Which congruence criterion do you use in the following ?
(a) Given: AC = DF, AB = DE, BC = EF
By SSS congruence criterion, since it is given that AC = DF, AB = DE, BC = EF
The three sides of one triangle are equal to the three corresponding sides of another triangle.
So ,ABC DEF
(b) Given: RP = ZX , RQ = ZY , ∠PRQ = ∠XZY
By SAS congruence criterion, since it is given that RP = ZX, RQ = ZY andPRQ = XZY
The two sides and one angle in one of the triangle are equal to the corresponding sides and the angle of other triangle.
(c) Given:MLN = FGH, NML = HFG, ML = FG
By ASA congruence criterion, since it is given thatMLN = FGH, NML = HFG, ML = FG.
The two angles and one side in one of the triangle are equal to the corresponding angles and side of other triangle.
(d) Given: EB = BD, AE = CB,A = C = 90°
By RHS congruence criterion, since it is given that EB = BD, AE = CB,A = C =
Hypotenuse and one side of a right angled triangle are respectively equal to the hypotenuse and one side of another right angled triangle.
You want to show thatART PEN:
(a) If you have to use SSS criterion, then you need to show:
(i) AR = (ii) RT = (iii) AT =
Sol : (a) Using SSS criterion,ART PEN
(i) AR = PE (ii) RT = EN (iii) AT = PN
(b) If it is given that ∠T = ∠N and you are to use SAS criterion, you need to have:
(i) RT = (ii) PN =
Sol : (b) Given:T = N
Using SAS criterion,ART PEN
(i) RT = EN (ii) PN = AT
(c) If it is given that AT = PN and you are to use ASA criterion, you need to have:
(i) ? (ii) ?
Sol : (c) Given: AT = PN
Using ASA criterion,ART ≅ PEN
(i)RAT = EPN (ii) RTA = ENP
You have to show thatAMP AMQ. In the following proof, supply the missing reasons:
|1. PM = QM||1. Given|
|PMA = QMA||2. Given|
|3. AM = AM||3. Common|
|4.AMP AMQ||4. SAS congruence rule|
InABC, ∠A = ∠B = and ∠C = 110º
InPQR, ∠P = 30º ∠Q = and ∠R =
A student says thatABC PQR by AAA congruence criterion. Is he justified? Why or why not?
No, because the two triangles with equal corresponding angles need not be congruent. In such a correspondence, one of them can be an enlarged copy of the other.
In the figure, the two triangles are congruent. The corresponding parts are marked. We can writeRAT ?
It can be observed that ,
∠RAT = ∠WON (Given)
AR = OW (Given)
∠ART = ∠OWN (Given)
By using ASA congruence criteria , ΔRAT = ΔWON
Complete the congruence statement:
BCA ? and QRS ?
Given that BC = BT
TA = CA
BA is common
∴ ΔBCA ≅ ΔBTA
Similarly PQ = RS
TQ = QS
PT = QR
∴ ΔQRS ≅ ΔTPQ
InBAT and BAC, given triangles are congruent so the corresponding parts are:
BB, A A, T C
Thus,BCA BTA [By SSS congruence rule]
InQRS and TPQ, given triangles are congruent so the corresponding parts are:
PR, T Q, Q S
Thus,QRS TPQ [By SSS congruence rule]
In a squared sheet, draw two triangles of equal area such that:
(i) the triangles are congruent.
(ii) the triangles are not congruent.
What can you say about their perimeters?
These triangles are congruent, i.e.,ABC PQR [By SSS congruence rule]
Here, ΔABC and ΔPQR have the same area and the perimeter of both the triangles will be the same because they are congruent .
Here, the two triangles have the same height and base. Thus, their areas are equal. However, these triangles are not congruent to each other. Also, the perimeter of both the triangles will not be the same.
Draw a rough sketch of two triangles such that they have five pairs of congruent parts but still the triangles are not congruent.
Let us draw two triangles ΔPQR and ΔABC.
All angles are equal, two sides are equal except one side. Hence,PQR are not congruent to ABC.
IfABC and PQR are to be congruent, name one additional pair of corresponding parts. What criterion did you use?
ABC and PQR are congruent. Then one additional pair is
Given:B = Q =
C = R
Therefore,ABC PQR [By ASA congruence rule]
Explain, whyABC FED
Given that , ∠ABC = ∠FED …(i)
∠BAC = ∠EFD …(ii)
The two angles of ΔABC are equal to the two respective angles of ΔFED . Also , the sum of all interior angles of a triangle is 180º . Therefore , third angle of both triangles will also be equal in measure .
∠BCA = EDF …(iii)
Also , given that BC = ED …(iv)
by using equation (i) , (ii) , (iii) and (iv) , we obtain
ΔABC≅ΔFED (ASA criteria)