Congruence and Similarity of Triangles
What is Congruence and Similarity of Triangles?
Two triangles are congruent if they have the same size and shape. All corresponding sides and angles are equal.
Key formula / rule: SSS Congruence
Key points
- To differentiate between congruent and similar triangles.
- To apply congruence postulates (SSS, SAS, ASA, AAS, RHS) to prove triangle congruence.
- To apply similarity postulates (AAA, SSS, SAS) to prove triangle similarity.
- To use CPCTC to find unknown sides and angles.
Common exam trap
Confusing congruence with similarity.
Definitions
- Term
Congruent Triangles
- Meaning
Two triangles are congruent if they have the same size and shape. All corresponding sides and angles are equal.
- Term
Similar Triangles
- Meaning
Two triangles are similar if they have the same shape but not necessarily the same size. Their corresponding angles are equal, and the ratio of their corresponding sides is constant.
- Term
CPCTC
- Meaning
Corresponding Parts of Congruent Triangles are Congruent. This principle is used after proving triangles congruent to equate their corresponding sides and angles.
- Term
Ratio of Similarity
- Meaning
The constant ratio between the lengths of corresponding sides of two similar triangles.
Learning objectives
To differentiate between congruent and similar triangles.
To apply congruence postulates (SSS, SAS, ASA, AAS, RHS) to prove triangle congruence.
To apply similarity postulates (AAA, SSS, SAS) to prove triangle similarity.
To use CPCTC to find unknown sides and angles.
To calculate ratios of sides, perimeters, and areas of similar triangles.
Formulae
- Name
SSS Congruence
- Note
∆ABC ≅ ∆PQR if AB=PQ, BC=QR, CA=RP.
- Expression
If three sides of one triangle are equal to the three corresponding sides of another triangle, then the triangles are congruent.
- Name
SAS Congruence
- Note
∆ABC ≅ ∆PQR if AB=PQ, ∠B=∠Q, BC=QR.
- Expression
If two sides and the included angle of one triangle are equal to the two corresponding sides and the included angle of another triangle, then the triangles are congruent.
- Name
ASA Congruence
- Note
∆ABC ≅ ∆PQR if ∠B=∠Q, BC=QR, ∠C=∭R.
- Expression
If two angles and the included side of one triangle are equal to the two corresponding angles and the included side of another triangle, then the triangles are congruent.
- Name
AAS Congruence
- Note
∆ABC ≅ ∆PQR if ∠A=∠P, ∠B=∠Q, AC=PR (or BC=QR).
- Expression
If two angles and a non-included side of one triangle are equal to the two corresponding angles and the non-included side of another triangle, then the triangles are congruent.
- Name
RHS Congruence
- Note
For ∆ABC and ∆PQR, if ∠B=∠Q=90°, AC=PR, and AB=PQ (or BC=QR).
- Expression
In two right-angled triangles, if the hypotenuse and one side of one triangle are equal to the hypotenuse and the corresponding side of the other triangle, then the triangles are congruent.
- Name
AAA Similarity
- Note
∆ABC ~ ∆PQR if ∠A=∠P, ∠B=∠Q, ∠C=∭R.
- Expression
If all three corresponding angles of two triangles are equal, then the triangles are similar.
- Name
SSS Similarity
- Note
∆ABC ~ ∆PQR if AB/PQ = BC/QR = CA/RP.
- Expression
If the three corresponding sides of two triangles are in the same ratio, then the triangles are similar.
- Name
SAS Similarity
- Note
∆ABC ~ ∆PQR if AB/PQ = BC/QR and ∠B=∠Q.
- Expression
If two corresponding sides of two triangles are in the same ratio and the included angles are equal, then the triangles are similar.
- Name
Ratio of Areas of Similar Triangles
- Note
If ∆ABC ~ ∆PQR, then Area(∆ABC)/Area(∆PQR) = (AB/PQ)² = (BC/QR)² = (CA/RP)².
- Expression
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
- Name
Ratio of Perimeters of Similar Triangles
- Note
If ∆ABC ~ ∆PQR, then Perimeter(∆ABC)/Perimeter(∆PQR) = AB/PQ = BC/QR = CA/RP.
- Expression
The ratio of the perimeters of two similar triangles is equal to the ratio of their corresponding sides.
Prerequisites
Basic geometry concepts (lines, angles, polygons).
Properties of triangles (sum of angles, types of triangles).
Understanding of ratios and proportions.
Common mistakes
Confusing congruence with similarity.
Incorrectly identifying corresponding sides or angles.
Applying congruence criteria where similarity is required, or vice-versa.
Forgetting to square the ratio of sides when calculating the ratio of areas.
Keywords
Congruence
Similarity
Triangles
SSS
SAS
ASA
AAS
RHS
AAA
CPCTC
Ratio of Areas
Ratio of Perimeters
Practice preview
If two triangles are similar, which of the following statements is always true?…
easy
In a right-angled triangle ABC, right-angled at B, BD is perpendicular to AC. If AB = 6 cm and BC = 8 cm, what is the length of BD?…
medium
The areas of two similar triangles are 81 sq cm and 49 sq cm respectively. If the altitude of the larger triangle is 9 cm, what is the corresponding altitude of the smaller triangle?…
medium