Question 11 Mark
The following statements are true $(T)$ and which are false $(F)?$If two adjacent angles are equal, then each angle measures 90°.
Answer
View full question & answer→As the statement is incomplete in itself.
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Question 21 Mark
The following statements are true $(T)$ and which are false $(F)?$ Angles forming a linear pair are supplementary.
Answer
View full question & answer→As the sum of the angles forming a linear pair is $180^\circ $.
Question 31 Mark
The following statements are true $(T)$ and which are false $(F)?$ Give reasons. Two lines perpendicular to the same line are perpendicular to each other.
Answer
The figure can be drawn as follows:
Here, $\text{l}\perp\text{n}$ and $\text{m}\perp\text{n}$ It is given that $\text{l}\perp\text{n},$ therefore, $\angle{1}=90^\circ\dots(\text{i})$
Similarly, we have $\text{m}\perp\text{n},$ therefore, $\angle{2}=90^\circ\dots(\text{ii})$
From $(i)$ and $(ii),$ we get: $\angle{1}=\angle{2}$
But these are the pair of corresponding angles.
Theorem states: If a transversal intersects two lines in such a way that a pair of correspondung angles is equal,
then the two lines are parallel. Thus, we can say that $l || m.$
View full question & answer→The figure can be drawn as follows:
Here, $\text{l}\perp\text{n}$ and $\text{m}\perp\text{n}$ It is given that $\text{l}\perp\text{n},$ therefore, $\angle{1}=90^\circ\dots(\text{i})$
Similarly, we have $\text{m}\perp\text{n},$ therefore, $\angle{2}=90^\circ\dots(\text{ii})$
From $(i)$ and $(ii),$ we get: $\angle{1}=\angle{2}$
But these are the pair of corresponding angles.
Theorem states: If a transversal intersects two lines in such a way that a pair of correspondung angles is equal,
then the two lines are parallel. Thus, we can say that $l || m.$
Question 41 Mark
The following statements are true $(T)$ and which are false $(F)?$ Give reasons. If two lines are intersected by a transversal, then corresponding angles are equal.
Answer
View full question & answer→The above statement holds good if the lines are parallel only.
Question 51 Mark
The following statements are true $(T)$ and which are false $(F)?$ Give reasons. Two lines parallel to the same line are parallel to each other.
Answer
View full question & answer→The figure is given as follows:
It is given that $l || m$ and $m || n$ We need to show that $l || m$ We have $l || m,$ thus, corresponding angles should be equal. That is, $\angle{1}=\angle{2}$ Similarly, $\angle{3}=\angle{2}$ Therefore, $\angle{1}=\angle{3}$ But these are the pair of corresponding angles. Therefore, $l || m.$
It is given that $l || m$ and $m || n$ We need to show that $l || m$ We have $l || m,$ thus, corresponding angles should be equal. That is, $\angle{1}=\angle{2}$ Similarly, $\angle{3}=\angle{2}$ Therefore, $\angle{1}=\angle{3}$ But these are the pair of corresponding angles. Therefore, $l || m.$
Question 61 Mark
The following statements are true $(T)$ and which are false $(F)?$ If angles forming a linear pair are equal, then each of these angles is of measure $90^\circ .$
Answer
View full question & answer→Let one of the angle in the linear pair be $x^\circ .$
Then, other angle also becomes equal to $x^\circ .$
Therefore, by the defination of linear pair,
we get: $x + x = 180^\circ$
$ 2x = 180^\circ $
$\text{x}=\frac{180^\circ}{2} x = 90^\circ $
Hence, if angles forming a linear pair are equal,
then each of these angles is of measure $90^\circ .$
Then, other angle also becomes equal to $x^\circ .$
Therefore, by the defination of linear pair,
we get: $x + x = 180^\circ$
$ 2x = 180^\circ $
$\text{x}=\frac{180^\circ}{2} x = 90^\circ $
Hence, if angles forming a linear pair are equal,
then each of these angles is of measure $90^\circ .$
Question 71 Mark
The following statements are true $(T)$ and which are false $(F)?$ Give reasons. If two parallel lines are intersected by a transversal, then alternate interior angles are equal.
Answer
View full question & answer→Let $l$ and $m$ are two parallel lines. And transversal t intersects $l$ and $m$ making two pair of alternate interior angles, $\angle{1},\angle{2}$ and $\angle{3},\angle{4}.$
We need to prove that $\angle{1}=\angle{2}$ and $\angle{3}=\angle{4}.$ We have, $\angle{2}=\angle{5}$ [Vertically opposite angles] Again, $\angle{3}=\angle{6}$ [Corresponding angles] Hence, $\angle{1}=\angle{2}$ and $\angle{3}=\angle{4}.$
We need to prove that $\angle{1}=\angle{2}$ and $\angle{3}=\angle{4}.$ We have, $\angle{2}=\angle{5}$ [Vertically opposite angles] Again, $\angle{3}=\angle{6}$ [Corresponding angles] Hence, $\angle{1}=\angle{2}$ and $\angle{3}=\angle{4}.$
Question 81 Mark
The following statements are true $(T)$ and which are false $(F)?$ Angles forming a linear pair can both be acute angles.
Answer
View full question & answer→Let us assume one of the angle in a linear pair be $x;$ Such that $x^\circ < 90^\circ ,$
that is, an acute angle.
Therefore, the other angle in the linear pair becomes $(180 - x)^\circ ,$ which clearly cannot be acute.
that is, an acute angle.
Therefore, the other angle in the linear pair becomes $(180 - x)^\circ ,$ which clearly cannot be acute.
Question 91 Mark
The following statements are true $(T)$ and which are false $(F)?$ Give reasons. If two parallel lines are intersected by a transversal, then the interior angles on the same side of the transversal are equal.
Answer
View full question & answer→Theoram states: If a transversal intersects two parallel lines then the pair of alternate interior angles is equal.