Telangana SCERT Solution Class VII (7) Math Chapter 8 Congruency of Triangles
Exercise – 1
1.) Decide whether the SSS congruence is true with the following figures. Give reasons
Answer:
Here we have to decide the given triangles are SSS congruence or not.
We know,
Side-Side-Side (SSS) criterion for congruence of triangles: If three sides of a triangle are equal to the corresponding three sides of another triangle, then the triangles are congruent.
By observing ΔACRandΔNBE
AC = BR
CR = NB
AR = NE
From this SSS congruence is true.
Answer:
Here we have to decide the given triangles are SSS congruence or not.
We know,
Side-Side-Side (SSS) criterion for congruence of triangles: If three sides of a triangle are equal to the corresponding three sides of another triangle, then the triangles are congruent.
By observing ΔLSDandΔLAD
LA = SD
LS ≠ AD
LD = LD
From this SSS congruence is false.
2.) For the following congruent triangles, find the pairs of corresponding angles
Answer:
Here we have to find the pairs of corresponding angles given that triangles are congruent.
We know,
When 2 triangles are congruent there corresponding angles are equal.
By observing ΔPTQandΔRSQ
PQ = QR
PT = RS
QT = QS
Δ PQT=ΔRSQ are congruent by Side-Side-Side (SSS) criterion.
Corresponding angles are
∟P = ∟R
∟TQP = ∟SQR
∟T = ∟S
Answer:
Here we have to find the pairs of corresponding angles given that triangles are congruent.
We know,
When 2 triangles are congruent there corresponding angles are equal.
By observing ΔPOQ andΔROS
PO = RO
OQ = OS
PQ = RS
ΔPOQ =ΔROSare congruent by Side-Side-Side (SSS) criterion.
Corresponding angles are
∟ROS = ∟POQ
∟R = ∟P
∟S = ∟Q
3.) In adjacent figure, choose the correct answer!
(i) ΔPQR =ΔPQS
(ii) ΔPQR =ΔQPS
(iii) ΔPQR =ΔSQP
(iv) ΔPQR =ΔSPQ
Answer:
From the given figure,
PR = SQ = 5 cm
PQ =PQ = common side
PS = RQ = 3.5 cm
From this
All combinations of triangles are congruent by Side-Side-Side (SSS) criterion.
4.) In the figure given below, AB = DC and AC = DB. Is ΔABC =ΔDCB
Answer:
Given that,
AB = DC and AC = DB.
We have to show that ΔABC=ΔDCB
By comparing ΔABC andΔDCB
AB = DC
BC = CB = common side
AC = DB.
From this
ΔABC andΔDCB are congruent by Side-Side-Side (SSS) criterion.
ΔABC =ΔDCB
Exercise – 2
1.) What additional information do you need to conclude that the two triangles given here under are congruent using SAS rule?
Answer:
Here, we have to show that the given two triangles are congruent by Side-Angle-Side (SAS) rule.
In ΔGHJandΔRTS
∟H = ∟T
To prove triangles ΔGHJ andΔRTSare congruent by Side-Angle-Side (SAS) rule we required more information such as GH = TR and HJ = TS
2.) The map given below shows five different villages. Village M lies exactly halfway between the two pairs of villages A and B as well as and P and Q. What is the distance between village A and village P. (Hint: check if ΔPAM =ΔQBM)
Answer:
Given map of five different villages.
Village M lies exactly halfway between the two pairs of villages A and B as well as and P and Q.
We have to find the distance between village A and village P.
Given that ΔPAM =ΔQBM
From ΔPAM =ΔQBM
AM = BM
PM = QM
AP = QB
But,
QM = 6 km
PM = 6 Km
Now,
AP = QB
QB = 4 km
Then
AP = 4 km.
The distance between village A and village P is 4 km.
3.) Look at the pairs of triangles given below. Are they congruent? If congruent write the corresponding parts.
Answer:
From figure, we have to show that triangles are congruent or not. If congruent then writing corresponding parts.
From figure,
In ΔACB and ΔSRT
∟A = ∟S
AC = SR = 6 cm
AB = ST = 4 cm
ΔACB and ΔSRT are congruent by Side-Angle-Side (SAS) rule.
Answer:
From figure, we have to show that triangles are congruent or not. If congruent then writing corresponding parts.
From figure,
In ΔPOQ and ΔROS
∟ POQ = ∟ROS
QO = SO = 4 cm
PO = RO = 3 cm
ΔPOQ and ΔROS are congruent by Side-Angle-Side (SAS) rule.
4.) Which corresponding sides do we need to know to prove that the triangles are congruent using the SAS criterion?
Answer:
We have to find which corresponding sides we need to prove that the triangles are congruent using the SAS criterion.
From figure,
In ΔABC and ΔRQP
∟ B = ∟Q = 400
BC =QP = 5 cm
To show triangle are congruent using the SAS criterion,
We need to know that AB = RQ
Answer:
We have to find which corresponding sides we need to prove that the triangles are congruent using the SAS criterion.
From figure,
In ΔADC and ΔABC
∟DAC = BAC = 350
AC =AC = common side
To show triangle are congruent using the SAS criterion,
We need to know that AB = AD
Exercise – 3
1.) In following pairs of triangles, find the pairs which are congruent? Also, write the criterion of congruence.
Answer:
From given figures,
In ΔABC and ΔRPQ
AB = RP = 6.5 cm
∟B = ∟P = 700
∟ C = ∟Q = 600
Given triangles ΔABC and ΔRPQ are congruent by Angle- Angle – Side rule.
Answer:
From given figures,
In ΔABD and ΔCDB
BD = BD = common side
∟ABD = ∟CDB = 200
∟ DBC = ∟ADB = 300
Given triangles ΔABD and ΔCDB are congruent by Angle – Side – Angle rule.
2.) In the adjacent figure.
(i) Are ΔABC and ΔDCB congruent?
(ii) Are ΔAOB congruent to ΔDOC?
Also identify the relation between corresponding elements and give reason for your answer.
Answer:
In given figure,
In ΔABC and ΔDCB
∟BAC = ∟CDB = 300
∟ACB = ∟DBC = 500
AC = BD = same length
ΔABC and ΔDCB are congruent by Angle – Side – Angle rule.
Exercise-4
1.) Which congruence criterion do you use in the following?
(i) Given:
AC = DF
AB = DE
BC = EF
So, ΔABC =ΔDEF
Answer:
Here,
In ΔABC andΔDEF
AC = DF
AB = DE
BC = EF
ΔABC and ΔDEF are congruent by Side – Side – Side rule.
(ii) Given:
ZX = RP
RQ = ZY
∟PRQ =∟XZY
So, ΔPQR =ΔXYZ
Answer:
Here,
In ΔPQR andΔXYZ
ZX = RP
RQ = ZY
∟PRQ =∟XZY
ΔPQR and ΔXYZare congruent by Side – Angle – Side(S-A-S) rule.
2.) You want to show that ΔART =ΔPEN,
(i) If you have to use SSS criterion, then you need to show
(a) AR =
(b) RT =
(c) AT =
Answer:
Here, we have to show ΔART =ΔPEN by Side – Side – Side (SSS) rule.
For this
We have to show 3 sides are equal to their corresponding sides.
(a)Side AR = Side PE
(b) SideRT =Side EN
(c) SideAT =Side PN
(ii) If it is given that ∟T = given that AT = PN N and you are to use SAS criterion, you need to have
(a) RT =
(ii) PN =
Answer:
Here, we have to show ΔART =ΔPEN by Side – Angle – Side (SAS) rule.
For this
We have to show 2 sides are equal and 1 angle is equal.
- a) SideRT = SideEN
b)Side PN =Side AT
(iii) If it is given that AT = PN and you are to use ASA criterion, you need to have
(a) ?
(b) ?
Answer:
Here, we have to show ΔART =ΔPEN by Angle – Side – Angle (ASA) rule.
Given that AT = PN
We have to show that,
∟A = ∟P
∟T =∟N
4.) In ΔABC, ∟A = 30o,∟B = 40o and ∟C = 110o, In ΔPQR, ∟P = 30o,∟Q = 40o and ∟R = 110o A student says that ΔABC = ΔPQR by AAA congruence criterion. Is he justified? Why or why not?
Answer:
Given that,
In ΔABC, ∟A = 30o, ∟B = 40o and ∟C = 110o
In ΔPQR, ∟P = 30o,∟Q = 40o and ∟R = 110o
We have to show that,
ΔABC = ΔPQRby AAA congruence criterion.
In ΔABC = ΔPQR,
∟A =∟P = 30o
∟B =∟Q = 40o
∟C =∟R = 110o
From this we cannot say that ΔABC andΔPQR are congruent but they are similar.
9.) Explain, why ΔABC =ΔFED
Answer:
We have to show that,
ΔABC =ΔFED
In ΔABC andΔFED
AB = EF
∟ABC = ∟FED = 900
BC = ED
ΔABC and ΔFED are congruent by Side – Angle –Side (SAS) rule.
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