Question

In given figure, AD and CE are two altitudes of $$\vartriangle ABC$$
A) $\vartriangle AEF \sim \vartriangle CDF$
B) $\vartriangle ABD \sim \vartriangle CBE$
C) $\vartriangle AEF \sim \vartriangle ADB$
D) $\vartriangle FDC \sim \vartriangle BEC$

Answer 1

Hint:We have three criteria to prove two triangles similar which are as follows:
AA : In this criteria, if we have two different angles in the two triangles equal then we can say the triangles are similar.
SAS : In this criteria, if we have two sides and one angle in the two triangles equal then we can say the triangles are similar.
SSS : In this criteria, if we have three sides in the two triangles equal then we can say the triangles are similar.

Complete step-by-step answer:
Now, In $\vartriangle AEFand\vartriangle CDF$, we have
$\angle AEF = \angle CDF = {90^o}$ [$\because CE \bot AB,AD \bot BC$]
$\angle AFE = \angle CFD$(vertically opposite angles)
Thus, by AA criteria of similarity, we have
$\vartriangle AEF \sim \vartriangle CDF$
Now, In $\vartriangle ABDand\vartriangle CBE$, we have
$\angle ADB = \angle CEB = {90^o}$ [$\because CE \bot AB,AD \bot BC$]
$\angle ABD = \angle CBE$ (Common angle)
Thus, by AA criteria of similarity, we have
$\vartriangle ABD \sim \vartriangle CBE$
Now, In $\vartriangle AEFand\vartriangle ADB$, we have
$\angle AEF = \angle ADB = {90^o}$ [$\because CE \bot AB,AD \bot BC$]
$\angle FAE = \angle DAB$ (Common angle)
Thus, by AA criteria of similarity, we have
$\vartriangle AEF \sim \vartriangle ADB$
Now, In $\vartriangle FDCand\vartriangle BEC$, we have
$\angle FDC = \angle BEC = {90^o}$ [$\because CE \bot AB,AD \bot BC$]
$\angle FCD = \angle ECB$ (Common angle)
Thus, by AA criteria of similarity, we have
$\vartriangle FDC \sim \vartriangle BEC$

Note:To prove the triangles similar, we have 3 criteria, don’t confuse it with congruence of triangle, in similarity, we have 1 criteria in which we can prove by AA rule (angle angle rule.) which is not applicable in congruence. Two figures are said to be congruent if they have the same shape or size.