Content Curator
Congruence of triangles is a property of two triangles. Any two triangles are said to be congruent if all the 3 corresponding sides and angles of triangles are equal in measure. Read this entire article to know about the congruence of triangles, conditions for congruence of two triangles, corresponding parts of congruent triangles and solved examples related to congruence of triangles. Two shapes are said to be congruent if their size and shape are the same. Images of two shapes are in a way that the mirror image of one should coincide with the other.
| Table of Content |
Keyterms: Geometry, Triangle, Congruence in the triangles, Angle, Area of Triangle, SSS, SAS, ASA, AAS, RHS
Read more: Nature of roots of quadratic equations
Congruent Triangles
[Click Here for Sample Questions]
Two triangles are said to be congruent if their all-corresponding sides and angles are equal.
Sides: AB=BC=AC=PQ=QR=PR
Angles: ∠A=∠B=∠C=∠P=∠Q=∠R
So, we can say both triangles ABC and PQR are congruent.
Conditions for congruence of Triangles
[Click Here for Sample Questions]
If there are two triangles A and B then if they fulfil any of the below mentioned conditions then they are said to be congruent and they are mentioned like below:
A\(\cong\) B
- SSS (Side-Side-Side) – Two triangles are said to be congruent by SSS condition if all three sides are equal.
- SAS(Side-Angle-Side)- Two triangles are said to be congruent by SAS condition if their two sides and 1 angle are equal.
- ASA(Angle-Side-Angle)- Two triangles are said to be congruent by ASA condition if their two angles and 1 side are equal.
- AAS(Angle-Angle-Side)- Two triangles are said to be congruent by AAS condition if their two angles and 1 side are equal.
- RHS (Right angle- Hypotenuse-Side)- If the hypotenuse and side of one right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, the two triangles are congruent
Read more: Pythagoras Theorem
SSS: Side-Side- Side Rule
[Click Here for Sample Questions]
If all the three corresponding sides of two triangles are equal then they are said to be congruent by SSS rule.
So, in this case,
AB=PQ
AC=PR
BC=QR
Then triangles ABC and PQR are congruent by SSS rule.
SAS: Side-Angle- Side
[Click Here for Sample Questions]
If any two corresponding sides of two triangles and one angle are equal then two triangles are said to be congruent with SAS rule.
So, in this case,
AB=PQ
BC=QR and
∠A=∠P
Then triangles ABC and PQR are congruent by SAS rule.
ASA: Angle – Side - Angle
[Click Here for Sample Questions]
If any two corresponding angles of two triangles and one side is equal then two triangles are said to be congruent with ASA rule.
So, in this case,
∠B=∠Q
BC=QR and
∠A=∠P
Then triangles ABC and PQR are congruent by ASA rule
RHS: (Right angle- Hypotenuse-Side)
Two triangles are said to be congruent by RHS If the hypotenuse and a side of a right- angled triangle is equivalent to the hypotenuse and a side of the second right- angled triangle.
Also Read:
Things to Remember
- Congruent triangles are those triangles whose sides and angles are exactly equal.
- There are 4 rules to determine if two triangles are congruent: SSS, SAS, ASA, RHS
- SSS congruence rule states that if all sides of a triangle are equal, the triangles are congruent.
- SAS congruence rule states that if two sides and an angle in the middle of the two sides are equal, the two triangles are congruent.
- ASA congruence rule states that if two angles and a side in the middle of the two angle are equal, the triangles are congruent
- RHS rule states that if in a right angled triangle hypotenuse and one side are equal, the two triangles are congruent.
Read More: Euclidean plane geometry
Sample Questions
Ques1: In triangle ABC, if AB=AC and ∠B= 70°, then what is the value of ∠A? (4 Marks)
Ans. In a Δ ABC
AB=AC (Given)
∠B=∠C= 70°, (Angle opposite to equal sides are equal)
As per angle sum property some of 3 angles of a triangle is 180 degrees.
∠A + ∠B +∠C= 180°
∠A+70°+70°= 180°
∠A = 180° – 140°
∠A = 40°
Ques2: What is CPCT? (2 Marks)
Ans. CPCT stands for Corresponding Parts of Congruent Triangles. It is a theorem which states that if two or more triangles which are congruent to each other then the corresponding angles and sides are also congruent to each other.
Ques3: Is it possible two equilateral triangles are always congruent? (2 Marks)
Ans. No it is not always possible. Equilateral triangle angles are of 60 degrees each but it is noy necessary that their sides are always equal.
Ques4: The length of two sides of an isosceles triangle are 5cm and 8 cm, find perimeter of triangle. (3 Marks)
Ans. If we consider two sides of isosceles triangles are 5 cm and third one 8 cm then perimeter will be 5 cm+5 cm+8 cm= 18 cm
If we consider two sides of isosceles triangles are 8 cm and third one 5 cm then perimeter will be 8 +8 +5 = 21 cm.
Ques 5: In the given figure, find these two triangles LMN and ONM are congruent and by which rule? (4 Marks)
Ans. In ΔLMN and ΔONM
LM=ON
LN=OM
MN=NM
By SSS rule, ΔLMN and ΔONM are congruent.
Ques 5: The triangle ABC is an isosceles triangle. In the given figure, AB = AC and AP = AQ. Show that CP = BQ. (4 Marks)
Ans.
Given: In ΔABQ and ΔACP,
AB = AC (Stated in question)
∠BAQ = ∠CAP (Common angle)
AQ = AP (Stated in question)
Hence using SAS congruence,
ΔABQ \(\cong\)ΔACP
So, CP = BQ (Corresponding parts of congruent triangles)
Ques 6: ΔABC and ΔDBC are 2 isosceles triangles on the same base, as shown in the figure. The line segment BC and vertices A and D are on the same side of BC and AD has been extended to intersect BC at P. Prove that : (i) ΔABD \(\cong\) ΔACD (ii) ΔABP \(\cong\) ΔACP (4 Marks)
Ans: (i) In the triangles ABD & ACD,
It is given that AB = AC and BD = CD.
AD is common to both triangles.
So, using SSS congruence,
ΔABD \(\cong\) ΔACD
(ii) In the triangles ABP & ACP,
It is given that AB = AC
∠BAP = ∠CAP [corresponding parts of congruent triangles, ΔABD \(\cong\) ΔACD]
AP is common to both triangles.
So SAS congruence rule,
ΔABP \(\cong\) ΔACP
Ques 7: In the figure given below, AE = AD & BD = CE. Show that ΔAEB \(\cong\) ΔADC. (4 Marks)
Ans: It is given that AE = AD … (a)
& CE = BD … (b)
Adding (a) and (b),
We get AE + CE = AD + BD
So, AC = AB … (c)
In ΔAEB & ΔADC,
It is given that AE = AD
AB = AC (from (c))
∠A is common in both the triangles.
So, using SAS congruence criteria,
ΔAEB is congruent to ΔADC
Check-Out:
Comments