Given:Lines $AB$ and $CD$ intersect at $O$.$\angle AOC + \angle BOE = 70^o$ and $\angle BOD = 40^o$To do:We have to find $\angle BOE$ and reflex $\angle COE$.Solution:$AOB$ is a line.Therefore, $\angle AOC + \angle COE + \angle BOE = 180^o$$(\angle AOC + \angle BOE) + \angle COE = 180^o$$70^o ... Read More

Given: Lines $XY, MN$ intersect at $O$ , $\angle POY=90^o$ and $a:b=2:3$. To do: We have to find $c$.Solution:Given, $\angle POY=90^o$ and $a:b=2:3$We know that, The sum of the measures of the angles in linear pairs is always $180^o$.This implies, $\angle POY+a+b=180^o$By substituting $\angle POY=90^o$ in the above equation We get, $90^o+a+b=180^o$$a+b=180^o-90^o$$a+b=90^o$Let $a$ be ... Read More

Given:$AC=BD$.To do:We have to prove that $AB=CD$.Solution:Given, $AC=BD$According to the Figure, We get, $AC=AB+BC$$BD=BC+CD$This implies, $AB+BC=BC+CD$....(i)  (Since, $AC=BD$According to Euclid's Axiom, When equals are subtracted from equals, reminders are also equal.Therefore, Let us subtract $BC$ from both sides of (i)We get, $AB+BC-BC=BC+CD-BC$This implies, $AB=CD$Hence proved.Read More

Given:Point $C$ is the midpoint of $\overline{AB}$.To do:We have to prove that every line segment has one and only one midpoint.Solution:Let us assume points $C$ and $D$ are two mid-points of $\overline{AB}$.Since, $C$ and $D$ are midpoints of $\overline{AB}$.We get, $AC=CB$ and $AD=BD$According to Euclid's AxiomWe get, $AC+CB=AB$  (Since, $AC+CB$ ... Read More

To do:We have to give the definition for each of the given terms and also the other terms that are needed to be defined first.Solution:Yes, there are other terms and point which are needed to be defined first before we define the given terms.Line: Line can defined as collection of ... Read More

To do:We have to consider the given postulates.Solution:Yes, these postulates contain undefined terms.The position of $C$ is not defined. Whether the point lies on the line segment which joins $AB$ or not.Whether all the points mentioned in postulates lie in the same plane or not.Yes, these postulates are consistent as ... Read More

Given:A point $C$ lies between two points $A$ and $B$ such that $AC=BC$.To do:We have to prove that $AC=\frac{1}{2}AB$.Solution:Given, $AC=BC$By adding $AC$ on both sides we get, $AC+AC=BC+AC$This implies, $2AC=BC+AC$    ($BC+AC$ coincides with $AB$)According to Euclid's Axiom $4$$BC+AC=AB$.Therefore, $2AC=AB$This implies, $AC=\frac{1}{2}AB$ Read More

To do:We have to find which of the given statements are true and which are false.Solution:(i) We know that, There can be an infinite number of lines that can be drawn through a single point.For instance, Therefore, The statement, that only one line can pass through a single point is ... Read More

To do:We have to write the geometric representation of $2x+9=0$ as an equation(i) in one variable and(ii) in two variables.Solution:(i) We know that, To draw a graph of a linear equation in one variable, we need only one solution.Given, $2x+9=0$This implies, $2x=-9$$x=\frac{-9}{2}$$x=-4.5$Therefore, $x=-4.5$ and $y=0$The geometric representation of $2x+9=0$ in ... Read More

To do:We have to write the geometric representation of $y=3$ as an equation(i) in one variable and(ii) in two variables.Solution:(i) We know that, To draw a graph of a linear equation in one variable, we need only one solution.Given, $y=3$This implies, $x=0$ and $y=3$The geometric representation of $y=3$ in one ... Read More