Hinge Theorem

Definition

The Hinge Theorem states that if two triangles have two sides of one triangle that 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 will be longer than the third side of the second triangle. This theorem is significant in understanding triangle inequalities and serves as a basis for indirect proofs involving side lengths and angles.

5 Must Know Facts For Your Next Test

1. The Hinge Theorem can be used to compare the lengths of sides in two triangles when given two pairs of congruent sides.
2. If two triangles have the same two sides but different included angles, the triangle with the larger included angle will always have a longer third side.
3. This theorem emphasizes the relationship between angles and side lengths in triangles, reinforcing concepts of triangle inequality.
4. The Hinge Theorem is particularly useful in solving problems related to triangle congruence and proving indirect results about triangles.
5. It can also aid in establishing inequalities among multiple triangles, showcasing how angles affect side lengths.

Review Questions

• How does the Hinge Theorem illustrate the relationship between angles and side lengths in triangles?
  • The Hinge Theorem shows that when two triangles share two sides of equal length, the size of their included angle affects the length of their opposite sides. If one triangle has a larger included angle than another with equal corresponding sides, its opposite side must be longer. This principle reinforces how angles influence not just shape but also dimensions within triangular structures.
• In what situations would you apply the Hinge Theorem to prove a statement about triangle side lengths?
  • The Hinge Theorem is applied when comparing two triangles that share two sides with known congruences. For instance, if you know both triangles have two equal sides but differing angles, you can utilize the theorem to determine which triangle has a longer third side based on the relative sizes of their included angles. This method can also assist in proving indirect relationships between various geometric figures.
• Evaluate a scenario where using the Hinge Theorem could lead to an indirect proof. What steps would you take?
  • Consider a scenario where you're tasked with proving that one triangle is larger than another based on given congruent sides. First, assume for contradiction that both triangles have equal third sides despite having differing included angles. According to the Hinge Theorem, this assumption would lead to an inconsistency since a triangle with a larger angle must inherently have a longer opposite side. This contradiction validates the assertion that one triangle indeed has a longer third side than the other, illustrating effective use of indirect proof through geometric reasoning.

Related terms

Triangle Inequality Theorem: A principle stating that in any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

Congruent Triangles: Triangles that are identical in shape and size, having corresponding sides and angles that are equal.

Indirect Proof: A method of proof where one assumes that the conclusion is false, leading to a contradiction, thus proving that the conclusion must be true.