Answer:
By changing the order of vertices,
We have four conditions,
[tex]\triangle ABE\sim \triangle ACD[/tex]
[tex]\triangle ABE\sim \triangle ADC[/tex]
[tex]\triangle AEB\sim\triangle ACD[/tex]
[tex]\triangle AEB \sim \triangle ADC[/tex]
When we have [tex]\triangle ABE\sim \triangle ACD[/tex] or [tex]\triangle AEB \sim \triangle ADC[/tex],
Then by the property of similar angles,
[tex]\frac{AE}{AD}=\frac{AB}{AC}[/tex]
⇒ [tex]\frac{15}{18} = \frac{10}{10+x}[/tex]
⇒ [tex]150+15x= 180[/tex]
⇒ [tex]15x = 180 -150[/tex]
⇒ [tex]15x = 30[/tex]
⇒ x = 2
But, [tex]\triangle ABE\sim \triangle ADC[/tex] or [tex]\triangle AEB\sim\triangle ACD[/tex] is given,
Then by the property of similar triangles,
[tex]\frac{AB}{AD}=\frac{AE}{AC}[/tex]
⇒ [tex]\frac{10}{18} = \frac{15}{10+x}[/tex]
⇒ [tex]100+10x= 15\times 18[/tex]
⇒ [tex]10x = 270 -100[/tex]
⇒ [tex]10x = 170[/tex]
⇒ x = 17
Note: [tex]\angle A[/tex] is reflexive angle in both triangles this is why, in each condition we took A is corresponding to A.
