Exercise 5.1 - Chapter 5 Geometry Term 1 7th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Ex $5.1$
Question 1 .
Name the pairs of adjacent angles.
Solution:
(i) $\angle \mathrm{ABG}$ and $\angle \mathrm{GBC}$ are adjacent angles.
(ii) $\angle \mathrm{BCF}$ and $\angle \mathrm{FCD}$ are adjacent angles.
(iii) $\angle \mathrm{BCF}$ and $\angle \mathrm{FCE}$ are adjacent angles.
(iv) $\angle \mathrm{FCE}$ and $\angle \mathrm{ECD}$ are adjacent angles.

Question $2 .$
Find the angle $\angle \mathrm{JIL}$ from the given figure.
Solution:
$\angle \mathrm{LIK}$ and $\angle \mathrm{KIJ}$ are adjacent angles.
$\therefore \angle \mathrm{JIL}=\angle \mathrm{LIK}+\angle \mathrm{KIJ}$
$\begin{aligned}
&=38^{\circ}+27^{\circ} \\
&=65^{\circ}
\end{aligned}$

Question $3 .$
Find the angles $\angle \mathrm{GEH}$ from the given figure.

Solution:
$\begin{aligned}
&\angle \mathrm{HEF}=\angle \mathrm{HEG}+\angle \mathrm{GEF} \\
&120^{\circ}=\angle \mathrm{HEG}+34^{\circ} \\
&120^{\circ}-34^{\circ}=\angle \mathrm{GEH}+34^{\circ}-34^{\circ} \\
&\angle \mathrm{GEH}=86^{\circ}
\end{aligned}$

Question $4 .$
Given that $\mathrm{AB}$ is a straight line. Calculate the value of $\mathrm{x}^{\circ}$ in the following cases.

Solution:
(i) Since the angles are linear pair $\angle \mathrm{AOC}+\angle \mathrm{BOC}=180^{\circ}$
$\begin{aligned}
&72^{\circ}+x^{\circ}=180^{\circ} \\
&72^{\circ}+x^{\circ}-12^{\circ}=180^{\circ}-72^{\circ} \\
&x^{\circ}=108^{\circ}
\end{aligned}$
(ii) Since the angles are linear pair
$\angle \mathrm{AOC}+\angle \mathrm{BOC}=180^{\circ}$

$\begin{aligned}
3 x+42^{\circ}=180^{\circ} & \\
3 x^{\circ}+42^{\circ}-42^{\circ} &=180^{\circ}-42^{\circ} \\
3 x^{\circ} &=138^{\circ} \\
x^{\circ} &=\frac{138^{\circ}}{3}=46^{\circ} \\
x^{\circ} &=46^{\circ}
\end{aligned}$
(iii) Since the angles are linear pair
$\begin{aligned}
&\angle \mathrm{AOC}+\angle \mathrm{BOC}=180^{\circ} \\
&4 \mathrm{x}^{\circ}+2 \mathrm{x}^{\circ}=180^{\circ} \\
&6 \mathrm{x}^{\circ}=180^{\circ} \\
&\mathrm{x}^{\circ}=180^{\circ}
\end{aligned}$

Question 5 .
One angle of a linear pair is a right angle. What can you say about the other angle?
Solution:
If the angle are linear pair, then their sum is $180^{\circ}$.
Given one angle is right angle ie $90^{\circ}$.
$\therefore$ The other angle $=180^{\circ}-90^{\circ}=90^{\circ}$
$\therefore$ The other angle also a right angle
 

Question 6 .
If the three angles at a point are in the ratio $1: 4: 7$, find the value of each angle?
Solution:
We know that the sum of angles at a point is $360^{\circ}$.
Given the three angles are in the ratio $1: 4: 7$.
Let the three angles be $1 \mathrm{x}, 4 \mathrm{x}, 7 \mathrm{x}$.

$\begin{aligned}
\therefore x^{\circ}+4 x^{\circ}+7 x^{\circ} &=360^{\circ} \\
12 x^{\circ} &=360^{\circ} \\
x^{\circ} &=\frac{360^{\circ}}{12} \\
x &=30^{\circ} \\
\therefore 1 x^{\circ} &=30^{\circ} \\
4 x^{\circ} &=4 \times 30^{\circ}=120^{\circ} \\
7 x^{\circ} &=7 \times 30^{\circ}=210^{\circ} .
\end{aligned}$
$\therefore$ The three angles are $30^{\circ}, 120^{\circ}$ and $210^{\circ}$.

Question 7.
Three are six angles at a point. One of them is $45^{\circ}$ and the other five angles are all equal. What is the measure of all the five angles.
Solution:

We know that the sum of angles at a point is $360^{\circ}$.
One angle $=45^{\circ}$
Let the equal angles be $x^{\circ}$ each
$\begin{aligned}
&\therefore \mathrm{x}^{\circ}+\mathrm{x}^{\circ}+\mathrm{x}^{\circ}+\mathrm{x}^{\circ}+\mathrm{x}^{\circ}+45^{\circ}=360^{\circ} \\
&5 \mathrm{x}^{\circ}+45^{\circ}-45^{\circ}=360^{\circ}-45^{\circ}
\end{aligned}$

$\begin{aligned}
&5 x^{\circ}=315^{\circ} \\
&x^{\circ}=\frac{315^{\circ}}{5} \\
&x^{\circ}=63^{\circ}
\end{aligned}$
$\therefore$ Measure of all 5 equal angles $=63^{\circ}$.

Question $8 .$
In the given figure, identify
(i) Any two pairs of adjacent angles.
(ii) Two pairs of vertically opposite angles.
Solution:
(i) (a) $\angle \mathrm{PQT}$ and $\angle \mathrm{TOS}$ are adjacent angles.
(b) $\angle \mathrm{PQU}$ and $\angle \mathrm{UQR}$ are adjacent angles.
(ii) (a) $\angle \mathrm{PQT}$ and $\angle \mathrm{RQU}$ are vertically opposite angles.
(b) $\angle \mathrm{TQR}$ and $\angle \mathrm{PQU}$ are vertically opposite angles.

Question $9 .$
The angles at a point are $x^{\circ}, 2 x^{\circ}, 3 x^{\circ}, 4 x^{\circ}$ and $5 x^{\circ}$. Find the value of the largest angle?
Sum of angles at a point $=360^{\circ}$
$\begin{aligned}
&\therefore \mathrm{x}^{\circ}+2 \mathrm{x}^{\circ}+3 \mathrm{x}^{\circ}+4 \mathrm{x}^{\circ}+5 \mathrm{x}^{\circ}=360^{\circ} \\
&15 \mathrm{x}^{\circ}=360^{\circ} \\
&\mathrm{x}^{\circ}=\frac{360^{\circ}}{15} \\
&\mathrm{x}^{\circ}=24^{\circ} .
\end{aligned}$
$\therefore$ The largest angle $=5 \mathrm{x}^{\circ}$
$=5 \times 24^{\circ}=120^{\circ}$
The largest angle is $120^{\circ}$
 

Question 10 .
From the given figure, find the missing angle.
Solution:
Lines $\overleftrightarrow{R P}$ and $\overleftrightarrow{S Q}$ are interesting at ' $\mathrm{O}$ '

$\therefore$ Vertically opposite angles are equal.
$\therefore \mathrm{x}=105^{\circ}$
$\therefore$ Missing angle $=105^{\circ}$

Question $11 .$
Find the angles $x^{\circ}$ and $y^{\circ}$ in the figure shown.
Solution:
Consider the line $m$.
$\mathrm{x}^{\circ}$ and $3 \mathrm{x}^{\circ}$ are linear pair of angles
$\therefore \mathrm{x}^{\circ}+3 \mathrm{x}^{\circ}=180^{\circ}$
$\begin{aligned}
4 x^{\circ} &=180^{\circ} \\
x^{\circ} &=\frac{180^{\circ}}{4} \\
x^{\circ} &=45^{\circ}
\end{aligned}$

Also lines $\mathrm{l}$ and $\mathrm{m}$ intersects.
Vertically opposite angles are equal.
ie $3 x^{\circ}=y^{\circ}$
$3 \times 45^{\circ}=\mathrm{y}^{\circ}$
$\mathrm{y}=135^{\circ}$
$\mathrm{x}^{\circ}=15^{\circ}$ and
$\mathrm{y}^{\circ}=135^{\circ}$
 

Question 12 .
Using the figure, answer the following questions.

(i) What is the measure of angle $x^{\circ}$ ?
(ii) What is the measure of angle $\mathrm{y}^{\circ}$ ?
Solution:
From the figure $\mathrm{x}^{\circ}$ and $125^{\circ}$ are vertically opposite angles. So they are equal ie ie $\mathrm{x}^{\circ}=125^{\circ}$
Also $\mathrm{y}^{\circ}$ and $125^{\circ}$ are linear pair of angles.
$\begin{aligned}
&\therefore \mathrm{y}^{\circ}+125^{\circ}=180^{\circ} \\
&\mathrm{y}^{\circ}+125^{\circ}-125^{\circ}=180^{\circ}-125^{\circ} \\
&\mathrm{y}^{\circ}=55^{\circ} \\
&\mathrm{x}^{\circ}=125^{\circ}, \\
&\mathrm{y}^{\circ}=55^{\circ}
\end{aligned}$
 

Question $13 .$
Adjective angles have
(i) No common interior, no common arm, no common vertex.
(ii) One common vertex, one common arm, common interior
(iii) One common arm, one common vertex, no common interior.
(iv) One common arm, no common vertex, no common interior.
Solution:
(iii) one common arm, one common vertex, no common interior
 

Question $14 .$
In the given figure the angles $\angle 1$ and $\angle 2$ are
(i) Opposite angles
(ii) Adjacent angles
(iii) Linear angles
(iv) Supplementary angles

Solution:
(iii) Linear pair

Question $15 .$
Vertically opposite angles are
(i) not equal in measure
(ii) Complementary
(iii) supplementary
(iv) equal in measure
Solution:
(iv) equal in measure

Question $16 .$
The sum of all angles at a point is
(i) $360^{\circ}$
(ii) $180^{\circ}$
(iii) $90^{\circ}$
(iv) $0^{\circ}$
Solution:
(i) $360^{\circ}$

Question $17 .$
The measure of $\angle B O C$ is
(i) $90^{\circ}$
(ii) $180^{\circ}$
(iii) $80^{\circ}$
(iv) $100^{\circ}$

Solution:
(iii) $80^{\circ}$