Question

In fig. 3.94 $\angle$ABD = 30$°$, $\angle$ADC = 70$°$, $\angle$ACD = 25$°$
Arrange sides AB, AD, AC in descending order of their lengths.

Answer 1

$$\begin{gathered} \mathrm{In}△\mathrm{ADC}, \\ \therefore \angle \mathrm{ADC}>\angle \mathrm{ACD}, \\ \because \mathrm{AC}>\mathrm{AD}...\left(\mathrm{i}\right) \\ \mathrm{In}△\mathrm{ABD}, \\ \angle \mathrm{ADB}+70°=180°(\mathrm{linear}\mathrm{pair}) \\ \Rightarrow \angle \mathrm{ADB}=110° \\ \because \angle \mathrm{ADB}>\angle \mathrm{ABD} \\ \therefore \mathrm{AB}>\mathrm{AD}.....\left(\mathrm{ii}\right) \\ \mathrm{Also},\mathrm{in}△\mathrm{ABC}, \\ \because \angle \mathrm{B}>\angle \mathrm{C} \\ \therefore \mathrm{AC}>\mathrm{AB}.....\left(\mathrm{iii}\right) \\ \mathrm{From}\left(\mathrm{i}\right),\left(\mathrm{ii}\right)\mathrm{and}\left(\mathrm{iii}\right), \\ \mathrm{AC}>\mathrm{AB}>\mathrm{AD}. \end{gathered}$$