Question

3. In the given figure, T and M are two points inside a parallelogram PQRS such that
PT = MR and PT || MR. Then prove that:
(a)trianglePTR = triangleRMP

(b) RT || PM and RT = RM​

Answer 1

GIVEN:-

• T and M are two points inside a parallelogram PQRS.

• PT = MR and PT || MR.

TO FIND:-

• ∆PTR = ∆RMP

• RT || PM and RT = RM

CONCEPT USED:-

• When the two angles of transversal lines are alternate interior then the lines are Parallel.

• When two triangles are Congurent then Corresponding part of Congurent triangles are also equal.

Now,

In ∆PTR and ∆RMP.

$\implies\rm{\angle{TRP} = \angle{MPR}}$(Alt.int.angle).

$\implies\rm{PT = MR }$(Given).

$\implies\rm{ PR = RP }$ (common sides).

So, By S-A-S Congurence criteria ∆PTR ≈ ∆RMP.

Therefore,

$\implies\rm{PT = MR }$ (By CPCT)

$\implies\rm{ PT || MR }$ (Concept Used).

MORE TO KNOW

There are 5 Congurency Criteria.

• S - A - S = Side angle side.

• A - S - A = Angle Side Angle

• A - A- S = Angle Angle Side

• R- H - S = Right Hand Side.

• S - S - S = Side Angle Side.

Answer 2

Answer:

(i) In triangle PTR and triangle RMP

PT=MR (Given)

angle RPT=MRP ( Alternate Interior Angles )

PR=PR (Common)

So, it is prove that triangle PTR is congruent to triangle RMP .

(ii) By CPCT rule it is also prove that RT is parallel and equal to PM.