Question

Solve this question from the image ​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

ABCD is parallelogram and side AD is produced to E and BE intersects CD at F.

Now, We know that,

In parallelogram, opposite angles are equal.

$\rm \implies\:\boxed{ \tt{ \: \angle BAD \: = \: \angle BCD \: }}$

$\rm \implies\:\boxed{ \tt{ \: \angle BAE \: = \: \angle BCF \: }} - - - - (1)$

As ABCD is a parallelogram, so AD || BC

Further, AD is produced to E, so AE || BC

Now, AE || BC and BE is transversal.

We know, alternate interior angles are equal.

$\rm \implies\:\boxed{ \tt{ \: \angle AEB \: = \: \angle CBF \: }} - - - (2)$

$\red{\rm :\longmapsto\:Now, \: In \: \triangle \: EAB \: and \: \triangle \: CBF}$

$\rm :\longmapsto\:\angle BAE \: = \: \angle BCF \: \: \: \{proved \: above \}$

$\rm :\longmapsto\:\angle AEB \: = \: \angle CBF \: \: \: \{proved \: above \}$

$\bf\implies \:\triangle \: ABE \: \sim \: \triangle \: CFB \: \: \{AA \: Similarity \}$

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More to know :-

1. Pythagoras Theorem :-

This theorem states that : In a right-angled triangle, the square of the longest side is equal to sum of the squares of remaining sides.

2. Converse of Pythagoras Theorem :-

This theorem states that : If the square of the longest side is equal to sum of the squares of remaining two sides, angle opposite to longest side is right angle.

3. Area Ratio Theorem :-

This theorem states that :- The ratio of the area of two similar triangles is equal to the ratio of the squares of corresponding sides.

4. Basic Proportionality Theorem :-

If a line is drawn parallel to one side of a triangle, intersects the other two lines in distinct points, then the other two sides are divided in the same ratio.