Triangle Inequalities

Summary

This video explains various triangle inequality theorems and concepts. It covers Triangle Inequality Theorems 1 and 2, which describe how the lengths of sides and the sizes of angles are related in a triangle. The Exterior Angle Inequality Theorem and the third Triangle Inequality Theorem, which discusses the sum of two sides being greater than the third, are also explained. Additionally, the Hinge Theorem and its converse are introduced to compare triangles with congruent sides. Practical examples illustrate how to use these theorems to determine the relative sizes of angles and sides in triangles.

Takeaways

• 😀 The Triangle Inequality Theorem 1 states that the longest side of a triangle is opposite the largest angle.
• 😀 Triangle Inequality Theorem 1 can be used to compare angles and sides in a triangle, helping to list angles from smallest to largest.
• 😀 Triangle Inequality Theorem 2 asserts that if one angle in a triangle is larger, the side opposite that angle is longer.
• 😀 To list sides from shortest to longest using Triangle Inequality Theorem 2, identify the angles and match them to the opposite sides.
• 😀 The Exterior Angle Inequality Theorem states that the measure of an exterior angle is always greater than either of the two non-adjacent interior angles.
• 😀 Triangle Inequality Theorem 3 explains that the sum of the lengths of any two sides in a triangle must be greater than the length of the third side.
• 😀 If the sum of two sides equals or is smaller than the third side, the sides cannot form a valid triangle.
• 😀 To find the possible lengths for the third side in a triangle, add the two known side lengths and subtract the smaller from the larger.
• 😀 The Hinge Theorem compares two triangles with two congruent sides, stating that the triangle with the larger included angle will have the longer third side.
• 😀 The Converse of the Hinge Theorem works in reverse: if the third side of one triangle is longer, the angle opposite it will be larger than in the other triangle.

Q & A

What does Triangle Inequality Theorem 1 state about side lengths and angles in a triangle?

Triangle Inequality Theorem 1 states that if one side of a triangle is longer than another, the angle opposite the longer side is larger than the angle opposite the shorter side. This means the longest side is opposite the largest angle.

How can you use Triangle Inequality Theorem 1 to order angles from smallest to largest?

To order angles from smallest to largest, identify the shortest side, as it is opposite the smallest angle. Then list the angles in increasing order based on the side lengths, with the longest side corresponding to the largest angle.

What does Triangle Inequality Theorem 2 explain about the relationship between angles and sides?

Triangle Inequality Theorem 2 explains that if one angle of a triangle is larger than another, the side opposite the larger angle is longer than the side opposite the smaller angle. It describes a reverse relationship from Theorem 1.

How would you apply Triangle Inequality Theorem 2 to determine the longest side in a triangle?

To determine the longest side using Theorem 2, compare the angles. The side opposite the largest angle will be the longest, as the larger angle corresponds to a longer side.

What is the Exterior Angle Inequality Theorem, and how does it relate to interior angles?

The Exterior Angle Inequality Theorem states that the measure of an exterior angle of a triangle is greater than the measure of either of the two non-adjacent interior angles. This implies that the exterior angle is always larger than the interior angles it is adjacent to.

How do you check if a set of three sides can form a triangle using Triangle Inequality Theorem 3?

To check if three sides can form a triangle, add any two sides together and see if their sum is greater than the third side. If this condition is satisfied for all three combinations of sides, the sides can form a triangle.

What happens if the sum of two sides is equal to the third side in a triangle?

If the sum of two sides is equal to the third side, the sides do not form a triangle. The sum must be strictly greater than the third side for a valid triangle.

How do you determine the possible lengths of the third side when two sides are known in a triangle?

To find the possible lengths of the third side, subtract the smaller side from the larger side to get the minimum possible length, and add the two sides together to get the maximum possible length. The third side must be greater than the difference and less than the sum of the two given sides.

What does the Hinge Theorem describe about the relationship between angles and side lengths in triangles?

The Hinge Theorem states that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first triangle is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.

How is the Converse of the Hinge Theorem different from the original Hinge Theorem?

The Converse of the Hinge Theorem states that if two sides of one triangle are congruent to two sides of another triangle, and the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second.