Engaging in this kind of logical evaluation helps strengthen problem-solving skills.