NCERT Solutions for Class 7 Maths Chapter 5 Lines and Angles -Download PDF

NCERT Solutions for Class 7 Maths Chapter 5 Lines and Angles Topic 5.2.1

1. Can two acute angles be a complement to each other?

Answer: Yes, two acute angles can be complementary to each other.

For e.g. Acute angles ${20}^{\circ }$ and ${70}^{\circ }$ are complementary angle as their sum is ${90}^{\circ }$ .

1. Are the angles marked 1 and 2 adjacent? (Fig). If they are not adjacent, say, ‘why’.

Answer: The condition for being adjacent angles are:-

(a) they have a common vertex

(b) they have common arm

Hence in the given figures:-

(i) These angles are adjacent angles as they agree above conditions.

(ii) The angles are adjacent angles.

(iii) These angles are not adjacent as their vertices are different.

(iv) These are adjacent angles.

(v) The angles are adjacent angles.

1.(v) Name the pairs of angles in each figure:

(v)

Answer: The angles shown are pair of alternate interior angles.

1.(vi) Name the pairs of angles in each figure:

Answer: The given angles are linear pairs of angles as they form a straight line.

NCERT Solutions for Class 7 Maths Chapter 5 Lines and Angles Exercise 5.1

1.(i) Find the complement of each of the following angles:

Answer: The sum of the complementary angle is ${90}^{\circ }$ .

Thus the complementary angle to the given angle is : $={90}^{\circ }-{20}^{\circ }={70}^{\circ }$

1.(ii) Find the complement of each of the following angles:

Answer: The sum of the complementary angle is ${90}^{\circ }$ .

Thus the complementary angle to the given angle is : $={90}^{\circ }-{63}^{\circ }={27}^{\circ }$

1.(iii) Find the complement of each of the following angles:

Answer: The sum of the complement angles is ${90}^{\circ }$ .

Thus the complement of the angle is given by : $={90}^{\circ }-{57}^{\circ }={33}^{\circ }$

2.(i) Find the supplement of each of the following angles:

Answer: We know that sum of supplement angles is ${180}^{\circ }$

The supplement of the given angle is : $={180}^{\circ }-{105}^{\circ }={75}^{\circ }$

2.(ii) Find the supplement of each of the following angles:

Answer: We know that the sum of angles of supplementary pair is ${180}^{\circ }$ .

Thus the supplement of the given angle is : $={180}^{\circ }-{87}^{\circ }={93}^{\circ }$

2.(iii) Find the supplement of each of the following angles:

Answer: We know that the sum of angles of supplementary pair is ${180}^{\circ }$ .

Thus the supplement of the given angle is :

$={180}^{\circ }-{154}^{\circ }={26}^{\circ }$

3. Identify which of the following pairs of angles are complementary and which are supplementary.

$(i){65}^o,{115}^o$ $(ii){63}^o,{27}^o$ $(iii){112}^o,{68}^o$

$(iv){130}^o,{50}^o$ $(v){45}^o,{45}^o$ $(vi){80}^o,{10}^o$

Answer: We know that the sum of supplementary angles is ${180}^{\circ }$ and the sum of complementary angle is ${90}^{\circ }$ .

(i) Sum of the angles is : ${65}^{\circ }+{115}^{\circ }={180}^{\circ }$ . Hence these are supplementary angles.

(ii) Sum of the angles is : ${63}^{\circ }+{27}^{\circ }={90}^{\circ }$ . Hence these are complementary angles.

(iii) Sum of the angles is : ${112}^{\circ }+{68}^{\circ }={180}^{\circ }$ . Hence these are supplementary angles.

(iv) Sum of the angles is : ${130}^{\circ }+{50}^{\circ }={180}^{\circ }$ . Hence these are supplementary angles.

(v) Sum of the angles is : ${45}^{\circ }+{45}^{\circ }={90}^{\circ }$ . Hence these are complementary angles.

(vi) Sum of the angles is : ${80}^{\circ }+{10}^{\circ }={90}^{\circ }$ . Hence these are complementary angles.

4. Find the angle which is equal to its complement.

Answer: Let the required angle be $\mathrm{\Theta }$ .

Then according to question, we have :

$\mathrm{\Theta }+\mathrm{\Theta }={90}^{\circ }$

or $2\mathrm{\Theta }={90}^{\circ }$

or $\mathrm{\Theta }={45}^{\circ }$

5. Find the angle which is equal to its supplement.

Answer: Let the required angle be $\mathrm{\Theta }$ .

Then according to the question :

$\mathrm{\Theta }+\mathrm{\Theta }={180}^{\circ }$

or $2\mathrm{\Theta }={180}^{\circ }$

or $\mathrm{\Theta }={90}^{\circ }$

Hence the angle is ${90}^{\circ }$ .

6. In the given figure, $\mathrm{\angle }1$ and $\mathrm{\angle }2$ are supplementary angles. If $\mathrm{\angle }1$ is decreased, what changes should take place in $\mathrm{\angle }2$ so that both the angles still remain supplementary.

Answer: Since it is given that $\mathrm{\angle }1$ and $\mathrm{\angle }2$ are supplementary angles, i.e. the sum of both angles is ${180}^{\circ }$ .

Thus if $\mathrm{\angle }1$ is decreased then to maintain the sum $\mathrm{\angle }2$ needs to be increased.

7. Can two angles be supplementary if both of them are:

(i) acute ? (ii) obtuse ? (iii) right ?

Answer: We know that the sum of supplementary angles is ${180}^{\circ }$ .

(i) The maximum value of the sum of two acute angles is less than ${180}^{\circ }$ . Thus two acute angles can never be supplementary.

(ii) The minimum value of the sum of two obtuse angles is more than ${180}^{\circ }$ . Thus two obtuse angles can never be supplementary.

(iii) Sum of two right angles is ${180}^{\circ }$ . Hence two right angles are supplementary.

8. An angle is greater than ${45}^o$ . Is its complementary angle greater than ${45}^o$ or equal to ${45}^o$ or less than ${45}^o$ ?

Answer: We know that the sum of two complementary angles is ${90}^{\circ }$ .

Thus if one of the angles is greater than ${45}^{\circ }$ then the other angle needs to be less than ${45}^{\circ }$ .

9. In the adjoining figure:

(i) Is $\mathrm{\angle }1$ adjacent to $\mathrm{\angle }2$ ?
(ii) Is $\mathrm{\angle }AOC$ adjacent to $\mathrm{\angle }AOE$ ?

(iii) Do $\mathrm{\angle }COE$ and $\mathrm{\angle }EOD$ form a linear pair?
(iv) Are $\mathrm{\angle }BOD$ and $\mathrm{\angle }DOA$ supplementary?
(v) Is $\mathrm{\angle }1$ vertically opposite to $\mathrm{\angle }4$ ?
(vi) What is the vertically opposite angle of $\mathrm{\angle }5$ ?

Answer: (i) Yes, $\mathrm{\angle }1$ adjacent to $\mathrm{\angle }2$ as these have the same vertex and have one common arm.

(ii) No, $\mathrm{\angle }AOC$ is not adjacent to $\mathrm{\angle }AOE$ . This is because $\mathrm{\angle }AOE$ contains $\mathrm{\angle }AOC$ .

(iii) Yes the given angles form a linear pair as they are pair of supplementary angles.

(iv) Since BOA is a straight line thus the given angles are supplementary.

(v) Yes, $\mathrm{\angle }1$ and $\mathrm{\angle }4$ are vertically opposite angles as they are the angles formed by two intersecting straight lines.

(vi) The vertically opposite angle to $\mathrm{\angle }5$ is _{$(\mathrm{\angle }2+\mathrm{\angle }3)$} .

10. (i) Indicate which pairs of angles are:

(i) Vertically opposite angles.

Answer: The vertically opposite pairs are :

(a) $\mathrm{\angle }1$ and $\mathrm{\angle }4$

(b) $\mathrm{\angle }5$ and _{$(\mathrm{\angle }2+\mathrm{\angle }3)$}

10.(ii) Indicate which pairs of angles are:

(ii) Linear pairs

Answer: The sum of angles in linear pair is ${180}^{\circ }$ .

Thus the linear pairs are :

(a) $\mathrm{\angle }1and\mathrm{\angle }5$

(b) $\mathrm{\angle }4and\mathrm{\angle }5$

11. In the following figure, is $\mathrm{\angle }1$ adjacent to $\mathrm{\angle }2$ ? Give reasons.

Answer: No, $\mathrm{\angle }1$ and $\mathrm{\angle }2$ are not adjacent angles as their vertex is not same/common.

For being adjacent angles the pair must have a common vertex and have a common arm.

12.(i) Find the values of the angles x, y, and z in each of the following:

Answer: From the figure :

(i) $\mathrm{\angle }x={55}^{\circ }$ (Vertically opposite angle)

(ii) $\mathrm{\angle }y={180}^{\circ }-{55}^{\circ }={125}^{\circ }$ (Linear pair)

(iii) $\mathrm{\angle }y=\mathrm{\angle }z={125}^{\circ }$ (Vertically opposite angle)

12.(ii) Find the values of the angles x, y, and z in each of the following:

Answer: From the figure we can observe that :

(i) $\mathrm{\angle }z={40}^{\circ }$ (Vertically opposite angle)

(ii) $\mathrm{\angle }x={180}^{\circ }-{40}^{\circ }-{25}^{\circ }={115}^{\circ }$ (Linear pair/straight line)

(iii) $\mathrm{\angle }y={180}^{\circ }-\mathrm{\angle }z={140}^{\circ }$ (Vertically opposite angle).

13. Fill in the blanks:

(i) If two angles are complementary, then the sum of their measures is _______.

(ii) If two angles are supplementary, then the sum of their measures is ______.

(iii) Two angles forming a linear pair are _______________.

(iv) If two adjacent angles are supplementary, they form a ___________.

(v) If two lines intersect at a point, then the vertically opposite angles are always _____________.

(vi) If two lines intersect at a point, and if one pair of vertically opposite angles are acute angles, then the other pair of vertically opposite angles are __________.

Answer: (i) ${90}^{\circ }$

(ii) ${180}^{\circ }$

(iii) Supplementary angles

(iv) Straight line

(v) equal

(vi) obtuse angles (as they form a line).

14. In the adjoining figure, name the following pairs of angles.

(i) Obtuse vertically opposite angles
(ii) Adjacent complementary angles
(iii) Equal supplementary angles
(iv) Unequal supplementary angles
(v) Adjacent angles that do not form a linear pair

Answer: (i) $\mathrm{\angle }AODand\mathrm{\angle }BOC$ are the vertically obtuse angles.

(ii) $\mathrm{\angle }AOBand\mathrm{\angle }AOE$ are the complementary angles.

(iii) $\mathrm{\angle }BOEand\mathrm{\angle }DOE$ are the equal supplementary angles.

(iv) $\mathrm{\angle }BOCand\mathrm{\angle }COD$ are the unequal pair of supplementary angle.

(v) $\mathrm{\angle }AOBand\mathrm{\angle }AOE$ , $\mathrm{\angle }EODand\mathrm{\angle }COD$ and $\mathrm{\angle }AOEand\mathrm{\angle }EOD$ are adjacent angles but are not supplementary angles.

NCERT Solutions for Class 7 Maths Chapter 5 Lines and Angles Exercise 5.2

1.(i) State the property that is used in each of the following statements?

(i) If $a\parallel b$ , then $\mathrm{\angle }1=\mathrm{\angle }5$ .

Answer: The statement "If _{$a\parallel b$} , then _{$\mathrm{\angle }1=\mathrm{\angle }5$} " is true using the corresponding angles property .

1.(ii) State the property that is used in each of the following statements?

(ii) If $\mathrm{\angle }4=\mathrm{\angle }6$ , then $a\parallel b.$

Answer: The property used here is 'alternate interior angle property'.

1.(iii) State the property that is used in each of the following statements?

(iii) If $\mathrm{\angle }4+\mathrm{\angle }5={180}^o$ , then $a\parallel b.$

Answer: The property used here is ' Interior angles on the same side of the transversal are a pair of supplementary angles '.

2. In the adjoining figure, identify

(i) the pairs of corresponding angles.
(ii) the pairs of alternate interior angles.
(iii) the pairs of interior angles on the same side of the transversal.
(iv) the vertically opposite angles.

Answer: (i) Corresponding angles :- $\mathrm{\angle }1and\mathrm{\angle }5$ , $\mathrm{\angle }2and\mathrm{\angle }6$ , $\mathrm{\angle }3and\mathrm{\angle }7$ , $\mathrm{\angle }4and\mathrm{\angle }8$

(ii) Alternate interior angles :- $\mathrm{\angle }2and\mathrm{\angle }8$ , $\mathrm{\angle }3and\mathrm{\angle }5$ ,

(iii) Alternate angles on the same side of traversal :- $\mathrm{\angle }2and\mathrm{\angle }5$ , $\mathrm{\angle }3and\mathrm{\angle }8$

(iv) Vertically opposite angles :- $\mathrm{\angle }1and\mathrm{\angle }3$ , $\mathrm{\angle }2and\mathrm{\angle }4$ , $\mathrm{\angle }5and\mathrm{\angle }7$ , $\mathrm{\angle }6and\mathrm{\angle }8$

3. In the adjoining figure, $p\parallel q$ . Find the unknown angles.

Answer: The angles can be found using different properties:

(a) $\mathrm{\angle }e={180}^{\circ }-{125}^{\circ }={55}^{\circ }$ (The angles are linear pair)

(b) $\mathrm{\angle }e=\mathrm{\angle }f={55}^{\circ }$ (Vertically opposite angle)

(c) $\mathrm{\angle }d={125}^{\circ }$ (Corresponding angle)

(d) $\mathrm{\angle }d=\mathrm{\angle }b={125}^{\circ }$ (Vertically opposite angle)

(e) $\mathrm{\angle }a=\mathrm{\angle }c={180}^{\circ }-{125}^{\circ }={55}^{\circ }$ (Vertically opposite angel, linear pair).

4.(i) Find the value of $x$ in each of the following figures if $l\parallel m$ .

Answer: The linear pair of the ${110}^{\circ }$ is : $\mathrm{\Theta }={180}^{\circ }-{110}^{\circ }={70}^{\circ }$

Thus the value of x is : $x={70}^{\circ }$ (Corresponding angles of parallel lines are equal).

4.(ii) Find the value of $x$ in each of the following figures if $l\parallel m.$

Answer: The value of x is ${100}^{\circ }$ , as these are the corresponding angles.

5. In the given figure, the arms of two angles are parallel. If $\mathrm{\angle }ABC={70}^o$ , then find

$(i)\mathrm{\angle }DGC$

$(ii)\mathrm{\angle }DEF$

Answer: (i) Since side AB is parallel to DG.

Thus : $\mathrm{\angle }DGC={70}^{\circ }$ (Corresponding angles of parallel arms are equal.)

(ii) Further side BC is parallel to EF.

We have : $\mathrm{\angle }DEF=\mathrm{\angle }DGC={70}^{\circ }$ (Corresponding angles of parallel arms are equal.)

6. In the given figures below, decide whether $l$ is parallel to $m$ .

Answer: (i) In this case. the sum of the interior angle is ${126}^{\circ }+{44}^{\circ }={170}^{\circ }$ thus l is not parallel to m.\

(ii) In this case, also l is not parallel to m as the corresponding angle cannot be ${75}^{\circ }$ (Linear pair will not form).

(iii) In this l and m are parallel. This is because the corresponding angle is ${57}^{\circ }$ and it forms linear pair with ${123}^{\circ }$ .

(iv) The lines are not parallel as the linear pair not form. (Since the corresponding angle will be ${72}^{\circ }$ otherwise.)