In the given figure, ${\text{XY}}\parallel {\text{BC}}$. Find the length of XY, given BC = 6 cm

Hint: We use the transversal of parallel lines, where we equal the angles using corresponding angles. So that we can find the relation between the sides using AA similarity as the corresponding sides will be proportional. So that we can find the length of the line parallel to BC, that is XY.

Complete step by step solution:
Given: ${\text{XY}}\parallel {\text{BC}}$
That is XY is parallel to the base/side of the triangle BC.
To find: XY
Since ${\text{XY}}\parallel {\text{BC}}$and AB is transversal, then,
$\vartriangle {\text{AXY}} \cong {\text{ }}\vartriangle {\text{ABC}}$[By the corresponding angles are congruent].
Transversal of parallel line: It is a line that intersects two or more other (often parallel) lines.
Since ${\text{XY}}\parallel {\text{BC}}$and AB is transversal, then
$\vartriangle {\text{AYX}} \cong {\text{ }}\vartriangle {\text{ABC}}$[By the corresponding angles are congruent].
Therefore$\angle AXY = \angle ABC$
And $\angle AYX = \angle ACB$
$\vartriangle AXY$and$\vartriangle ABC$are equiangular and hence they are similar [By AA Similarity]
AA Similarity: If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. Or we can say, in two triangles, if two pairs of corresponding angles are congruent, then the triangles are similar. AA similarity is Angle-Angle Similarity.
Since, Triangles are similar
Hence corresponding sides will be proportional.
$\therefore \dfrac{{AX}}{{AB}} = \dfrac{{XY}}{{BC}} = \dfrac{{AY}}{{AC}}$
$XY = \dfrac{6}{4} \because [AB = AX + BX]$
XY = 1.5 cm
Therefore the length of XY is 1.5 cm.

Note: If transversal intersects with two or more parallel lines then the formed corresponding angles, alternate interior angles and alternate exterior angles are congruent. The pairs of consecutive interior angles formed by the transversal are supplementary. If two pairs of corresponding angles are congruent, then it can be shown that all three pairs of corresponding angles are congruent, by the Angle Sum Theorem.