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Is the following statement true or false?

$M$ and $N$ are points on the sides $A B$ and $A C$ of a triangle $Δ A B C$ such that $B M = C N$ . If $∠ B = ∠ C$ then $M N$ is not parallel to $B C$ .

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True

False

answer is B.

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Detailed Solution

We must know that the angle sum property of a triangle which is that in a triangle the sum of interior angles is

180 °

.
Sides opposite to equal angles in a triangle are equal.
Angles opposite to equal sides in a triangle are equal.
Given that

M

and

N

are points on the sides

A B

and

A C

such that

B M = C N

In the triangle

Δ ABC

∠ B = ∠ C

and

AB = AC

…..i) (sides opposite to equal angle are equal)
Subtracting

B M

from both sides of the equation i),

⇒ AB − BM = AC − BM ⇒ AB − BM = AC − CN (∵ BM = CN) AM = AN

∴ ∠ AMN = ∠ ANM

(Angles opposite to equal sides are equal)
In

Δ ABC

∠ A + ∠ B + ∠ C = 180 °

…..ii) (The sum of interior angles of a triangle is

180 °

)
In

Δ AMN,

∠ A + ∠ A M N + ∠ A N M = 180 °

…..iii)
On solving the equations (ii) and (iii) we get,

∠ B + ∠ C = ∠ A M N + ∠ A N M ⇒ 2 ∠ B = 2 ∠ A M N ⇒ ∠ B = ∠ A M N

∠ B

and

∠ A M N

are corresponding angles,

∴ MN ‖ BC

Thus, the given statement is false and the correct statement is,
‘

M

and

N

are points on the sides

A B

and

A C

of a triangle

Δ A B C

such that

B M = C N

. If

∠ B = ∠ C

then

M N

is parallel to

B C

.’
Therefore the correct option is 2.

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