Question

Solve both the question s
the matter is urgent ​

Answer 1

1)

∆ PQR and ∆ LMN are two Isosceles triangles.

PQ = PR

LM = LN

and

PQ = LM

=> PQ = LM = PR = LN

QR = MN

We have

PQ ≅ LM

QR ≅ MN

PR ≅LN

Therefore, ∆ PQR and ∆ LMN are congruent triangles by SSS Congruency property.

∆ PQR ≅ ∆ LMN

SSS Congruency property:-

In two triangles , if three sides in the first triangle are equal to the corresponding three sides in the second triangle respectively then two triangles are congruent triangles.

2)

In ∆ PQR , PO is the perpendicular to QR

PO bisects the angle P

Now,

In ∆ PQO and ∆ PRO

< QOP ≅ < ROP

< QPO ≅ < RPO

PO ≅PO ( common side)

∆ PQO is congruent to ∆ PRO

∆ PQO ≅ ∆ PRO

We know that

∆ PQO is congruent to ∆ PQR

∆ PRO is congruent to ∆ PQR

∆ PRO ≅ ∆ PQR

=> PQ ≅ PR

Since , Corresponding parts are congruent in the Congruent triangles

Two sides are equal in the triangle

Therefore, ∆ PQR is an Isosceles triangle

Answer 2

Step-by-step explanation:

1)

∆ PQR and ∆ LMN are two Isosceles triangles.

PQ = PR

LM = LN

and

PQ = LM

=> PQ = LM = PR = LN

QR = MN

We have

PQ ≅ LM

QR ≅ MN

PR ≅LN

Therefore, ∆ PQR and ∆ LMN are congruent triangles by SSS Congruency property.

∆ PQR ≅ ∆ LMN

SSS Congruency property:-

In two triangles , if three sides in the first triangle are equal to the corresponding three sides in the second triangle respectively then two triangles are congruent triangles.

2)

In ∆ PQR , PO is the perpendicular to QR

PO bisects the angle P

Now,

In ∆ PQO and ∆ PRO

< QOP ≅ < ROP

< QPO ≅ < RPO

PO ≅PO ( common side)

∆ PQO is congruent to ∆ PRO

∆ PQO ≅ ∆ PRO

We know that

∆ PQO is congruent to ∆ PQR

∆ PRO is congruent to ∆ PQR

∆ PRO ≅ ∆ PQR

=> PQ ≅ PR

Since , Corresponding parts are congruent in the Congruent triangles

Two sides are equal in the triangle

Therefore, ∆ PQR is an Isosceles triangle

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