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 degree measures of two angles in a right triangle could be (A) \(20^{\circ}\) and \(70^{\circ}\), (B) \(30^{\circ}\) and \(60^{\circ}\), and (C) \(45^{\circ}\) and \(45^{\circ}\).

Step-by-step solution

Checking through the option

Start with option A, adding \(20^{\circ}\) and \(70^{\circ}\) gives us \(90^{\circ}\). So, this pair could be the angles in a right triangle.

Comparing with the sum required

For option B, the sum of \(30^{\circ}\) and \(60^{\circ}\) is \(90^{\circ}\), again meeting the requirement for a pair of angles in a right triangle.

Further check through the options

For options C, D, and E, the sum of each pair of angles is \(90^{\circ}\), \(110^{\circ}\), and \(150^{\circ}\) respectively. Only option C could therefore be angles in a right triangle.

Key concepts

Degree Measures

The degree measure of an angle is a way to express the size of the angle using a unit called degrees. In a triangle, the angles always add up to a specific total. For a right triangle, two of the angles must have a sum of \(90^{\circ}\). This is because the right angle itself is \(90^{\circ}\) in measure, making the combined total for all angles in the triangle \(180^{\circ}\).

For example, if we take option A with angles \(20^{\circ}\) and \(70^{\circ}\), adding them gives \(20^{\circ} + 70^{\circ} = 90^{\circ}\). Therefore, these angles fulfill the requirement to be part of a right triangle. It is crucial to ensure that these measures complement the \(90^{\circ}\) right angle in any right triangle settings.

Triangle Properties

Understanding the properties of triangles helps to identify and solve problems involving different types of triangles, including right triangles. In a triangle:

• The sum of the interior angles is always \(180^{\circ}\).
• A right triangle contains one angle that is exactly \(90^{\circ}\).
• The other two angles in a right triangle must add up to \(90^{\circ}\), making them complementary.

For right triangles, knowing these properties helps you solve exercises that involve finding unknown angles. Option C illustrates this perfectly with two angles of \(45^{\circ}\) each. They add up to \(90^{\circ}\), fitting the criteria for a right triangle's additional angles.

In summary, recognizing these properties allows for easier navigation through exercises involving right triangle angles.

Geometry Solutions

Geometry solutions require applying understanding and skills to figure out correct configurations and measures. When faced with angle measures in right triangles, check:

• If the sum of two angles equals \(90^{\circ}\), they can be part of a right triangle.
• Ensure measurements logically fit within the known properties of right triangles.

To decide which pairs from a list can be angles of a right triangle, follow this approach. For option B, adding \(30^{\circ}\) and \(60^{\circ}\) results in a \(90^{\circ}\) sum. This verifies they can indeed pair with a \(90^{\circ}\) right angle to form a proper right triangle.

Being able to spot these solutions enables one to tackle more complex geometry challenges, solidifying the foundational skills in understanding triangle properties and their measures.