Conditions for the SSS - Side Side Side congruence

Two triangles are said to be congruent if three sides of one triangle are respectively equal to the three sides of the other triangle.

**Experiment to prove Congruence with SSS: **

Draw ∆LMN with LM = 3 cm, LN = 4 cm, MN = 5 cm.

Also, draw another ∆XYZ with XY = 3cm, XZ = 4cm, YZ= 5cm.

We see that LM = XY, LN = XZ and MN = YZ.

Make a trace copy of ∆XYZ and try to make it cover ∆LMN with X on L, Y on M and Z on N.

We observe that: two triangles cover each other exactly.

Therefore ∆LMN ≅ ∆XYZ

Worked-out problems on side side side congruence triangles (SSS postulate):

**1.** LM = NO and LO = MN. Show that ∆ LON ≅ ∆ NML.

**Solution: **

In ∆LON and ∆NML

LM = NO → given

LO = MN → given

LN = NL → common

Therefore, ∆ LON ≅ ∆ NML, by side-side-side (SSS) congruence condition

**2.** In the given figure, apply SSS congruence condition and state the result
in the symbolic form.

**Solution:**

In ∆LMN and ∆LON

LM = LO = 8.9cm

MN = NO = 4cm

LN = NL = 4.5 cm

Therefore, ∆LMN ≅ ∆LON, by side side side (SSS) congruence condition

**3.** In the adjoining figure, apply S-S-S congruence condition and state the result in the symbolic form.

**Solution:**

In ∆LNM and ∆OQP

LN = OQ = 3 cm

NM = PQ = 5cm

LM = PO = 8.5cm

Therefore, ∆LNM ≅ ∆OQP, by Side Side Side (SSS) congruence condition

**4.** ∆OLM and ∆NML have common base LM, LO = MN and OM = NL. Which of the
following are true?

(i) ∆LMN ≅ ∆LMO

(ii) ∆LMO ≅ ∆LNM

(iii) ∆LMO ≅ ∆MLN

**Solution: **

LO = MN and OM = NL → given

LM = LM → common

Thus, ∆MLN ≅ ∆LMO, by SSS congruence condition

Therefore, statement (iii) is true. So, (i) and (ii) statements are false.

**5.** By Side Side Side congruence prove that 'Diagonal of the rhombus bisects each other at right
angles'.

**Solution:** Diagonal LN and MP of the rhombus LMNP intersect
each other at O.

It is required to prove that LM ⊥ NP and LO = ON and MO = OP.

**Proof:** LMNP is a rhombus.

Therefore, LMNP is a parallelogram.

Therefore, LO = ON and MO = OP.

In ∆LOP and ∆LOM; LP = LM, [Since, sides of a rhombus are equal]

Side LO is common

PO = OM, [Since diagonal of a parallelogram bisects each other]

Therefore, ∆LOP ≅ ∆LOM, [by SSS congruence condition]

But, ∠LOP + ∠MOL = 2 rt. angle

Therefore, 2∠LOP = 2 rt. angle

or, ∠LOP = 1 rt. angle

Therefore, LO ⊥ MP

i.e., LN ⊥ MP (Proved)

[**Note:** Diagonals of a square are
perpendicular to each other]

**6.** In a quadrilateral LMNP, LM = LP and MN = NP.

Prove that LN ⊥ MP and MO = OP [O is the point of intersection of MP and LN]

**Proof:**

In ∆LMN and ∆LPN,

LM = LP,

MN = NP,

LN = NL

Therefore, ∆LMN ≅ ∆LPN, [by SSS congruence condition]

Therefore, ∠MLN = ∠PLN -------- (i)

Now in ∆LMO and ∆LPO,

LM = LP;

LO is common and

∠MLO = ∠PLO

∆LMO ≅ ∆LPO, [by SAS congruence condition]

Therefore, ∠LOM = ∠LOP and

MO = OP, [*Proved*]

But ∠LOM + ∠LOP = 2 rt. angles.

Therefore, ∠LOM = ∠LOP = 1 rt. angles.

Therefore, LO ⊥ MP

i.e., LN ⊥ MP, [*Proved*]

**7.** If the opposite sides of a quadrilateral are equal, prove that the quadrilateral will be parallelogram.

LMNO is a parallelogram quadrilateral, whose sides LM = ON and LO = MN. It is required to prove that LMNO is a parallelogram.

**Construction:** Diagonal LN is drawn.

**Proof:** In ∆LMN and ∆NOL,

LM = ON and MN = LO, [By hypothesis]

LN is common side.

Therefore, ∆LMN ≅ ∆NOL, [by Side Side Side congruence condition]

Therefore, ∠MLN = ∠LNO, [Corresponding angles of congruent triangles]

Since, LN cuts LM and ON and the both alternate angles are equal.

Therefore, LM ∥ ON

Again, ∠MNL = ∠OLN [Corresponding angles of congruent triangles]

But LN cuts LO and MN, and the alternate angles are equal.

Therefore, LO ∥ MN

Therefore, In quadrilateral LMNO,

LM ∥ ON and

LO ∥ MN.

Therefore, LMNO is a parallelogram. [Proved]

[**Note:** Rhombus is parallelogram.]

**Conditions for the Congruence of Triangles**

**Right Angle Hypotenuse Side congruence**

**Converse of Pythagorean Theorem**

**7th Grade Math Problems**

**8th Grade Math Practice**

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