Question

Choose the numbers that are not measures of the three angles of a triangle. a. \(10,20,150\) b. \(30,60,90\) c. \(40,70,80\) d. \(45,55,80\)

Short answer

Option c: 40, 70, 80 is not a valid set of triangle angles.

Step-by-step solution

Recall Triangle Angle Sum Property

The sum of the measures of the angles in a triangle must equal 180 degrees.

Check Each Option

For each set of numbers, add the three angles together to see if they equal 180 degrees.

Evaluate Option a

Add the angles: \(10 + 20 + 150 = 180\). Option a is a valid triangle.

Evaluate Option b

Add the angles: \(30 + 60 + 90 = 180\). Option b is a valid triangle.

Evaluate Option c

Add the angles: \(40 + 70 + 80 = 190\). Option c does not form a triangle since the sum exceeds 180 degrees.

Evaluate Option d

Add the angles: \(45 + 55 + 80 = 180\). Option d is a valid triangle.

Key concepts

Understanding Angle Measures

When talking about triangles, understanding the measures of angles is essential. A triangle is a shape made up of three lines, or sides, connected at three distinct points called vertices. Each vertex has an interior angle, and these three angles together determine the shape and type of the triangle.

• An acute triangle has all angles less than 90 degrees.
• A right triangle has one angle exactly equal to 90 degrees.
• An obtuse triangle has one angle greater than 90 degrees.

Knowing the exact measures of the angles helps in identifying and categorizing the type of triangle at hand. In geometry, this understanding is foundational for solving many related problems.

Criteria for a Valid Triangle

To determine whether three given angles form a valid triangle, they must satisfy a very specific condition: their sum must equal 180 degrees. This is often referred to as the "Triangle Angle Sum Property." If the sum is less or more than 180 degrees, the angles cannot form a triangle. Here’s a simple way to check if a set of angle measures constitutes a valid triangle:

• Add all three angles together.
• Verify if the total is exactly 180 degrees.

For example, in a set of angles measuring 30, 60, and 90 degrees:

• Adding them results in 180 degrees, hence, they can certainly form a valid triangle.
• However, in a set like 40, 70, and 80 degrees, the sum is 190 degrees, which exceeds 180, making it an invalid set for a triangle.

Using this criteria, one can quickly discern if a given trio of angles can build the desired triangular shape.

The Sum of Angles in a Triangle

One of the most pivotal properties in geometry is the sum of angles in a triangle. According to this property, the interior angles of any triangle always add up to 180 degrees no matter the type or size of the triangle. This property serves as a cornerstone for many geometric proofs and problem solving activities:

• It helps to understand the relationship between the angles, allowing us to find unknown angles when two are known.
• It aids in deducing whether a specific combination of angles can form a triangle.

Take, for example, a triangle with angles of 50, 60, and 70 degrees: no matter how these angles are arranged spatially, their sum will still be 180 degrees, confirming that they define a valid triangle. Thus, ensuring that the angle sums to 180 is crucial in verifying the triangle's existence.