Question

15. Two lines l and m intersect at the point O and P is a point on a line n passingthrough the point O such that P is equidistant from l and m. Prove that n is the bisector of the angle formed by l and m.

16. Line segment joining the mid-points M and N of parallel sides AB and DC,respectively of a trapezium ABCD is perpendicular to both the sides AB and DC.Prove that AD = BC.

solve it.....​

Answer 1

Ans 15.. in pics

Ans 16

ANSWER

Construct AN and BN at the point N

Consider △ANM and ∠BNM

We know that N is the midpoint of the line AB

So we get

AM=BM

From the figure we know that

∠AMN=∠BMN=90∘

MN is common i.e. MN=MN

By SAS congruence criterion

△ANM≅△BNM

AN=BN(c.p.c.t)…(1)

We know that

∠ANM=∠BNM(c.p.c.t)

Subtracting LHS and RHS by 90 °

90∘ −∠ANM=90∘ −∠BNM

So we get

∠AND=∠BNC…(2)

Now, consider △AND and △BNC

AN=BN

∠AND=∠BNC

We know that N is the midpoint of the line DC

DN=CN

By SAS congruence criterion

△AND≅△BNC

AD=BC(c.p.c.t)

Therefore, it is proved that AD=BC.