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.
More Related Articles:
- Classification of Triangles
- Properties of Angles of a Triangle
- An Altitude of an Equilateral Triangle is also a Median
- Point on the Bisector of an Angle
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.
Frequently Asked Question’s
1. What is the use of congruence in maths?
Two figures are congruent means they have the same shape and size or one is a mirror image of others. The use of congruency is to say they occupy the same space and the same quantity of material is used to make them.
2. Where do we use triangles in everyday life?
We can see triangles as food items like chips, samosas and traffic signs.
3. What is the application of congruent triangles?
Two triangles are congruent if they have corresponding angles, sides. The applications of congruency of triangles are rectangle with one diagonal, rhombus with diagonal, square with anyone diagonal and so on.