 Triangles are said to be congruent if they satisfy the conditions of CPCT. CPCT means corresponding parts of congruent triangles. While learning the geometry and measurement topic, you must know the criteria for congruency of triangles and applications of congruence of triangles. Read this complete page to get more useful details about the conditions for triangle congruence applications.

Congruency of Triangles

Two triangles are said to be congruent when their corresponding sides, corresponding angles are congruent. Congruent means figures or shapes can be flipped, rotated to coincide with the other shapes. The following are the criteria for congruency that are used to prove that two triangles are congruent.

• SSS (Side-Side-Side): If any two triangles have three pairs of congruent sides, then those triangles are congruent to each other.
• SAS(Side-Angle-Side): If two triangles have two pairs of congruent sides, congruent included angle (the angle between the congruent sides), then the triangles are said to be congruent.
• AAS (Angle-Angle-Side): If two triangles have two pairs of congruent angles, and a non-common side in one triangle is congruent to the corresponding side in another triangle, then those are congruent.
• ASA(Angle-Side-Angle): If two triangles have two pairs of angles that are congruent, the common side of the congruent angles also congruent in both triangles, then triangles are congruent.
• RHL(Right angled-Hypotenuse-Leg): If two right triangles have one pair of congruent legs, hypotenuses congruent, then the triangles are congruent.

Applications of Congruency of Triangles

The below-mentioned are some of the applications of congruency of triangles.

Application 1:
By drawing one diagonal on a rectangle, it divides the rectangle into two congruent triangles. Diagonal BC, divides the rectangle ABCD into two triangles △ABC, △BCD
Congruent sides in both triangles are AB ≅ CD, AC ≅ BD [opposite sides of the rectangle have the same length], , BC ≅ BC [Common side in both triangles]
By using the SSS criterion, △ABC ≅ △BCD.
We can prove this by using RHL or SAS.

Application 2:
Show that both triangles are congruent in the following figure. Here, diagonal BC divides the quadrilateral ABCD into two triangles ABC, BCD.
AC ≅ BC [It is given]
∠BCD ≅ ∠CBA [It is given]
BC ≅ BC [Common side in both triangles]
By using SAS congruency criteria, △ABC ≅ △BCD.

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Examples on Congruency of Triangles Applications

Example 1:
Prove that the following two triangles are congruent. Solution:
Given that,
Two triangles are △ABC, △ACD
Congruent angles are ∠BAC ≅ ∠DAC, ∠BCA ≅ ∠DCA
AC ≅ AC [Common side]
By ASA criterion, △ABC ≅ △ACD

Example 2:
In the given figure, triangle XYZ is right-angled at Y. XMNZ and YOPZ are squares. Show that XP = YN. Solution:
Given that,
∆XYZ is a right angles triangle at Y, ∠Y = 90°, XMNZ and YOPZ are squares.
To show: XP = YN

Statement Reason
∠XZN = 90° Angle of square XMNZ.
∠YZN = ∠YZX + ∠XZN = x° + 90° From the image
∠YZP = 90° Angle of square YOPZ
∠XZP = ∠XZY + ∠YZP = x° + 90° Using the above statement
In ∆XZP and ∆YZN,
∠XZP = ∠YZN
ZP = YZ
XZ = ZN
Using statements 2 and 4
Sides of square YOPZ
Sides of square XMNZ
∆XZP ≅ ∆YZN By SAS Criterion of congruency
XP = YN (Shown) CPCT

Example 3:
G is the midpoint of EH. Are the following triangles congruent? Solution:
Given that,
G is the midpoint of EH
So, EG = GH
∠FEG ≅ ∠GHI
∠EGF ≅ ∠HGI as they are vertical angles
By using ASA, ∆EFG ≅ ∆GHI.
Yes, the triangles are congruent.