The $9$ steps in paper plane making are given in a catalogue.

Lets expand the figure for Step $\mathrm{III}$, as shown below:

If in the given figure, $\angle 1=\angle 2$ and $\angle 3=\angle 4$, and you have to prove that $\angle \mathrm{ABF}=\angle \mathrm{CBF}$, then state the Euclid's axiom which is to be used here.

Three lighthouse towers are located at points $A, B$ and $C$ on the section of a national forest to protect animals from hunters in the forest department as shown in the figure. Since Wildlife is a part of our environment and conversation of each of its elements is important for ecological balance, for which the department locates extra towers. How many straight lines can be drawn from $A$ to $C$. State the Euclid Axiom which states the required result.

In the figure below, $AN\times BN=B{M}^2-M{N}^2$, as per Euclid's elements Book $\mathrm{II}$, proposition, $5$. $M$ is the midpoint of line segment, $AB$. $N$ is any point on $AB$

A student is trying to use the above proposition to prove that, given the sum of two numbers, their product is the highest when the numbers are equal.

Prove and verify Euclid's proposition $5$ and then show how it can be used to verify student's findings.

The strength of a circular ring is reinforced by adding two straight elements $AB$ and $CD$ as shown in the figure. $AB$ and $CD$ Intersect at $O$.

Prove that, $OA\times OB=OC\times OD$. Given, $OA=20 \mathrm{cm}, \mathrm{OB}=5 \mathrm{cm}, \mathrm{OC}=16 \mathrm{cm}$, find $OD$

A student constructed an angle using compass and ruler. The steps of construction are given below.

$\left(1\right)$ Segment $AB$ was drawn

$\left(2\right)$ An arc was drawn with $A$ as centre and radius $AB$

$\left(3\right)$ Another arc was drawn with $B$ as the centre and radius $AB$, intersecting the previously drawn arc at $C$

$\left(4\right)$ A line segment was drawn joining $B$ and $C$

Show that $\angle ABC$ measures $60°$

Three square shaped lands share their boundary with a right-angled triangle as shown in the figure below. Show that sum of area of the smaller squares is equal to the area of the largest square.