Question

Tell whether the given measures could be the angle measures of a triangle. \(60^{\circ}, 90^{\circ}, 30^{\circ}\)

Short answer

Yes, the given measures could be the angle measures of a triangle.

Step-by-step solution

Sum of angles in a triangle

For any set of angles to be the measures of a triangle's angles, the sum of those angles must be 180 degrees. Check if the given angles add up to 180 degrees.

Add the given angles

Calculate the sum of the given angles: \(60^{\circ} + 90^{\circ} + 30^{\circ}e 180^{\}\text\))

Conclusion

Since the sum of the given angles is 180 degrees, they could indeed be the measures of the angles of a triangle.

Key concepts

Triangle Properties

When studying triangles, there are some key properties that are important to understand. A triangle has three angles and three sides. This applies to all triangles, regardless of their type. Triangles can be categorized based on their angles (acute, obtuse, or right) or their sides (equilateral, isosceles, or scalene).

For example:

• An equilateral triangle has three equal sides and three equal angles, each measuring 60 degrees.
• An isosceles triangle has at least two equal sides and two equal angles.
• A scalene triangle has all different sides and angles.

Understanding these properties helps in recognizing and analyzing triangles. One core property that all triangles share is that the sum of their interior angles is always 180 degrees.

Angle Sum Theorem

The Angle Sum Theorem is a fundamental rule in geometry. This theorem states that the sum of the interior angles of any triangle is always 180 degrees. It applies universally, meaning it holds true for all types of triangles, whether they are right, acute, or obtuse.

Let's break it down:

• For a triangle with angles A, B, and C, the equation would be: \( A + B + C = 180^{\circ} \).
• This property is essential for determining if a set of angles can form a triangle.

Using our example of 60°, 90°, and 30°, we can apply the Angle Sum Theorem: \( 60^{\circ} + 90^{\circ} + 30^{\circ} = 180^{\circ} \). Since the angle sum is 180 degrees, these angles form a valid triangle.

Geometric Proof

To understand how angles relate within a triangle, we can use geometric proofs. A geometric proof involves a set of logical statements backed by geometric principles which demonstrate the truth of a theorem or proposition.

For the Angle Sum Theorem, one common geometric proof involves using parallel lines. If you extend one side of the triangle and draw a line parallel to another side of the triangle, you create alternate interior angles. By adding these angles together, you can show that their sum is 180 degrees.

Another way to understand this is to break down any triangle into two right triangles. Since each right triangle has a sum of 90 degrees for its angles, two right triangles together make 180 degrees, reinforcing the Angle Sum Theorem.

These proofs help demonstrate why the sum of the angles in a triangle is always 180 degrees, providing a solid mathematical foundation for this property.