**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 ****and****ΔDCB 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 ****and****ΔRTSare 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 = 40^{0}

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 = 35^{0}

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 = 70^{0}

∟ C = ∟Q = 60^{0}

**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 = 20^{0}

∟ DBC = ∟ADB = 30^{0}

**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 = 30^{0}

∟ACB = ∟DBC = 50^{0}

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 = 30 ^{o},∟B = 40^{o} and ∟C = 110^{o}, In ΔPQR, ∟P = 30^{o},∟Q = 40^{o} and ∟R = 110^{o} A student says that ΔABC **

**=**

**ΔPQR by AAA congruence criterion. Is he justified? Why or why not?**

Answer:

Given that,

In ΔABC, **∟**A = 30^{o}, **∟**B = 40^{o} and **∟**C = 110^{o}

In ΔPQR, **∟**P = 30^{o},**∟**Q = 40^{o} and **∟**R = 110^{o}

We have to show that,

ΔABC = ΔPQRby AAA congruence criterion.

In ΔABC = ΔPQR,

**∟**A =**∟**P = 30^{o}

**∟**B =**∟**Q = 40^{o}

**∟**C =**∟**R = 110^{o}

**From this we cannot say that ΔABC and****ΔPQR 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 = 90^{0}

BC = ED

**ΔABC and ****ΔFED are congruent by Side – Angle –Side (SAS) rule.**

Also See: Telangana class 7 Data Handling