In this chapter we will learn the SAS congruency rule with solved examples.
Let us first review the basics of congruency.
What are congruent triangles?
Two triangles are congruent when they have equal sides and angles.
This means that congruent triangle will perfectly overlap when placed against each other.
What is SAS postulate?
The full form of SAS is side – angle – side.
It says that two triangles are congruent when they have two equal sides and angles between the sides are also equal.
For Example;
Consider the below triangle ABC and PQR
We can see that;
AB = PQ = 5 cm
∠A = ∠Q = 45 degree
AC = RQ = 4 cm
By SAS congruency, both the triangles are congruent.
Hence, ABC \mathtt{\cong } ▵PQR
I hope the concept is clear, let us look at some examples for better understanding.
SAS Postulate – Solved Problems
(01) In the below figure find value of ∠ACB
Solution
Comparing triangle ACB and DBC
AC = DB = 4 cm
∠BAC = ∠CDB = 86 degree
AB = CD = 3 cm
By SAS congruency, triangles ACB and DBC are congruent.
Hence, ▵ACB \mathtt{\cong } ▵DBC
We know that congruent triangles have equal sides and angles.
We can say that;
∠ACB = ∠BDC = 40 degree.
(02) check if the triangles ABC and CDA are congruent or not.
Solution
Taking triangle ABC and CDA.
AB = CD = 4 cm
∠BAC = ∠ACD = 90 degree
AC = CA (common side)
By SAS congruency, both the triangles are congruent.
Hence, ▵ABC \mathtt{\cong } ▵CDA
(03) Check if triangle AOB and COD are congruent.
Solution
Taking triangle OAB and OCD
OA = OC = 3.1 cm
∠AOB = ∠COD = 87 degree
OB = OD = 3.4 cm
By SAS congruency, both the triangles are congruent.
Hence, ▵AOB \mathtt{\cong } ▵COD
(04) Check if the triangles ABC and PQR are congruent or not.
Solution
Let us first find value of ∠PQR.
We know that sum of angles of triangle equals 180 degree.
∠P + ∠Q + ∠R = 180
51 + ∠Q + 63 = 180
∠Q + 114 = 180
∠Q = 180 – 114
∠Q = 66 degree.
Now comparing the triangles BAC and PQR
AB = PQ = 4 cm
∠A = ∠Q = 66 degree
AC = QR = 3.5 cm
By SAS congruency, both triangles are congruent.
Hence, ▵BAC \mathtt{\cong } ▵PQR
(05) Which of the following statement is true.
(i) ▵ABD \mathtt{\cong } ▵CBD
(ii) ▵ABD \mathtt{\cong } ▵CDB
(iii) ▵ABD \mathtt{\cong } ▵DBC
Solution
Taking the triangles ABD and CDB
AB = CD = 3 cm
∠BAD = ∠BCD
AD = BC = 4 cm
Both the triangles are congruent by SAS congruency.
So, ▵ABD \mathtt{\cong } ▵CDB
Hence option (ii) is correct among the given three options.
Note:
While mentioning the congruency of triangles the sequence of alphabet is important.
Foe example, In ABD and CDB;
AB = CD { first two alphabet of both triangles}
∠B = ∠D { second alphabets }
AD = BC { first and last alphabet }
Note that the letters are synchronized as per congruency.
Now consider the third option;▵ABD \mathtt{\cong } ▵DBC
\mathtt{AB\ \neq \ DB} { first two letters }
Hence, the naming is not properly synchronized.
Next chapter : Understand ASA congruency in detail