Name: Theorems to Prove That Given Lines are Parallel
Uploaded: 2019-07-19
Duration: 5 min 15 s
Description: A theorem checks whether two lines are parallel. It involves finding corresponding angles. Know how to determine these angles and whether the lines are paral...

Question

If two parallel lines are intersected by a transversal, prove that the bisectors of the two pairs interior angles enclose a rectangle.

Accepted Answer

Let us consider $A B | | C D$ and are intersected by a transversal $E F$ in points $G$ and $H$ respectively as shown below:

To prove: $X Q Y P$ is a rectangle.

Proof:

We know that,

$∠ A X Y = ∠ X Y D$ [Alternate angle]

$\Rightarrow \frac{1}{2} ∠ A X Y = \frac{1}{2} ∠ X Y D$

$\Rightarrow ∠ 1 = ∠ 4$

These form a pair of equal alternate interior angles.

So,

$X P | | Y Q . . . . \dots (1)$

Similarly,

$Y P | | X Q . . . . . . . . . (2)$

From equation $(1)$ and $(2)$ , we have

$X Q Y P$ is a parallelogram $. . . . . . . . (3)$

A quadrilateral is a parallelogram if it both the pairs of opposite side are parallel.

Now,

$∠ B XY + ∠ X Y D = 180 °$

$\Rightarrow \frac{1}{2} ∠ B XY + \frac{1}{2} ∠ X Y D = \frac{1}{2} (180 °)$

So, $∠ 4 + ∠ 2 = 90 ° . . . . . . . . . . (4)$

In $Δ X Y Q$ ,

$∠ 4 + ∠ 2 + ∠ X Y Q = 180 °$

$\Rightarrow 90 ° + ∠ X Y Q = 180 °$

From equation $(3)$ , we have

$∠ X Y Q = 90 ° . . . . . . . . . . (5)$

From equation $(3)$ and $(5)$ , we have, $XQPY$ is a parallelogram with one of its angles of measure $90 °$ .

Hence, $XQPY$ is a rectangle.

If two parallel lines are intersected by a transversal, prove that the bisectors of the two pairs interior angles enclose a rectangle.

Important Questions on Lines and Angles

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