In this chapter we will learn the concept of ASA congruency condition with solved examples.
Let us first review the basics of congruency of triangles.
What are congruent triangles?
Two triangles are congruent when they have equal sides and angles.
Hence, two congruent triangles can fully superimpose over each other.
What is ASA postulate?
ASA stands for Angle – Side- Angle.
The postulate states that of two triangles have equal angles and the side between them is also equal then the triangles are congruent.
For example, consider the two triangles ABC and PQR given below;
Here;
∠A = ∠R = 50 degree
AC = PR = 3.4 cm
∠C = ∠P = 80 degree
By ASA congruency, both the triangles are congruent.
Hence,
▵ ABC \mathtt{\cong } ▵RQP
Note:
The equal side should always present between the set of equal angles otherwise this postulate will be invalid.
I hope you understood the concepts, let us solve some problems for better understanding.
ASA Postulate – Solved Problems
(01) find the ∠Q in triangle PQR.
Solution
Comparing the above triangles;
∠ A = ∠ P = 60 degree
AB = PR = 4.5 cm
∠ B = ∠ R = 50 degree
By ASA postulate, both the triangles are congruent.
Hence, ▵ ABC \mathtt{\cong } ▵PRQ.
When triangles are congruent then all corresponding sides and angles are equal.
So, ∠C = ∠Q = 30 degree
(02) Find the length of side BD in below image.
Solution
Consider the triangle BAC and CDB.
∠ABC = ∠DCB = 40 degree
BC = CB = 6 cm
∠ACB = ∠DBC = 60 degree
By ASA congruency, both the triangles are congruent.
Hence, ▵ BAC \mathtt{\cong } ▵CDB
We know that in congruent triangles all corresponding sides and angles are equal.
AC = BD = 3.5 cm
(03) Given below is the equilateral triangle ABC such that line OA bisect ∠A. Also ∠ABO = ∠ACO. Check if triangle AOB and AOC is congruent.
Solution
We know that in equilateral triangle, side lengths are equal.
AB = BC = CA
Taking triangle AOB and AOC.
∠OAB = ∠OAC { bisected angles are equal }
AB = AC {equilateral triangle sides}
∠ABO = ∠ACO { given in question }
So by ASA postulate we can say that both triangles are congruent.
Hence, ▵ AOB \mathtt{\cong } ▵AOC
(04) Check if the triangles ABC and CDE are congruent.
Solution
Taking the triangles ABC and CDE
∠ABC = ∠ECD = 108 degree
BC = CD = 4 cm
∠ACB = ∠EDC = 30 degree
So by ASA postulate, both the triangles are congruent.
Hence, ▵ABC \mathtt{\cong } ▵ECD
(05) Using ASA postulate, prove that diagonal of parallelogram bisect each other.
Solution
Given is the parallelogram ABCD such that AC and BD are the diagonals.
We have to prove is diagonals bisect each other.
i.e. DO = OB and AO = OC
Take triangle AOB and DOC
∠OAB = ∠OCD {alternate interior angle}
CD = AB { opposite side of parallelogram are equal}
∠ODC = ∠OBA { alternate interior angle }
By ASA congruency both the triangles are congruent.
Hence, ▵AOB \mathtt{\cong } ▵COD
We know that congruent triangles have equal sides and angles.
So, DO = OB and AO = OC
Hence proved.
Next chapter : Understand AAS congruency in detail