Congruence

State the condition (SSS or SAS or ASA or RHS) under which △ABC ≅ △PQR in each of the following cases.

(i)

(ii)

(iii)

(iv)

Answer

(i)

AB = QP = 2.5 cm

BC = QR = 4 cm

AC = PR = 3.5 cm

Condition: SSS

(ii)

∠B = ∠Q = 90° $\quad$[Right angle]

AC = PR $\quad$[Hypotenuse]

AB = PQ $\quad$[Side]

Condition: RHS

(iii)

∠A = ∠P = 35° $\quad$[Angle]

AB = PQ $\quad$[Side]

∠B = ∠Q = 90° $\quad$[Angle]

Condition: ASA

(iv)

AC = QR $\quad$[Side]

∠C = ∠R = 90° $\quad$[Angle]

BC = RP $\quad$[Side]

Condition: SAS

State giving reasons, which of the following are pairs of congruent triangles. Write down the names of triangles with congruent sign in each case, wherever they are congruent.

(i)

(ii)

(iii)

(iv)

(v)

Answer

(i)

Consider △ABC,

∠A = 180° - (60° + 50°) = 70°.

Consider △FDE,

∠F = 180° - (60° + 50°) = 70°.

In △ABC and △FDE:

∠A = ∠F = 70° $\quad$[Angle]

AB = FD $\quad$[Side]

∠B = ∠D = 60° $\quad$[Angle]

Since two angles and the included side match, they are congruent by ASA condition of congruence.

△ABC ≅ △FDE

(ii)

In △ABC and △PQR:

AB = PR $\quad$[Side]

∠B = ∠R = 35° $\quad$[Angle]

BC = QR $\quad$[Side]

Since two sides and the included angle match, they are congruent by SAS condition of congruence.

△ABC ≅ △PRQ

(iii)

In △DEH and △PQR:

DH = RP $\quad$[Side]

∠H = ∠P = 90° $\quad$[Right-angle]

HE = PQ $\quad$[Side]

Since two sides and the included right angle match, they are congruent by SAS condition of congruence.

△DEH ≅ △RQP

(iv)

In △ABC and △PQR:

AB = RQ $\quad$[Side]

∠B = ∠Q = 90° $\quad$[Right-angle]

CB = PQ $\quad$[Side]

Since two sides and the included right angle match, they are congruent by SAS condition of congruence.

△ABC ≅ △RQP

(v)

Consider △XYZ,

∠X = 180° - (30° + 80°) = 70°.

In △LMN and △XYZ:

LM = ZX $\quad$[Side]

∠M = ∠X = 70° $\quad$[Angle]

MN = XY $\quad$[Side]

Since two sides and the included angle match, they are congruent by SAS condition of congruence.

△LMN ≅ △ZXY

State, giving reasons, whether the following pairs of triangles are congruent or not:

△ABC in which ∠A = 50°, ∠B = 60°, BC = 4.5 cm and △DEF in which ∠E = 60°, EF = 4.5 cm, ∠F = 70°.

Answer

△ABC in which ∠A = 50°, ∠B = 60°, BC = 4.5 cm and △DEF in which ∠E = 60°, EF = 4.5 cm, ∠F = 70°.

Consider △ABC,

∠C = 180° - (50° + 60°) = 70°.

In △ABC and △DEF:

∠C = ∠F = 70° $\quad$[Angle]

BC = EF = 4.5 cm $\quad$[Side]

∠B = ∠E = 60° $\quad$[Angle]

Since two angles and the included side match,

They are congruent by ASA condition of congruence.

State, giving reasons, whether the following pairs of triangles are congruent or not:

△DEF in which ∠E = 48°, DE = 6 cm, EF = 8 cm and △MNR in which ∠R = 48°, MN = 6 cm, MR = 8 cm.

Answer

△DEF in which ∠E = 48°, DE = 6 cm, EF = 8 cm and △MNR in which ∠R = 48°, MN = 6 cm, MR = 8 cm.

In △DEF, the angle 48° is between the two given sides (SAS).

In △MNR, the angle 48° is at vertex R, which is not the included angle between side MN and MR (the included angle would be ∠M).

The given angle is not in the corresponding "included" position for both triangles.

∴ They are not congruent.

State, giving reasons, whether the following pairs of triangles are congruent or not:

△KLM in which KM = 4 cm, ∠K = 75°, ∠M = 40° and △PQR in which PR = 4 cm, ∠Q = 65°, ∠R = 40°.

Answer

△KLM in which KM = 4 cm, ∠K = 75°, ∠M = 40° and △PQR in which PR = 4 cm, ∠Q = 65°, ∠R = 40°.

Consider △PQR,

∠P = 180° - (65° + 40°) = 75°.

In △KLM and △PQR:

∠K = ∠P = 75° $\quad$[Angle]

KM = PR = 4 cm $\quad$[Side]

∠M = ∠R = 40° $\quad$[Angle]

Since two angles and the included side match,

△KLM ≅ △PRQ by ASA condition of congruence.

State, giving reasons, whether the following pairs of triangles are congruent or not:

△ABC in which AB = 3 cm, ∠A = 90°, BC = 5 cm and △KLM in which KM = 3 cm, ∠K = 90°, LM = 5 cm.

Answer

△ABC in which AB = 3 cm, ∠A = 90°, BC = 5 cm and △KLM in which KM = 3 cm, ∠K = 90°, LM = 5 cm.

In △ABC and △KML:

AB = KM = 3 cm $\quad$[Side]

∠A = ∠K = 90° $\quad$[Right angle]

BC = LM = 5 cm $\quad$[Hypotenuse]

Since one side, the hypotenuse and the right angle match,

△ABC ≅ △KML by RHS condition of congruence.

In the given figure, AB = CD and AD = CB. Prove that :

(i) △ABD ≅ △CDB

(ii) ∠A = ∠C

Answer

(i) △ABD ≅ △CDB

In △ABD and △CDB:

AB = CD $\quad$[Given]

AD = CB $\quad$[Given]

BD = DB $\quad$[Common side]

∴ △ABD ≅ △CDB $\quad$[By SSS condition of congruence]

(ii) ∠A = ∠C

Since △ABD ≅ △CDB, corresponding angles are equal.

∴ ∠A = ∠C $\quad$[CPCT]

⇒ Hence proved

In the given figure, ∠SPR = ∠QRP and ∠RSP = ∠PQR. Prove that :

(i) PQ = RS

(ii) PS = QR

Answer

To prove that the sides are equal, we first show that the two triangles formed by diagonal PR are congruent.

In △SPR and △QRP:

∠SPR = ∠QRP $\quad$[Given]

∠RSP = ∠PQR $\quad$[Given]

∴ ∠SRP = ∠QPR $\quad$[Third angles of triangles]

PR = RP $\quad$[Common side]

△SPR ≅ △QRP $\quad$[By ASA condition of congruence]

Therefore, by CPCT:

(i) PQ = RS

(ii) PS = QR

⇒ Hence proved

In the given figure, we have AO = BO and CO = DO. Prove that :

(i) △AOC ≅ △BOD

(ii) AC = BD

Answer

(i) To prove △AOC ≅ △BOD

In △AOC and △BOD:

AO = BO $\quad$[Given]

∠AOC = ∠BOD $\quad$[Vertically opposite angles are always equal]

CO = DO $\quad$[Given]

△AOC ≅ △BOD $\quad$[By SAS condition of congruence]

(ii) To prove AC = BD

Since △AOC ≅ △BOD, we have:

AC = BD $\quad$[Corresponding Parts of Congruent Triangles]

⇒ Hence proved

In the given figure, AB ⊥ BD, CD ⊥ BD and AB = CD. Prove that :

(i) △ABD ≅ △CDB

(ii) AD = CB

Answer

(i) To prove △ABD ≅ △CDB

In △ABD and △CDB:

AB = CD $\quad$[Given]

∠ABD = ∠CDB = 90° $\quad$[Right-angle]

BD = BD $\quad$[Common side to both triangles]

△ABD ≅ △CDB $\quad$[By SAS condition of congruence]

(ii) To prove AD = CB

Since △ABD ≅ △CDB, we have

AD = CB $\quad$[Corresponding Parts of Congruent Triangles]

⇒ Hence proved

In the given figure, PL ⊥ OA and PM ⊥ OB such that OL = OM. Prove that :

(i) △OLP ≅ △OMP

(ii) PL = PM

(iii) ∠LOP = ∠MOP

Answer

(i) To prove △OLP ≅ △OMP

In △OLP and △OMP:

OL = OM $\quad$[Given]

∠OLP = ∠OMP = 90° $\quad$[Given PL ⊥ OA and PM ⊥ OB]

OP = OP $\quad$[Common side to both triangles, acting as hypotenuse]

△OLP ≅ △OMP $\quad$[By RHS condition of congruence]

(ii) To prove PL = PM

Since △OLP ≅ △OMP, we have:

PL = PM $\quad$[Corresponding Parts of Congruent Triangles]

(iii) To prove ∠LOP = ∠MOP

Again, using the property of CPCT from congruence proof in part (i):

∠LOP = ∠MOP

⇒ Hence proved

In the given figure, we have PQ = SR and PR = SQ. Prove that :

(i) △PQR ≅ △SRQ

(ii) ∠PQR = ∠SRQ

Answer

(i) To prove △PQR ≅ △SRQ

In △PQR and △SRQ:

PQ = SR $\quad$[Given]

PR = SQ $\quad$[Given]

QR = QR $\quad$[Common base for both triangles]

△PQR ≅ △SRQ $\quad$[By SSS condition of congruence]

(ii) To prove ∠PQR = ∠SRQ

Since △PQR ≅ △SRQ, we can conclude that all their corresponding parts are equal.

By the property of CPCT (Corresponding Parts of Congruent Triangles):

∠PQR = ∠SRQ

⇒ Hence proved

In the given figure, we have AC ⊥ CD, BC ⊥ CD and DA = DB. Prove that : CA = CB.

Answer

To prove CA = CB, first prove △DCA ≅ △DCB

In △DCA and △DCB:

DA = DB $\quad$[Given]

∠DCA = ∠DCB = 90° $\quad$[Given AC ⊥ CD and BC ⊥ CD]

CD = CD $\quad$[Common side]

△DCA ≅ △DCB $\quad$[By RHS condition of congruence]

Since △DCA ≅ △DCB, we have:

CA = CB $\quad$[Corresponding Parts of Congruent Triangles]

⇒ Hence proved

In the adjoining figure, △ABC is an isosceles triangle in which AB = AC and AD is a median. Prove that :

(i) △ADB ≅ △ADC

(ii) ∠BAD = ∠CAD

Answer

(i) To prove △ADB ≅ △ADC

In △ADB and △ADC:

AB = AC $\quad$[Given]

BD = CD $\quad$[Given that AD is a median, which by definition bisects the base BC]

AD = AD $\quad$[Common side to both triangles]

△ADB ≅ △ADC $\quad$[By SSS condition of congruence]

(ii) To prove ∠BAD = ∠CAD

Since △ADB ≅ △ADC, we have:

∠BAD = ∠CAD $\quad$[Corresponding angles of congruent triangles]

⇒ Hence proved

In the adjoining figure, △ABC is an isosceles triangle in which AB = AC and AD is the bisector of ∠A. Prove that :

(i) △ADB ≅ △ADC

(ii) ∠B = ∠C

(iii) BD = DC

(iv) AD ⊥ BC

Answer

(i) To prove △ADB ≅ △ADC

In △ADB and △ADC:

AB = AC $\quad$[Given]

∠BAD = ∠CAD $\quad$[Given, as AD is the bisector of ∠A]

AD = AD $\quad$[Common side to both triangles]

∴ △ADB ≅ △ADC $\quad$[By SAS condition of congruence]

(ii) To prove ∠B = ∠C

Since △ADB ≅ △ADC, we have:

∠B = ∠C $\quad$[Corresponding angles of congruent triangles]

(iii) To prove BD = DC

Since △ADB ≅ △ADC, we have:

BD = DC $\quad$[Corresponding parts of congruent triangles]

(iv) To prove AD ⊥ BC

From CPCT, we know that:

∠ADB = ∠ADC

Since BC is a straight line, these two angles form a linear pair:

∴ ∠ADB + ∠ADC = 180°

⇒ ∠ADB + ∠ADB = 180° $\quad$[∵∠ADB = ∠ADC]

⇒ 2∠ADB = 180°

⇒ ∠ADB = $\dfrac{180^\circ}{2}$

⇒ ∠ADB = 90°

Since the angle is 90°,

AD ⊥ BC

⇒ Hence proved

In the adjoining figure, △ABC is an isosceles triangle in which AB = AC. If BM ⊥ AC and CN ⊥ AB, prove that :

(i) △BMC ≅ △CNB

(ii) BM = CN

Answer

(i) To prove △BMC ≅ △CNB

Since AB = AC, the base angles of isosceles triangle ABC are equal.

∴ ∠ABC = ∠ACB

As N lies on AB and M lies on AC,

∠CBN = ∠BCM

Also,

∠BMC = ∠CNB = 90° $\quad$[Given, BM ⊥ AC and CN ⊥ AB]

∴ ∠MBC = ∠BCN $\quad$[Third angles of triangles]

In △BMC and △CNB:

∠MBC = ∠BCN $\quad$[Angle]

BC = CB $\quad$[Common side]

∠BCM = ∠CBN $\quad$[Angle]

∴ △BMC ≅ △CNB $\quad$[By ASA condition of congruence]

(ii) To prove BM = CN

Since △BMC ≅ △CNB, we have:

BM = CN $\quad$[Corresponding parts of congruent triangles]

⇒ Hence proved

Two figures are said to be congruent if they have the same

1. shape
2. size
3. shape and size
4. area

Answer

Congruence implies that one figure can be placed exactly over the other. To achieve this, both the shape (angles and proportions) and the size (actual measurements) must be identical.

Hence, Option 3 is the correct option.

The symbol for congruency is

1. =
2. ≡
3. ≈
4. ≅

Answer

= → means equal in value or area.

≡ → denotes identity or equivalence.

≈ → means approximately equal.

≅ → is the specific mathematical symbol for congruence.

Hence, Option 4 is the correct option.

Two line segments are congruent only if

1. they have at least one end point common
2. they are equal in length
3. they are equal in length and parallel to each other
4. they have coincident end points

Answer

For 1D objects like line segments, the only "size" they have is length. If two segments have the same length, they are congruent, regardless of where they are positioned or if they are parallel.

Hence, Option 2 is the correct option.

Two angles are congruent if they have

1. a common vertex
2. a common arm
3. the same measure
4. equal lengths of their arms

Answer

Two angles are congruent if they have the same degree measure. The length of the arms does not matter because arms are rays that extend infinitely.

Hence, Option 3 is the correct option.

Two circles are said to be congruent only if

1. they have the same centre
2. they have the same radius
3. they have the same centre and same radius
4. they have the same radius and lie on different planes

Answer

Since all circles have the same shape, the only factor that determines their size is the radius.

If two circles have equal radii, they are congruent. They do not need to share the same center (that would make them concentric, not just congruent).

Hence, Option 2 is the correct option.

Which of the following is not a condition for triangles to be congruent?

1. SSS
2. SAS
3. ASA
4. AAA

Answer

SSS, SAS, and ASA are valid congruence criteria.

AAA (Angle-Angle-Angle) only proves that triangles are the same shape (similar), but not necessarily the same size. A small equilateral triangle and a giant equilateral triangle have the same angles but are not congruent.

Hence, Option 4 is the correct option.

If △ABC ≅ △DEF, then

1. AB = DF
2. BC = EF
3. AC = DE
4. All of these

Answer

In a congruence statement, the order of letters matters.

AB corresponds to DE

BC corresponds to EF

AC corresponds to DF

Therefore, only BC = EF is necessarily true based on the corresponding parts of congruent triangles (CPCT).

Hence, Option 2 is the correct option.

Two triangles are congruent. Between them, which of the following can be different?

1. area
2. perimeter
3. shape
4. orientation

Answer

Congruent triangles must have the same area, perimeter, and shape. However, one triangle can be rotated, flipped, or moved to a different position (orientation) and still remain congruent to the original.

Hence, Option 4 is the correct option.

Fill in the blanks :

(i) Two triangles having corresponding angles equal are always ............... .

(ii) Two squares are congruent if they have ............... .

(iii) If two triangles are similar, then they have corresponding ............... equal.

(iv) If two triangles are congruent, then they have corresponding ............... equal.

(v) Two congruent triangles are always ............... .

Answer

(i) Two triangles having corresponding angles equal are always similar.

(ii) Two squares are congruent if they have the same side length.

(iii) If two triangles are similar, then they have corresponding angles equal.

(iv) If two triangles are congruent, then they have corresponding sides and angles equal.

(v) Two congruent triangles are always similar.

Explanation

(i) Two triangles have the same shape, but they might be different sizes unless a corresponding side is also equal.

(ii) All squares have 90° angles, so the only factor that determines congruence is the length of the side.

(iii) The corresponding sides of two similar triangles are proportional, but not necessarily equal.

(iv) If two triangles are congruent, then they have corresponding sides and angles equal. This is often referred to as CPCT — Corresponding Parts of Congruent Triangles.

(v) Congruence is a special case of similarity where the ratio of corresponding sides is 1:1.

Write true (T) or false (F) :

(i) All squares are congruent.

(ii) All concentric circles are congruent.

(iii) If two squares have equal areas, they are congruent.

(iv) If two figures have equal areas, they are congruent.

(v) If two equilateral triangles are equal in area, they are congruent.

(vi) If the hypotenuse and an acute angle of a right triangle are equal to the hypotenuse and the corresponding acute angle of another right triangle, then the triangles are congruent.

Answer

(i) False
Reason — While all squares have the same shape (all angles are 90°), they can be different sizes. One square might have a side of 2 cm and another 5 cm; these are similar but not congruent.

(ii) False
Reason — Concentric circles share the same center but must have different radii to be distinct. Since their radii are different, their sizes are different, making them non-congruent.

(iii) True
Reason — The area of a square is side². If the areas are equal, the side lengths must also be equal ($s = \sqrt{Area}$). Since they have the same side lengths and same shape, they are congruent.

(iv) False
Reason — Figures can have the same area but completely different shapes. For example, a rectangle with sides 4 cm and 9 cm has an area of 36 cm², and a square with side 6 cm also has an area of 36 cm², but they are not congruent.

(v) True
Reason — For an equilateral triangle, the area is fixed by the side length ($Area = \dfrac{\sqrt{3}}{4}s^2$). If the areas are equal, the sides must be equal. Since all equilateral triangles have 60° angles, equal sides guarantee congruence by SSS or SAS.

(vi) True
Reason — If one acute angle is equal, and we know the right angle (90°) is equal, then the third angle must also be equal. Since the hypotenuse (a side) is also equal, the triangles are identical in shape and size. This is the ASA (Angle-Side-Angle) criterion.

Assertion: In △ABC, AB = 3.5 cm, AC = 5 cm, BC = 6 cm and in △PQR, PR = 3.5 cm, PQ = 5 cm and RQ = 6 cm. Then △ACB ≅ △PQR.

Reason: Two triangles are congruent if the three sides of one are equal to the three sides of the other.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

Answer

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Explanation

The sides of both triangles are equal (3.5 cm, 5 cm, 6 cm), so triangles are congruent by SSS.

So, Assertion is true.

The statement in reason is the formal definition of the SSS (Side-Side-Side) congruence criterion.

So, Reason is true and it correctly explains the assertion.

Hence, option 1 is the correct option.

Assertion: Two squares having same perimeter are congruent.

Reason: Any two squares are always similar.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

Answer

Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

Explanation

Perimeter of a square = 4 × side. If the perimeters are equal, the sides are equal. Since the sides are equal and all angles are 90°, the squares are congruent.

So, Assertion is true.

Every square has the same internal angles 90° and proportional sides, so all squares are indeed similar.

So, Reason is true.

The reason the squares are congruent is because their side lengths are equal. The similarity rule in Reason only explains why they have the same shape, not why they are the same size.

Hence, option 2 is the correct option.