Euclid's Axioms

Axioms are assumed

Euclid's second axiom is 

The things which are double of same the thing are

John is of the same age as Ravi. Shyam is also of the same age as Ravi. State the Euclid's axiom that illustrates the relative ages of John and Shyam.

It is known that if $x+y=10, \mathrm{then} x+y+z\mathit{=}10+z$. The Euclid's axiom that illustrates this statement is

If equals are subtracted from equals, the remainders are

If a point $C$ lies between $A \& B$, then

If $\mathrm{AB}=\mathrm{P}\mathrm{Q}$ and $\mathrm{P}\mathrm{Q}=\mathrm{}\mathrm{X}\mathrm{Y}$ then $\mathrm{AB}=X\mathrm{Y}$ is based on Euclids

If $AB=x+3, BC=2x \&\hspace{0.17em}AC=4x-5$, then for what value of $x$, $B$ lies on $AC?$

The things which coincide with one another are

Solve the equation $x-15=25$ and state Euclid’s axiom used here.

In the given Figure, $\mathrm{A}$ and $\mathrm{B}$ are the centres of the circles. Prove that $∆\mathrm{A}\mathrm{B}\mathrm{C}$ is equilateral.

In the given Figure, $\mathrm{A}\mathrm{B}=\mathrm{C}\mathrm{D}$ and $\mathrm{B}\mathrm{C}=\mathrm{D}\mathrm{E}$. By using Euclid's axiom, prove that $\mathrm{C}$ is the mid point of $\mathrm{A}\mathrm{E}$.

In a parallelogram $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$, diagonals $\mathrm{A}\mathrm{C}$ and $\mathrm{B}\mathrm{D}$ bisect at $\mathrm{O}$, show that $\mathrm{O}\mathrm{A}+\mathrm{O}\mathrm{B}=\mathrm{O}\mathrm{C}+\mathrm{O}\mathrm{D}.$

In a square $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$, diagonals $\mathrm{A}\mathrm{C}$ and $\mathrm{B}\mathrm{D}$ are diagonals as shown in Figure, show that $2\mathrm{O}{\mathrm{C}}^2=\mathrm{C}{\mathrm{D}}^2$.

In the figure, $\mathrm{P}$ and $\mathrm{Q}$ are the mid points of $\mathrm{}\mathrm{A}\mathrm{B}$ and $\mathrm{A}\mathrm{C},\mathrm{B}\mathrm{P}=\mathrm{A}\mathrm{Q},$ show that $\mathrm{A}\mathrm{B}=\mathrm{A}\mathrm{C}.$

Prove that a line segment has one and only one mid point.

In the figure, $\mathrm{A}\mathrm{B}=\mathrm{B}\mathrm{C}$, $\mathrm{P}$ is the mid point of $\mathrm{A}\mathrm{B}$ and $\mathrm{Q}$ is the mid point of $\mathrm{B}\mathrm{C}$. Show that $\mathrm{B}\mathrm{P}=\mathrm{B}\mathrm{Q}.$

Define perpendicular lines.

In Figure, $\mathrm{\angle }1=\mathrm{\angle }2$ and $\mathrm{\angle }3=\mathrm{\angle }4$, show that $\mathrm{}\mathrm{\angle }\mathrm{Q}\mathrm{P}\mathrm{S}=\mathrm{\angle }\mathrm{Q}\mathrm{R}\mathrm{S}.$