Question

in figure angle1=angle2 and NSQ is congruent to MTR, then prove that PTS is similar to PQR​

Answer 1

Answer:

If NSQ is congruent to MTR, then ST || MN

[Triangles on the same base between the same parallel lines are congruent, thus the converse is also valid.]

If ST || MN:

/_1 = /_SQR [corresponding angles]

/_2 = /_TRQ [corresponding angles]

Thus, in triangles PTS and PQR, two angles are equal.

By AA similarity criterion:

PTS is similar to PQR

proved.

___________________

Answer 2

Answer:

As,

angle 1 = angle 2

so, PS=PT....1

Also, NSQ is congruent to MTR

do, SQ=TR...2

divide eq. 1 and 2

PS/SQ=PT/TR

So, by converse BPT theorem,

ST//QR

corresponding angles are equal

angle 1 = angle SQR

angle 2 = angle TRQ

hence PTS is similar to PQR (by AA similarity criterion)