Question

If \[l\] and \[m\] are two parallel lines intersected by another pair of parallel lines \[p\] and \[q\]. Then

A.\[\Delta ABC \cong \Delta CDA\]
B.\[\Delta ABC \cong \Delta ADC\]
C.\[\Delta ABC \cong \Delta DCA\]
D.\[\Delta ABC \cong \Delta CAD\]

Answer 1

Hint: Here, we have to find the congruent triangles. We will find the congruent triangles using the Congruent criteria. Congruent triangles are those two triangles which are said to be congruent if and only if one of them can be made to superpose on the other so as to cover it exactly.

Complete step-by-step answer:
We are given that \[l\] and \[m\] are two parallel lines. These two parallel lines are intersected by another two parallel lines \[p\] and \[q\]. So, we have two triangles between these four parallel lines.

From the figure, we have two triangles \[\Delta ABC\] and \[\Delta ACD\].
Here, \[AC = CA\] which is common for both the triangles.
As \[p\] and \[q\] are parallel lines, so alternate interior angles will be equal. Therefore,
\[\angle BAC = \angle DCA\]
As \[l\] and \[m\] are parallel lines, alternate interior angles will be equal. Therefore,
\[\angle BCA = \angle DAC\]
Since two alternate interior angles and one of the sides are equal, therefore, the two triangles are congruent by ASA congruence rule.
So, \[\Delta ABC \cong \Delta CDA\]
Hence, option A is the correct option.
Note: We can make a mistake in understanding the symbols between similar and congruent. The order of letters in the name of two triangles will indicate the correspondence between the vertices of two triangles. Thus, two triangles are congruent only if there exists a correspondence between their vertices such that the correspondence sides and correspondence angles of two triangles are equal. Two triangles are congruent, if two angles and an included side of one triangle is equal to two angles and an included side of the other triangle, then the triangles are said to have ASA congruence.