M L Aggarwal Solutions for Chapter: Introduction to Euclid's Geometry, Exercise 1: Exercise 5

Does Euclid's fifth postulate imply the existence of parallel lines? Explain.

In the given figure, if $\mathrm{AB}=\mathrm{AD}$ and $\mathrm{AC}=\mathrm{AD}$, then prove that $\mathrm{AB}=\mathrm{AC}$. State Euclid's axiom to support this.

In a triangle $\mathrm{ABC}, \mathrm{X}$ and $Y$ are points on $\mathrm{AB}$ and $\mathrm{BC}$ such that $\mathrm{BX}=\mathrm{BY}$ and $AB=BC$. Show that $AX=CY$. State Euclid's axiom used.

In the given figure, if $X$ and $Y$ are mid-points of $AB$ and $BC$ and $AX=CY$ then show that $\mathrm{AC}=\mathrm{BC}$.

In the given figure, if $\mathrm{AC}=\mathrm{DC}$ and $\mathrm{CB}=\mathrm{CE}$ then show that $\mathrm{AB}=\mathrm{DE}$.

In the given figure, if $\angle \mathrm{ABC}=\angle \mathrm{ACB}$ and $\angle 3=\angle 4$ then show that $\angle 1=\angle 2$.

In the given figure, $\angle 1=\angle 2$ and $\angle 3=\angle 4$. Show that $\angle \mathrm{ABD}=\angle \mathrm{CBD}$ and state Euclid's axiom used.

In the given figure, show that $\mathrm{AH}>$ sum of length $\mathrm{AB}+\mathrm{BC}+\mathrm{CD}$. Also, state Euclid's axiom used.