Triangle Inequality Theorem and the Hinge Theorem

Front

What is the Triangle Inequality Theorem?

Back

The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Front

If a triangle has sides of lengths 5, 7, and 10, does it satisfy the Triangle Inequality Theorem?

Back

Yes, because 5 + 7 > 10, 5 + 10 > 7, and 7 + 10 > 5.

Front

What is the Hinge Theorem?

Back

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.

Front

How can you determine the order of angles in a triangle based on side lengths?

Back

The angle opposite the longest side is the largest angle, and the angle opposite the shortest side is the smallest angle.

Front

Given sides 3, 4, and 5, order the angles from smallest to largest.

Back

∠A, ∠B, ∠C where ∠C is opposite the longest side (5).

Front

What lengths cannot form a triangle?

Back

Lengths that do not satisfy the Triangle Inequality Theorem, such as 2, 2, and 4.

Front

If two sides of a triangle are 8 and 10, what is the range of possible lengths for the third side?

Back

The third side must be greater than 2 (10 - 8) and less than 18 (10 + 8). So, it must be between 2 and 18.

Front

What is the relationship between the sides and angles of a triangle?

Back

In a triangle, larger angles are opposite longer sides, and smaller angles are opposite shorter sides.

Front

How do you apply the Hinge Theorem to compare two triangles?

Back

Compare the lengths of the third sides based on the congruent sides and the included angles.

Front

What is the significance of the longest side in a triangle?

Back

The longest side is always opposite the largest angle.