Question

Tell whether the given measures could be the angle measures of a triangle. \(72^{\circ}, 72^{\circ}, 36^{\circ}\)

Short answer

Yes, these angles can be the measures of a triangle.

Step-by-step solution

Recall the Triangle Angle Sum Property

For any triangle, the sum of the interior angles must always be equal to 180 degrees. This is known as the Triangle Angle Sum Property.

Sum the Given Angles

Add the given measures: 72° + 72° + 36°.

Perform the Addition

Calculate the sum: 72° + 72° = 144° 144° + 36° = 180°.

Verify the Sum

Check if the sum of the angles is 180 degrees. Since 72° + 72° + 36° equals 180°, these measures can be the angles of a triangle.

Key concepts

triangle properties

Triangles are one of the most basic geometric shapes. They have three sides, three vertices, and three interior angles. Triangles are classified based on their angles and sides. For example, an equilateral triangle has all sides and angles equal, while a scalene triangle has all sides and angles different.

Triangles follow a set of rules and properties that determine their shapes and sizes. One crucial rule is that the sum of any two sides must always be greater than the third side. Another important property is the Triangle Angle Sum Property, which states that the sum of all interior angles in a triangle must always be 180 degrees. Understanding these properties helps students solve problems involving triangles and ensures correct calculations of their angles and sides.

interior angles

The interior angles of a triangle are the angles inside the triangle. Each triangle has three interior angles, and these play a key role in determining the shape and type of the triangle.

Here are some points to remember about interior angles in a triangle:

• The sum of the interior angles is always 180 degrees (Triangle Angle Sum Property).
• If one angle is a right angle (90 degrees), then the triangle is a right triangle.
• If all angles are less than 90 degrees, the triangle is acute.
• If one angle is greater than 90 degrees, the triangle is obtuse.

Calculations involving interior angles are crucial to correctly solving problems relating to triangles.
Remembering these points will help students easily identify and work with different types of triangles.

angle measures

Angle measures in a triangle are essential to understanding its geometry. In a triangle, each angle directly influences the shape and type of the triangle. Here's how you can work with angle measures in a triangle:

First, check the sum of the angles. According to the Triangle Angle Sum Property, they must add up to 180 degrees. For example, let's consider the angles 72°, 72°, and 36°. By adding these, we get: \(72° + 72° = 144°\) \(144° + 36° = 180°\). Since their sum is 180°, these could indeed be the measures of a triangle.

It is also helpful to recognize different types of triangles based on their angle measures:

• An equilateral triangle has three equal angles of 60° each.
• An isosceles triangle has two equal angles and one different angle.
• A scalene triangle has no equal angles.

Understanding angle measures is not just about knowing their values but also about how they define the triangle's type and properties.