SAS Congruence Rule Solved Examples
Example 1: Check if given triangles below are congruent or not using SAS congruency rule.
Solution:
Here, EF = MO = 3in [side 1]
FG = NO = 4.5in [side 2]
∠EFG = ∠MON = 110° [the angle between them]
Thus, △EFG ≅ △MNO ( By SAS rule ).
∴ These triangles are congruent by the SAS rule.
Hence Proved.
Example 2: ΔABC is an isosceles triangle and the line segment AD is the angle bisector of ∠A. Prove that ΔADB ≅ ΔADC by using SAS rule.
Solution:
ΔABC is an isosceles triangle, where it is given that AB=AC. [side 1]
Now the side AD is common in both the triangles ΔADB and ΔADC. [side 2]
As the line segment AD is the angle bisector of the angle A then it divides the ∠A into two equal parts.
Therefore, ∠BAD=∠CAD. [angle]
Hence, according to the SAS rule, the two triangles are congruent.
∴ ΔADB ≅ ΔADC.
Hence Proved.
Example 3: In isosceles triangle ΔPQR, point L is marked, M is the midpoint of the equal sides (PQ and QR) of the triangle and N as the midpoint of the third side. Is LN=MN?
Solution:
We need to prove: ΔLPN ≅ ΔMRN
Given that ΔPQR is an isosceles triangle and PQ=QR
Angles opposite to equal sides are equal. Thus, ∠QPR=∠QRP
Since L and M are the midpoints of PQ and QR respectively, hence, PL = LQ = QM = MR = QR/2
N is the midpoint of PR. Hence, PN = NR
In ΔLPN and ΔMRN:
LP = MR [side 1]
∠LPN = ∠MRN [angle]
PN = NR [side 2]
Thus, by SAS Criterion of Congruence, ΔLPN ≅ ΔMRN.
Since congruent parts of congruent triangles are equal, LN=MN.
Hence proved.
Example 4: AB is a line segment and line l is its perpendicular bisector. If a point P lies on l, show that P is equidistant from A and B.
Solution:
Line l ⊥ AB and passes through C which is the mid-point of AB.
We need to show that PA = PB.
So, let’s consider Δ PCA and Δ PCB.
C is the mid-point of AB. Hence, AC = BC
∠PCA = ∠PCB = 90°
PC is the common side for both triangles. Hence, PC = PC
Thus, by SAS Criterion of Congruence, ΔPCA ≅ ΔPCB
PA = PB, as they are corresponding sides of congruent triangles.
Hence proved.
Important Related Links:
SAS Congruence Rule
SAS Congruence Rule: SAS Congruence Rule is a principle in geometry that provides a method for determining if two triangles are congruent, meaning they have the same size and shape. SAS stands for Side-Angle-Side, indicating that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.
Condition of Congruency of Two Triangle: When the two sides of a triangle are equal to the two sides of another triangle, and the angles formed by these sides (the included angles) are also equal, then the two triangles are congruent Congruence between any two geometric objects is represented by “≅” which reads as ‘is congruent to’.
In this article, we will discuss the SAS Congruence Rule and criteria of congruence of right-angle triangles with examples and proof.
Table of Content
- What is Congruence?
- What is SAS Congruence Rule?
- SAS Congruence Rule Definition
- Criteria for SAS Congruence Rule
- SAS Congruence Rule Proof
- How to apply SAS Congruence Rule?
- SAS Similarity Criteria
- Congruence vs Similarity | Difference Between Congruence and Similarity
- SAS Congruence Rule Class 9
- SAS Congruence Rule Solved Examples
- SAS Congruence Rule – Practice Problems