State which pairs of triangles in Fig. 6.34 are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form
Solution:
(1) As we know If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar
This is referred to as the AAA (Angle - Angle - Angle) criterion of similarity of two triangles.
In ΔABC and ΔPQR
∠A = ∠P = 60
∠B = ∠Q = 80
∠C = ∠R = 40
All the corresponding angles of the triangles are equal.
By AAA criterion ΔABC ∼ ΔPQR
(2) As we know if in two triangles sides of one triangle are proportional to (i.e., in the same ratio of) the side of other triangles, then their corresponding angles are equal and hence the two triangles are similar.
This is referred as SSS (Side - Side - Side) similarity criterion for two triangles.
In ΔABC and ΔQRP
AB/QR = 2/4 = 1/2
BC/PR = 2.5/5 = 1/2
AC/PQ = 3/6 = 1/2
⇒ AB/QR = BC/PR = AC/PQ = 1/2
All the corresponding sides of the two triangles are in the same proportion. By SSS criterion ΔABC ∼ ΔQPR
(3) As we know if in two triangles sides of one triangle are proportional to (i.e., in the same ratio of) the side of other triangles, then their corresponding angles are equal and hence the two triangles are similar.
This is referred as SSS (Side - Side - Side) similarity criterion for two triangles.
In ΔLMP and ΔFED
LM/FE = 2.7/5
MPED = 2/4 = 1/2
LP/FD = 3/6 = 1/2
⇒ LM/FE ≠ MP = LP/FD
All the corresponding sides of the two triangles are not in the same proportion. Hence triangles are not similar. ΔLMP ∼ ΔFED
(4) As we know if one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.
This is referred as SAS (Side - Angle - Side) similarity criterion for two triangles.
In ΔNML and ΔPQR
NM/PQ = 2.5/5 = 1/2
M/QR = 5/10 = 1/2
⇒ NM/PQ = ML/QR = 1/2
∠M = ∠Q = 70°
One angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional.
By SAS criterion ⇒ ΔNML ∼ ΔPQR
(5) As we know if one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.
This is referred as SAS (Side - Angle - Side) similarity criterion for two triangles.
In ΔABC and ΔDFE
AB/DF = 2.5/5 = 1/2
BC/EF = 3/6 = 1/2
AB/DF = BC/EF = 1/2
∠A = ∠F = 80
But ∠B must be equal to 80°
The sides AB, BC includes ∠B , not ∠A)
Therefore, SAS criterion is not satisfied
Hence, the triangles are not similar, ΔABC ∼ ΔDFE
(6) As we know If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar
This is referred to as the AAA (Angle - Angle - Angle) criterion of similarity of two triangles.
In ΔDEF
∠D = 70°, ∠E = 80°
⇒ ∠F = 30° [∵ Sum of the angles in a triangle is 180°]
Similarly,
In ΔPQR
∠Q = 80° , ∠R = 30°
⇒ ∠P = 70°
In ΔDEF and ΔPQR
∠D = ∠P = 70°
∠E = ∠Q = 80°
∠F = ∠R = 30°
All the corresponding angles of the triangles are equal.
By AAA criterion ΔDEF ∼ ΔPQR
Alternate method:
As we are aware if two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.
This is referred to as the AA criterion for two triangles.
Solution:
In ΔDEF
∠D = 70°, ∠E = 80°
⇒ ∠F = 30° [Sum of the angles in a triangle is 180o]
Now,
In ΔDEF and ΔPQR
∠E = ∠Q = 80°
∠F = ∠R = 30°
The pair of corresponding angles of the triangles are equal.
By AA criterion ΔDEF ∼ ΔPQR
☛ Check: NCERT Solutions for Class 10 Maths Chapter 6
Video Solution:
State which pairs of triangles in Fig. 6.34 are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form:
Class 10 Maths NCERT Solutions Chapter 6 Exercise 6.3 Question 1
State which pairs of triangles in Fig. 6.34 are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form:
(1) Similar triangle with AAA criteria
(2) Similar triangle with SSS criteria
(3) Similar triangle with SSS criteria
(4) Similar triangle with SAS criteria
(5) Similar triangle with SAS criteria
(6) Similar triangle with AAA criteria
☛ Related Questions:
- Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using a similarity criterion for two triangles, show that.
- In Fig. 6.36 QR/QS = QT/PR and ∠1 = ∠2. Show that ΔPQS ~ ΔTQR.
- S and T are points on sides PR and QR of ∆PQR such that ∠P = ∠RTS. Show that ΔRPQ ~ ΔRTS.
- In Figure 6.37, if ∆ABE ≅ ∆ACD, show that ∆ADE ~ ∆ABC.