Euclid's Axioms and Postulates

If $S$ is a point lies in the interior of $\angle PQR$ such that, $\angle PQR=80°$ and $\angle PQS=35°$, determine the measure of $\angle RQS$.

In the given figure, $AC$ is a perpendicular,  $\angle 2=\angle 1$ and $\angle 3=\angle 4$. Find $\angle \theta$.

If $AB=(x+3), BC=2x$ and $AC=(4x-5)$, then for what value of $x, B$ lies on $AC$?

Read the following statements which are taken as axiom:

I. If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal.

II. If a transversal intersects two parallel lines, then alternate interior angles are equal.

Is this system of axioms consistent? Justify your answer.

Solve the equation $y-25=40$ and state which axiom will you use here?

It is known that, if $x+y=10$, then $x+y+z=10+z$. Which Euclid's axiom is used?

A point $C$ is said to lie between the points $A$ and $B$. Explain it.

$P$ lies in the interior of $\angle BAC$. If $\angle BAC=70°$ and $\angle BAP=42°$, then determine the measure of $\angle PAC$.

If $B$ lies between $A$ and $C, AC=19 \mathrm{cm}$ and $BC=8 \mathrm{cm}$. What is the measure of $AB$?

Define the condition of a line segment $AB$, such that point $C$ is called the mid-point of $AB$.

In the given figure, $C$ is the mid-point of the segment $AB$. $P$ and $Q$ are mid-points of the segments $AC$ and $BC$ respectively, then $AP$ is equal to

If $PQ$ is a line segment of length $12 \mathrm{cm}$ and $R$ is a point in its interior, then $P{R}^2+Q{R}^2+2PR.QR$ equals

Which of the following needs a proof?

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

If $A, B$ and $C$ are three points on a line such that $A$ lies in between $B$ and $C$, then

In the given figure, we have $\angle ABC=\angle ACB$ and $\angle 3=\angle 4$. Show that $BD=DC$.

Read the following axioms.

I: Things which are equal to the same things are equal to one another.

II: If equals are added to equals, then wholes are equal.

III: Things that are double of the same things are equal to one another.

Check whether the given system of axioms is consistent or inconsistent.

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

If $P$ and $Q$ are the centres of two intersecting circles, then prove that $PQ=QR=PR$.

$PQ$ is a line segment $12 \mathrm{cm}$ long and $R$ is a point in its interior such that $PR=8 \mathrm{cm}$. Then, find $QR, P{Q}^2-P{R}^2$ and $P{R}^2+Q{R}^2+2PR·QR$.