Question

please answer this question it is urgent​

Answer 1

Answer:

answer is by solving with sas criteria

Step-by-step explanation:

PQ=PR(BY GIVEN SIDE)

angle PTS=anglePRQ( by alternate angles)

ST||QR(. by given)

by SAS similarity criteria

PQ=PR

:-pqr is an isosceles triangle

Answer 2

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Question :-

In the given fig. , PQ = PR . If ST || QR , prove that ∆PST is an isosceles triangle .

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Answer :-

Given :-

In the given figure ,

PQ = PR ,

ST || QR

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Required to prove :-

• ∆PST is an isosceles triangle .

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Conditions used :-

1. If corresponding sides are equal , corresponding angles are also equal .

Similarly,

2. If corresponding angles are equal . then corresponding sides are also equal .

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Proof :-

Given that ,

PQ = PR

and similarly,

ST || QR

So, hereby using these two hints we can solve this whole problem .

Hence,

Let ,

Consider ∆ PQR

In ∆ PQR ,

It is given that ,

PQ = PR

Now recall the properties of triangle .

We know that ,

If corresponding sides are equal then corresponding angles are also equal .

Here,

The corresponding sides are ,

PQ = PR .

So,

Similarly,

The corresponding angles are ,

∠PQR & ∠PRQ

So,

Hence,

∠Q = ∠R

(Reason : If corresponding sides are equal , corresponding angles are equal )

Consider this equation 1 .

However,

Now consider the 2nd hint .

That is ,

ST || QR

Here, let's take SQ as the transversal .

So, according to the properties of parallel lines when a transversal passes through it .

∠PST = ∠SQR

( Reason : A pair of corresponding angles are equal )

Consider the above one as equation 2.

Now, let's take TR as the diagonal .

However,

∠PTS = ∠TRQ

( Reason : A pair of corresponding angles are equal )

But we know that,

∠Q = ∠R

So,

This can be written as ,

∠PTS = ∠SQR

Consider the above one as equation 3.

From ,

equation 2 and 3 we can conclude that ,

∠PST = ∠PTS

So,

Now consider ∆PST ,

From the above we got that,

∠PST = ∠PTS .

Again recall the properties of triangle which we used above .

So,

Here the corresponding angles are ,

∠PST & ∠PTS .

Similarly, corresponding sides are ,

PS & PT

We know that ,

If corresponding angles are equal corresponding sides are also equal .

So,

We can conclude that ,

PS = PT

Hence ,

We know that ,

If two sides of a triangle are equal then it is called as an isosceles triangle .

So,

∆PST is an isosceles triangle .

( Since , PS = PT )

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☑ Hence proved .

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