NCERT Solutions Class 7 Maths Chapter 5 Lines and Angles

4.0NCERT Questions with Solutions for Class 7 Maths Chapter 5 - Detailed Solutions

Exercise: 5.1

• Find the complement of each of the following angles:

• (i)

• (ii)

• (iii) Sol. The sum of the measures of complementary angles is $90^{\circ }$. (i) $20^{\circ }$ : Complement $=90^{\circ }-20^{\circ }=70^{\circ }$ (ii) $63^{\circ }$ : Complement $=90^{\circ }-63^{\circ }=27^{\circ }$ (iii) $57^{\circ }$ : Complement $=90^{\circ }-57^{\circ }=33^{\circ }$
• Find the supplement of each of the following angles:

• Sol. The sum of the measures of supplementary angles is $180^{\circ }$. (i) $105^{\circ }$ : Supplement $=180^{\circ }-105^{\circ }=75^{\circ }$ (ii) $87^{\circ }$ : Supplement $=180^{\circ }-87^{\circ }=93^{\circ }$ (iii) $154^{\circ }$ : Supplement $=180^{\circ }-154^{\circ }=26^{\circ }$
• Identify which of the following pairs of angles are complementary and which are supplementary. (i) $65^{\circ },115^{\circ }$ (ii) $63^{\circ },27^{\circ }$ (iii) $112^{\circ },68^{\circ }$ (iv) $130^{\circ },50^{\circ }$ (v) $45^{\circ },45^{\circ }$ (vi) $80^{\circ },10^{\circ }$ Sol. The sum of the measures of complementary angles is $90^{\circ }$ and that of supplementary angles is $180^{\circ }$. (i) $65^{\circ },115^{\circ }$ Sum of the measures of these angles $=65^{\circ }$ $+115^{\circ }=180^{\circ }$ $\therefore$ These angles are supplementary angles. (ii) $63^{\circ },27^{\circ }$ Sum of the measures of these angles $=63^{\circ }$ $+27^{\circ }=90^{\circ }$ $\therefore$ These angles are complementary angles. (iii) $112^{\circ },68^{\circ }$ Sum of the measures of these angles = $112^{\circ }+68^{\circ }=180^{\circ }$ $\therefore$ These angles are supplementary angles. (iv) $130^{\circ },50^{\circ }$ Sum of the measures of these angles $=$ $130^{\circ }+50^{\circ }=180^{\circ }$ $\therefore$ These angles are supplementary angles. (v) $45^{\circ },45^{\circ }$ Sum of the measures of these angles $=45^{\circ }$ $+45^{\circ }=90^{\circ }$ $\therefore$ These angles are complementary angles. (vi) $80^{\circ },10^{\circ }$ Sum of the measures of these angles $=80^{\circ }$ $+10^{\circ }=90^{\circ }$ $\therefore$ These angles are complementary angles.
• Find the angle which is equal to its complement. Sol. Let the angle be x. Complement of this angle is also $x$. The sum of the measures of a complementary angle pair is $90^{\circ }$. $$\begin{gathered} \therefore \mathrm{x}+\mathrm{x}=90^{\circ } \\ 2\mathrm{x}=90^{\circ } \\ \mathrm{x}=\frac{90^{\circ }}{2}=45^{\circ } \end{gathered}$$
• Find the angle which is equal to its supplement. Sol. Let the angle be x. Supplement of this angle is also x . The sum of the measures of a supplementary angle pair is $180^{\circ }$. $$\begin{gathered} \therefore \mathrm{x}+\mathrm{x}=180^{\circ } \\ 2\mathrm{x}=180^{\circ } \\ \mathrm{x}=90^{\circ } \end{gathered}$$
• In the given figure, $\angle 1$ and $\angle 2$ are supplementary angles. If $\angle 1$ is decreased, what changes should take place in $\angle 2$ so that both the angles still remain supplementary.

• Sol. $\angle 1$ and $\angle 2$ are supplementary angles. If $\angle 1$ is reduced, then $\angle 2$ should be increased by the same measure so that this angle pair remains supplementary.
• Can two angles be supplementary if both of them are: (i) Acute? (ii) Obtuse? (iii) Right? Sol. (i) No. Acute angle is always lesser than $90^{\circ }$. It can be observed that two angles, even of $89^{\circ }$, cannot add up to $180^{\circ }$. Therefore, two acute angles cannot be in a supplementary angle pair. (ii) No. Obtuse angle is always greater than $90^{\circ }$. It can be observed that two angles, even of $91^{\circ }$, will always add up to more than $180^{\circ }$. Therefore, two obtuse angles cannot be in a supplementary angle pair. (iii) Yes. Right angles are of $90^{\circ }$ and $90^{\circ }+90^{\circ }$ $=180^{\circ }$. Therefore, two right angles form a supplementary angle pair together.
• An angle is greater than $45^{\circ }$. Is its complementary angle greater than $45^{\circ }$ or equal to $45^{\circ }$ or less than $45^{\circ }$ ? Sol. Let A and B are two angles making a complementary angle pair and A is greater than $45^{\circ }$. $A+B=90^{\circ }$ $\mathrm{B}=90^{\circ }-\mathrm{A}$ Therefore, B will be lesser than $45^{\circ }$.
• In the adjoining figure:

• (i) Is $\angle 1$ adjacent to $\angle 2$ ? (ii) Is $\angle \mathrm{AOC}$ adjacent to $\angle \mathrm{AOE}$ ? (iii) Do $\angle \mathrm{COE}$ and $\angle \mathrm{EOD}$ form a linear pair? (iv) Are $\angle \mathrm{BOD}$ and $\angle \mathrm{DOA}$ supplementary? (v) Is $\angle 1$ vertically opposite to $\angle 4$ ? (vi) What is the vertically opposite angle of $\angle 5$ ? Sol. (i) Yes. Since they have a common vertex 0 and also a common arm OC. Also, their non-common arms, OA and OE, are on opposite side of the common arm. (ii) No. They have a common vertex 0 and also a common arm OA. However, their non common arms, OC and OE, are on the same side of the common arm. Therefore, these are not adjacent to each other. (iii) Yes. Since they have a common vertex 0 and a common arm OE. Also, their non common arms, OC and OD, are opposite rays. (iv) Yes. Since $\angle BOD$ and $\angle DOA$ have a common vertex 0 and their non-common arms are opposite to each other. (v) Yes. Since these are formed due to the intersection of two straight lines AB and CD. (vi) $\angle \mathrm{COB}$ is the vertically opposite angle of $\angle 5$ as these are formed due to the intersection of two straight lines, $AB$ and CD.
• Indicate which pairs of angles are: (i) Vertically opposite angles. (ii) Linear pairs.

• Sol. (i) $\angle 1$ and $\angle 4,\angle 5$ and $[\angle 2+\angle 3]$ are vertically opposite angles as these are formed due to the intersection of two straight lines. (ii) $\angle 1$ and $\angle 5,\angle 5$ and $\angle 4$ as these have a common vertex and also have noncommon arms opposite to each other
• In the following figure, is $\angle 1$ adjacent to $\angle 2$ ? Give reasons.

• Sol. $\angle 1$ and $\angle 2$ are not adjacent angles because their vertex is not common.
• Find the value of the angles $x,y$, and $z$ in each of the following:

• (i)

• Sol. (i) Since $\angle x$ and $\angle 55^{\circ }$ are vertically opposite angles, $\therefore \angle \mathrm{x}=55^{\circ }$ $\angle \mathrm{x}+\angle \mathrm{y}=180^{\circ }$ (Linear pair) $55^{\circ }+\angle \mathrm{y}=180^{\circ }$ $\angle \mathrm{y}=180^{\circ }-55^{\circ }=125^{\circ }$ $\angle \mathrm{y}=\angle \mathrm{z}$ (Vertically opposite angles) $\angle \mathrm{z}=125^{\circ }$ (ii) $\angle \mathrm{z}=40^{\circ }$ (Vertically opposite angles) $\angle \mathrm{y}+\angle \mathrm{z}=180^{\circ }$ (Linear pair) $\angle y=180^{\circ }-40^{\circ }=140^{\circ }$ $40^{\circ }+\angle \mathrm{x}+25^{\circ }=180^{\circ }$ (Angles on a straight line) $65^{\circ }+\angle \mathrm{x}=180^{\circ }$ $\angle \mathrm{x}=180^{\circ }-65^{\circ }=115^{\circ }$
• 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, then 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 ______ . Sol. (i) $90^{\circ }$ (ii) $180^{\circ }$ (iii) Supplementary (iv) Linear pair (v) Equal (vi) Obtuse angles
• 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 Sol. (i) $\angle \mathrm{AOD},\angle \mathrm{BOC}$ (ii) $\angle \mathrm{EOA},\angle \mathrm{AOB}$ (iii) $\angle$ EOB, $\angle$ EOD (iv) $\angle$ EOA, $\angle$ EOC (v) $\angle \mathrm{AOB}$ and $\angle \mathrm{AOE},\angle \mathrm{AOE}$ and $\angle \mathrm{EOD},\angle \mathrm{EOD}$ and $\angle$ COD

Exercise : 5.2

• State the property that is used in each of the following statements? (i) If a || b, then $\angle 1=\angle 5$ (ii) If $\angle 4=\angle 6$, then a || b (iii) If $\angle 4+\angle 5=180^{\circ }$, then a $\Vert$ b
  
  Sol. (i) Corresponding angles property. (ii) Alternate interior angles property. (iii) Interior angles on the same side of transversal are supplementary.
• 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
  
  Sol. (i) $\angle 1$ and $\angle 5,\angle 2$ and $\angle 6,\angle 3$ and $\angle 7,\angle 4$ and $\angle 8$ (ii) $\angle 2$ and $\angle 8,\angle 3$ and $\angle 5$ (iii) $\angle 2$ and $\angle 5,\angle 3$ and $\angle 8$ (iv) $\angle 1$ and $\angle 3,\angle 2$ and $\angle 4,\angle 5$ and $\angle 7,\angle 6$ and $\angle 8$
• In the adjoining figure, $\mathrm{p}\Vert \mathrm{q}$. Find the unknown angles.
  
  Sol. $\angle \mathrm{d}=125^{\circ }$ (Corresponding angles) $\angle \mathrm{e}=180^{\circ }-125^{\circ }=55^{\circ }$ (Linear pair) $\angle \mathrm{f}=\angle \mathrm{e}=55^{\circ }$ (Vertically opposite angles) $\angle \mathrm{c}=\angle \mathrm{f}=55^{\circ }$ (Corresponding angles) $\angle \mathrm{a}=\angle \mathrm{e}=55^{\circ }$ (Corresponding angles) $\angle$ b $=\angle$ d $=125^{\circ }$ (Vertically opposite angles)
• Find the value of $x$ in each of the following figures if $\ell \Vert \mathrm{m}$.
  
  (i)
  
  (ii) Sol. (i)
  
  $\angle \mathrm{y}=110^{\circ }$ (Corresponding angles) $\angle \mathrm{x}+\angle \mathrm{y}=180^{\circ }$ (Linear pair) $\angle \mathrm{x}=180^{\circ }-110^{\circ }=70^{\circ }$ (ii)
  
• In the given figure, the arms of two angles are parallel. If $\angle \mathrm{ABC}=70^{\circ }$, then find (i) $\angle \mathrm{DGC}$ (ii) $\angle \mathrm{DEF}$
  
  Sol. (i) Consider that $\mathrm{AB}\Vert \mathrm{DG}$ and a transversal BC is intersecting them. $$\begin{gathered} \angle \mathrm{DGC}=\angle \mathrm{ABC}\text{ (Corresponding angles) } \\ \angle \mathrm{DGC}=70^{\circ } \end{gathered}$$ (ii) Consider that $\mathrm{BC}|\mid \mathrm{EF}$ and a transversal DE is intersecting them. $\angle \mathrm{DEF}=\angle \mathrm{DGC}$ (Corresponding angles) $\angle \mathrm{DEF}=70^{\circ }$
• In the given figures below, decide whether $\ell$ is parallel to $m$.
  
  (i)
  
  (iii)
  
  Sol. (i)
  
  Consider two lines, $\ell$ and $m$, and $a$ transversal line n which is intersecting them. Sum of the interior angles on the same side of transversal $=126^{\circ }+44^{\circ }=170^{\circ }$ As the sum of interior angles on the same side of transversal is not $180^{\circ }$, therefore, $\ell$ is not parallel to m . (ii)
  
  $x+75^{\circ }=180^{\circ }($ Linear pair on line $\ell )$ $\mathrm{x}=180^{\circ }-75^{\circ }=105^{\circ }$ For $\ell$ and $m$ to be parallel to each other, corresponding angles ( $\angle \mathrm{ABC}$ and $\angle \mathrm{x}$ ) should be equal. However, here their measures are $75^{\circ }$ and $105^{\circ }$ respectively. Hence, these lines are not parallel to each other. (iii)
  
  $\angle \mathrm{x}+123^{\circ }=180^{\circ }$ (Linear pair) $\angle \mathrm{x}=180^{\circ }-123^{\circ }=57^{\circ }$ For $\ell$ and $m$ to be parallel to each other, corresponding angles ( $\angle \mathrm{ABC}$ and $\angle \mathrm{x}$ ) should be equal. Here, their measures are $57^{\circ }$ and $57^{\circ }$ respectively. Hence, these lines are parallel to each other. (iv)
  
  $98^{\circ }+\angle \mathrm{x}=180^{\circ }$ (Linear pair) $\angle \mathrm{x}=82^{\circ }$ For $\ell$ and $m$ to be parallel to each other, corresponding angles ( $\angle \mathrm{ABC}$ and $\angle \mathrm{x}$ ) should be equal. However, here their measures are $72^{\circ }$ and $82^{\circ }$ respectively. Hence, these lines are not parallel to each other.