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Condition for Two Lines to Be Parallel

In the given ...

Question

In the given figure, if $A D$ $=$ $B C$ and $A D$ $| |$ $B C$ , then __________.

A B

=

A D

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A B

=

D C

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B C

=

C D

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A D

=

C D

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Solution

The correct option is A $A B$ $=$ $D C$
Given that $A D | | B C$
$\Rightarrow ∠ D A C = ∠ B C A$ corresponding angles of parallel lines
Considering the triangles $Δ A B C$ and $Δ C D A$
$A D = B C$ .....Given
$∠ D A C = ∠ B C A$ .....proved above
$A C = A C$ .....Common Side

So by $S A S$ congruency

$Δ A B C$ $≅$ $Δ C D A$

Corresponding sides and angles of congruent triangles are also equal.
$\Rightarrow$ $A B = D C$

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Similar questions

In the given figure, if AD = BC and AD || BC, then

In a trapezium ABCD, if AB || CD then $(A C^{2} + B D^{2})$ = ?

(a) $B C^{2} + A D^{2} + 2 B C . A D$

(b) $A B^{2} + C D^{2} + 2 A B . C D$

(d) $B C^{2} + A D^{2} + 2 A B . C D$

∠ B A C = 90^{\circ} a n d A D ⊥ B C

B D . C D = B C^{2}

A B . A C = B C^{2}

B D . C D = A D^{2}

A B . A C = A D^{2}

Construct a quadrliateral ABCD; if:

(i) AB = 4.3 cm, BC = 5.4, CD = 5 cm, DA = 4.8 cm and $∠$ ABC = $75^{\circ}$ .

(ii) AB = 6 cm, CD = 4.5 cm, BC = AD = 5 cm and $∠ BCD = 60^{\circ}$

(iii) AB = 8 cm, BC = 5.4 cm, AD = 6 cm, $∠ A = 60^{\circ}$ and $∠ B = 75^{\circ}$ .

(iv) AB = 5 cm, BC = 6.5 cm, CD = 4.8 cm, $∠ B = 75^{\circ}$ and $∠ C = 120^{\circ}$

(v) AB = 6 cm = AC, BC = 4 cm, CD = 5 cm and AD = 4.5 cm.

(vi) AB = AD = 5 cm, BD = 7 cm and BC = DC = 5.5 cm

A B | | C D

B C | | A D .

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