In this chapter we will learn AAS postulate of congruency with solved examples.
Let us first review the basics of congruency.
What is congruency of triangles?
Two triangles are said to be congruent when they have equal sides and angles.
When the congruent triangles are placed against each other they will overlap fully.
What is AAS theorem ?
AAS stands for Angle – Angle – Side.
When the two angles of triangle and a non included sides are equal then the given triangles are congruent.
Non included side is the one which do not lie between the equal angles.
For Example;
Consider the two triangles below;
In triangle ABC and PQR;
∠B = ∠Q = 60 degree
∠ C = ∠R = 55 degree
AB = PQ = 3.5 cm
Hence, by AAS congruency both the triangles are congruent.
▵ABC \mathtt{\cong } ▵PQR.
Note:
Here the side AB & PQ are non included sides.
This means that the sides does not lie between the equal angles.
I hope you understand the AAS concept of congruency. Let us now solve some problems for better understanding.
AAS Postulate – Solved Problems
(01) Observe the below image. Prove that ▵ABD \mathtt{\cong } ▵ACD.
Solution
Taking triangle ABD and ACD;
∠ BAD = ∠CAD
∠ ABD = ∠ACD
BD = CD
By AAS congruency, both the triangle are congruent.
Hence, ▵ABD \mathtt{\cong } ▵ACD.
(02) Study the below image and prove ▵ABO \mathtt{\cong } ▵DCO
Solution
Taking triangle ABO and DCO.
∠ ABO = ∠DCO { given }
∠ AOB = ∠DOC { Vertically opposite angle }
AB = CD { given in image }
By AAS congruency, both the triangles are congruent.
▵ABO \mathtt{\cong } ▵DCO
(03) Prove triangle ABD and CBD are congruent.
Solution
Taking triangle ABD and CBD.
∠ BAD = ∠BCD { given }
∠ BDA = ∠BDC = 90 degree
BD = DB { common side }
By AAS congruency, both triangles are congruent.
Hence, ▵ADB \mathtt{\cong } ▵CDB
(04) In the below image line BO bisect ∠AOC. Prove that ▵ONW \mathtt{\cong } ▵OMW
Solution
As line BO bisect ∠AOC, we can write;
∠WON = ∠ WOM
Taking triangle ONW and OMW.
∠ONW = ∠OMW { both measure 90 degree }
∠WON = ∠ WOM
OW = WO { common side }
By AAS congruency, both the triangles are congruent.
▵ONW \mathtt{\cong } ▵OMW
(05) In the below image line AB || CD. Prove that triangle ABD is congruent to triangle CDB.
Solution
Taking triangle ABD and CDB.
∠ ADB = ∠ CBD { alternate interior angle }
∠ ABD = ∠ CDB { alternate interior angle }
AD = BC { given in image}
By AAS congruency theorem, both triangles are congruent.
Hence, ▵ABD \mathtt{\cong } ▵CDB
Next chapter : Understand RHS congruency in detail