Explain why knowing a combination of four pairs of equal sides or equal angles guarantees one of the congruence relationships.
Congruence of Geometric Shapes:
Congruent figures in geometry have the same shape and size. They may be mirror images of each other, and can be rotated and re positioned to coincide with each other. Several congruence postulates can be used to prove the congruence between two geometric shapes. For example, the postulates used to prove congruence between two or more triangles are:
1. SAS: Two sides and their included angle
2. SSS: Three sides
3. ASA: Two angles and their included side
4. AAS: Two angles and their non-included side
Answer and Explanation: 1
Let us assume that we are talking about triangles for the convenience of explanation.
The possible combinations of four pairs of equal and corresponding sides or angles that can be given between two triangles are:
1. Three angles and one side
2. One angle and three sides
3. Two angles and two sides
Taking them case by case and finding out the probable congruence rules that can be applied.
Case 1: Three angles and one side
If all three angles of two triangles are equal and one side is also equal, then that side can be the included side of any of the angle pairs, and at the same time it can be the non-included side of a different side of angle pair.
Hence, both ASA and AAS can be used to prove congruence.
Case 2: One angle and three sides
If all sides of two triangles are equal and one angle is also equal, then the given angle can be the included angle of any of the side pairs. Also, all three sides are also given to be equal.
Hence, both SAS and SSS can be used to prove congruence.
Case 3: Two angles and two sides
If two angles are given and two sides are given, at least one side will be either the non-included side or the included side for the two angles out of the two sides given.
Hence, either AAS or ASA can be used to prove congruence.
Thus by knowing a combination of either four pairs of equal sides or equal angles guarantees one of the congruence relationships.
Learn more about this topic:
fromChapter 5 / Lesson 2
This lesson introduces the idea of congruency applied to triangles. It brings examples of ASA, SSS, and SAS triangle postulates to check the triangles' congruency.