NCERT Solutions Class 7 Maths Chapter 5

Q.1

$\mathrm{Find}\mathrm{the}\mathrm{complement}\mathrm{of}\mathrm{the}\mathrm{following}\mathrm{angles}:$

Ans.

\begin{array}{l}\text{(i) 20}°\\ \text{Since the sum of complementary angle is 90}°\\ \text{So, we have}\\ \text{Complement}=\text{90}°-20°=\boxed{70°}\\ \text{(ii) 63}°\\ \text{Since the sum of complementary angle is 90}°\\ \text{So, we have}\\ \text{Complement}=\text{90}°-63°=\boxed{27°}\\ \text{(iii) 57}°\\ \text{Since the sum of complementary angle is 90}°\\ \text{So, we have}\\ \text{Complement}=\text{90}°-57°=\boxed{33°}\end{array}

Q.2 Find the supplement of the following angles:

Ans.

\begin{array}{l}\text{(i) 105}°\\ \text{Since the sum of supplementary angle is 180}°\\ \text{So, we have}\\ \text{Complement}=\text{180}°-105°=\boxed{75°}\\ \text{(ii) 87}°\\ \text{Since the sum of Supplementary angle is 180}°\\ \text{So, we have}\\ \text{Complement}=\text{180}°-87°=\boxed{93°}\text{(iii) 154}°\\ \text{Since the sum of Supplementary angle is 180}°\\ \text{So, we have}\\ \text{Complement}=\text{180}°-154°=\boxed{26°}\end{array}

Q.3

$\begin{array}{l}\mathrm{Identify}\mathrm{which}\mathrm{of}\mathrm{the}\mathrm{following}\mathrm{pairs}\mathrm{of}\mathrm{angles}\mathrm{are}\\ \mathrm{complementary}\mathrm{and}\mathrm{which}\mathrm{are}\mathrm{supplementary}.\\ \left(\mathrm{i}\right)65°,115°\left(\mathrm{ii}\right)63°,27°\left(\mathrm{iii}\right)112°,68°\\ \left(\mathrm{iv}\right)130°,50°\left(\mathrm{v}\right)45°,45°\left(\mathrm{vi}\right)80°,10°\end{array}$

Ans.

$\begin{array}{l}\left(\text{i}\right)\text{65}°,\text{115}°\\ \text{Since, the sum of complementary angle is 90}°\text{and sum of}\\ \text{supplementary angle is 180}°.\\ \text{So,}65°\text{+115}°=180°\\ \text{Therefore, given pair is supplemenatry}\text{.}\\ \left(\text{ii}\right)\text{63}°,\text{27}°\\ \text{Since, the sum of complementary angle is 90}°\text{and sum of}\\ \text{supplementary angle is 180}°.\\ \text{So,}63°\text{+27}°=90°\\ \text{Therefore, given pair is complementary}\text{.}\\ \left(\text{iii}\right)\text{112}°,\text{68}°\\ \text{Since, the sum of complementary angle is 90}°\text{and sum of}\\ \text{supplementary angle is 180}°.\\ \text{So,}112°\text{+68}°=180°\\ \text{Therefore, given pair is supplemenatry}\text{.}\\ \left(\text{iv}\right)\text{130}°,\text{50}°\\ \text{Since, the sum of complementary angle is 90}°\text{and sum of}\\ \text{supplementary angle is 180}°.\\ \text{So,}130°\text{+50}°=180°\\ \text{Therefore, given pair is supplemenatry}\text{.}\\ \left(\text{v}\right)\text{45}°,\text{45}°\\ \text{Since, the sum of complementary angle is 90}°\text{and sum of}\\ \text{supplementary angle is 180}°.\\ \text{So,}45°\text{+45}°=90°\\ \text{Therefore, given pair is complementary}\text{.}\\ \left(\text{vi}\right)\text{80}°,\text{10}°\\ \text{Since, the sum of complementary angle is 90}°\text{and sum of}\\ \text{supplementary angle is 180}°.\\ \text{So,}80°\text{+10}°=90°\\ \text{Therefore, given pair is complementary}\text{.}\end{array}$

Q.4

$\mathrm{Find}\mathrm{the}\mathrm{angle}\mathrm{which}\mathrm{is}\mathrm{equal}\mathrm{to}\mathrm{its}\mathrm{complement}.$

Ans.

$\begin{array}{l}\text{Let the angle be}\mathrm{x}.\\ \text{Since, it also equal to its complement.}\\ \text{So, complement angle}=\mathrm{x}.\\ \text{Sum of complementary angle is 90}°\\ \text{So, we get}\\ \mathrm{x}+\mathrm{x}=90°\\ 2\mathrm{x}=90°\\ \mathrm{x}=\frac{90°}{2}=45°\\ \text{Thus, the angle be}\boxed{\text{45}°}.\end{array}$

Q.5

$\mathrm{Find}\mathrm{the}\mathrm{angle}\mathrm{which}\mathrm{is}\mathrm{equal}\mathrm{to}\mathrm{its}\mathrm{supplement}.$

Ans.

$\begin{array}{l}\text{Let the angle be}\mathrm{x}\text{.}\\ \text{Since, it also equal to its supplement.}\\ \text{So, supplement angle}=\mathrm{x}\text{.}\\ \text{Sum of supplementary angle is 180}°\\ \text{So, we get}\\ \mathrm{x}+\mathrm{x}=180°\\ 2\mathrm{x}=180°\\ \mathrm{x}=\frac{180°}{2}=90°\\ \text{Thus, the angle be}\boxed{\text{90}°}.\end{array}$

Q.6

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

Ans.

\begin{array}{l}\text{Since,}\angle 1\text{and}\angle 2\text{are supplementary angles}\text{.}\\ \text{If}\angle 1\text{reduced, then}\angle 2\text{should be increased by the same}\\ \text{measure so that this angle remain supplementary}\text{.}\end{array}

Q.7

$\begin{array}{l}\mathrm{Can}\mathrm{two}\mathrm{angles}\mathrm{be}\mathrm{supplementary}\mathrm{if}\mathrm{both}\mathrm{of}\mathrm{them}\mathrm{are}\\ \left(\mathrm{i}\right)\mathrm{acute}?\\ \left(\mathrm{ii}\right)\mathrm{obtuse}?\\ \left(\mathrm{iii}\right)\mathrm{right}?\end{array}$

Ans.

$\begin{array}{l}\text{(i) No, if both angles are acute, that means that both angles}\\ \text{are less than 90}°\text{. In that case, their sum can not be equal to 180}°.\\ \\ \text{(ii) No, if both angles are obtuse, that means that both angles}\\ \text{greater than 90}°.\text{In that case, their sum will exceed 180}°.\\ \\ \text{(iii) Yes, if both angles are right angles, that is, 90}°,\text{then their}\\ \text{sum will be exact 180}°.\end{array}$

Q.8 An angle is greater than 45°. Is its complementary angle greater than 45° or equal to 45° or less than 45°?

Ans.

$\begin{array}{l}\text{Let}\mathrm{x}\text{and}\mathrm{y}\text{be two angles having complementary angle pair}\\ \text{and}\mathrm{x}\text{is greater than 45}°.\\ \text{Then,}\\ \mathrm{x}+\mathrm{y}=90°\\ \mathrm{y}=90°-\mathrm{x}\\ \text{Thus, y will be less than 45}°.\end{array}$

Q.9 

$\begin{array}{l}\mathrm{In}\mathrm{the}\mathrm{adjoining}\mathrm{figure}:\\ \left(\mathrm{i}\right)\mathrm{Is}\angle 1\mathrm{adjacent}\mathrm{to}\angle 2?\\ \left(\mathrm{ii}\right)\mathrm{Is}\angle \mathrm{AOC}\mathrm{adjacent}\mathrm{to}\angle \mathrm{AOE}?\\ \left(\mathrm{iii}\right)\mathrm{Do}\angle \mathrm{COE}\mathrm{and}\angle \mathrm{EOD}\mathrm{form}\mathrm{a}\mathrm{linear}\mathrm{pair}?\\ \left(\mathrm{iv}\right)\mathrm{Are}\angle \mathrm{BOD}\mathrm{and}\angle \mathrm{DOA}\mathrm{supplementary}?\\ \left(\mathrm{v}\right)\mathrm{Is}\angle 1\mathrm{vertically}\mathrm{opposite}\mathrm{to}\angle 4?\\ \left(\mathrm{v}\right)\mathrm{What}\mathrm{is}\mathrm{the}\mathrm{vertically}\mathrm{opposite}\mathrm{angle}\mathrm{of}\angle 5?\end{array}$

Ans.

\begin{array}{l}\text{(i) Yes, since they have a common vetex O and also a common}\\ \text{arm OC}\text{. Also, their non-common arms, OA}\text{}\text{and OB are on}\\ \text{either side of the common arm}\text{.}\\ \text{(ii) No}\text{. they have a common vertex O and also a common}\\ \text{arm OA}\text{. However, their non common arms, OC and OE are on}\\ \text{the same side of the common arm}\text{. Therefore, theses are not}\\ \text{adjacent to each other}\text{.}\\ \text{(iii) Yes, since they have a common vertex O and a common}\\ \text{arm OE}\text{. Also, their non common arms OC and OD, are}\\ \text{opposite rays}\text{.}\\ \text{(iv) Yes, since}\angle \text{BOD and}\angle \text{DOA have a common vertex O and}\\ \text{their non-common arms opposite to each other}\text{.}\\ \text{(v) Yes, since these are fromed dure to the intersection of two}\\ \text{straight lines (AB and CD)}\\ \text{(vi)}\angle \text{COB is the vertically opposite angle of}\angle 5\text{as these are}\\ \text{formed due to the intersection of two straight lines AB and CD}\text{.}\end{array}

Q.10

$\begin{array}{l}\mathrm{Indicate}\mathrm{which}\mathrm{pairs}\mathrm{of}\mathrm{angles}\mathrm{are}:\\ \left(\mathrm{i}\right)\mathrm{Vertically}\mathrm{opposite}\mathrm{angles}.\\ \left(\mathrm{ii}\right)\mathrm{Linear}\mathrm{pairs}.\end{array}$

Ans.

\begin{array}{l}\text{(i)}\angle 1\text{,}\angle 4\text{and}\angle 5,\left(\angle 2+\angle 3\right)\text{are vertically opposite angles}\\ \text{as these formed due to the intersection of straight lines}\\ \text{(ii)}\angle 1\text{and}\angle \text{5,}\angle 5\text{and,}\angle 4\text{as these have common vertex}\\ \text{and also have non-common arms opposite to each other}\text{.}\end{array}

Q.11 In the following figure, is

$\angle 1$

adjacent to

$\angle 2$

? Give reasons.

Ans.

$\angle 1$

and

$\angle 2$

are not adjacent angles because they don’t have common vertex.

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

Ans.

$\begin{array}{l}\text{(i) Since}\angle \mathrm{x}\text{\hspace{0.17em}and}\angle 55°\text{are vertically opposite angles.}\\ \text{So,}\mathrm{x}=\angle 55°\\ \angle \mathrm{x}°+\angle \mathrm{y}°=\angle 180°\\ \angle 55°+\angle \mathrm{y}°=\angle 180°\\ \angle \mathrm{y}°=\angle 180°-\angle 55°\\ =\angle 125°\\ \angle \mathrm{y}°=\angle \mathrm{z}°\left(\text{vertically\hspace{0.17em}opposite\hspace{0.17em}angles}\right)\\ \mathrm{So},\angle \mathrm{z}°=\angle 125°\\ \text{(ii)}\angle \mathrm{z}°=\angle 40°\left(\text{vertically\hspace{0.17em}opposite\hspace{0.17em}angles}\right)\\ \angle \mathrm{y}°+\angle \mathrm{z}°=\angle 180°\left(\text{Linear pair}\right)\\ \angle \mathrm{y}°=\angle 180°-\angle 40°\\ =\angle 140°\\ \angle 40°+\angle \mathrm{x}°+\angle 25°=\angle 180°\\ \angle \mathrm{x}°+\angle 65°=\angle 180°\\ \angle \mathrm{x}°=\angle 180°-\angle 65°\\ =\angle 115°\end{array}$

Q.13

$\begin{array}{l}\mathrm{Fill}\mathrm{in}\mathrm{the}\mathrm{blanks}:\\ \left(\mathrm{i}\right)\mathrm{If}\mathrm{two}\mathrm{angles}\mathrm{are}\mathrm{complementary},\mathrm{then}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{their}\\ \mathrm{measuresis}\_\_\_\_\_\_.\\ \left(\mathrm{ii}\right)\mathrm{If}\mathrm{two}\mathrm{angles}\mathrm{are}\mathrm{supplementary},\mathrm{then}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{their}\\ \mathrm{measuresis}\_\_\_\_\_.\\ \left(\mathrm{iii}\right)\mathrm{Two}\mathrm{angles}\mathrm{forming}\mathrm{a}\mathrm{linear}\mathrm{pair}\mathrm{are}\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.\\ \left(\mathrm{iv}\right)\mathrm{If}\mathrm{two}\mathrm{adjacent}\mathrm{angles}\mathrm{are}\mathrm{supplementary},\mathrm{they}\mathrm{form}\mathrm{a}\_\_\_\_\_\_\_\_\_\_.\\ \left(\mathrm{v}\right)\mathrm{If}\mathrm{two}\mathrm{lines}\mathrm{intersect}\mathrm{at}\mathrm{a}\mathrm{point},\mathrm{then}\mathrm{the}\mathrm{vertically}\mathrm{opposite}\\ \begin{array}{l}\mathrm{angles}\mathrm{are}\mathrm{always}\_\_\_\_\_\_\_\_\_\_\_\_\_.\\ \begin{array}{l}\left(\mathrm{vi}\right)\mathrm{If}\mathrm{two}\mathrm{lines}\mathrm{intersect}\mathrm{at}\mathrm{a}\mathrm{point},\mathrm{and}\mathrm{if}\mathrm{one}\mathrm{pair}\mathrm{of}\\ \mathrm{vertically}\mathrm{opposite}\mathrm{angles}\mathrm{are}\mathrm{acute}\mathrm{angles},\mathrm{then}\mathrm{the}\\ \mathrm{other}\mathrm{pair}\mathrm{of}\mathrm{vertically}\mathrm{opposite}\mathrm{angles}\mathrm{are}\_\_\_\_\_\_\_\_\_\_.\end{array}\end{array}\end{array}$

Ans.

\begin{array}{l}\text{Fill in the blanks}:\\ \left(\text{i}\right)\text{If two angles are complementary},\text{then the sum of their}\\ \text{measures is}\boxed{\text{90}°}.\\ \left(\text{ii}\right)\text{If two angles are supplementary},\text{then the sum of their}\\ \text{measures is}\boxed{180°}.\\ \left(\text{iii}\right)\text{Two angles forming a linear pair are}\boxed{\text{supplementary}}.\\ \left(\text{iv}\right)\text{If two adjacent angles are supplementary},\text{they}\\ \text{form a}\boxed{\text{Linear pair}}.\\ \left(\text{v}\right)\text{If two lines intersect at a point},\text{then the vertically opposite}\\ \text{angles are always}\boxed{\text{equal}}\text{}.\\ \left(\text{vi}\right)\text{If two lines intersect at a point},\text{and if one pair of vertically}\\ \text{opposite angles are acute angles},\text{then the other pair of}\\ \text{vertically opposite angles are}\boxed{\text{obtuse angles}}.\end{array}

Q.14

$\begin{array}{l}\mathrm{In}\mathrm{the}\mathrm{adjoining}\mathrm{figure},\mathrm{name}\mathrm{the}\mathrm{following}\mathrm{pairs}\mathrm{of}\mathrm{angle}.\\ \left(\mathrm{i}\right)\mathrm{Obtuse}\mathrm{vertically}\mathrm{opposite}\mathrm{angles}\\ \left(\mathrm{ii}\right)\mathrm{Adjacent}\mathrm{complementary}\mathrm{angles}\\ \left(\mathrm{iii}\right)\mathrm{Equal}\mathrm{supplementary}\mathrm{angles}\\ \left(\mathrm{iv}\right)\mathrm{Unequal}\mathrm{supplementary}\mathrm{angles}\\ \left(\mathrm{v}\right)\mathrm{Adjacent}\mathrm{angles}\mathrm{that}\mathrm{do}\mathrm{not}\mathrm{form}\mathrm{a}\mathrm{linear}\mathrm{pair}.\end{array}$

Ans.

$\begin{array}{l}\text{(i)}\angle \text{AOD},\angle \text{BOC}\\ \text{(ii)}\angle \text{EOA},\angle \text{AOB}\\ \text{(iii)}\angle \text{EOB},\angle \text{EOD}\\ \text{(iv)}\angle \text{EOA},\angle \text{EOC}\\ \text{(v)}\angle \text{AOB}\text{and}\angle \text{AOE},\angle \text{AOE}\text{\hspace{0.17em} and}\angle \text{EOD},\angle \text{EOD}\text{and}\angle \text{COD}\text{.}\end{array}$

Q.15

$\begin{array}{l}\mathrm{State}\mathrm{the}\mathrm{property}\mathrm{that}\mathrm{isused}\mathrm{in}\mathrm{each}\mathrm{of}\mathrm{the}\mathrm{following}\mathrm{statements}?\\ \begin{array}{l}\left(\mathrm{i}\right)\mathrm{If}\mathrm{a}\parallel \mathrm{b},\mathrm{then}\angle 1=\angle 5.\\ \left(\mathrm{ii}\right)\mathrm{If}\angle 4=\angle 6,\mathrm{then}\mathrm{a}\parallel \mathrm{b}.\\ \left(\mathrm{iii}\right)\mathrm{If}\angle 4+\angle 5=180°,\mathrm{then}\mathrm{a}\parallel \mathrm{b}.\end{array}\end{array}$

Ans.

$\begin{array}{l}\text{(i) Corresponding angles property}\\ \text{(ii) Alternate interior angles property}\\ \text{(ii) Interior angles on the same side of transversal are supplementary.}\end{array}$

Q.16 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.

Ans.

$\begin{array}{l}\text{(i)}\angle 1\text{and}\angle 5\text{,}\angle 2\text{and}\angle 6\text{,}\angle 3\text{and}\angle 7,\angle 4\text{and}\angle 8.\\ \text{(ii)}\angle 2\text{and}\angle 8,\angle 3\text{and}\angle 5\\ (\text{iii)}\angle 2\text{and}\angle 5,\angle 3\text{and}\angle 8\\ \text{(iv)}\angle 1\text{and}\angle 3,\angle 2\text{and}\angle 4,\angle 5\text{and}\angle 7,\angle 6\text{and}\angle 8.\end{array}$

Q.17

$\mathrm{In}\mathrm{the}\mathrm{adjoining}\mathrm{figure},\mathrm{p}\parallel \mathrm{q}.\mathrm{Find}\mathrm{the}\mathrm{unknown}\mathrm{angles}.$

Ans.

$\begin{array}{l}\angle \mathrm{d}=125°\left(\text{Corresponding\hspace{0.17em}angles}\right)\\ \angle \mathrm{e}=180°125°=55°\left(\text{Linear\hspace{0.17em}pair}\right)\\ \angle \mathrm{f}=\angle \mathrm{e}=55°\left(\text{Vertically\hspace{0.17em}opposite\hspace{0.17em}angles}\right)\\ \angle \mathrm{c}=\angle \mathrm{f}=55°\left(\text{Corresponding\hspace{0.17em}angles}\right)\\ \angle \mathrm{a}=\angle \mathrm{e}=55°\left(\text{Corresponding\hspace{0.17em}angles}\right)\\ \angle \mathrm{b}=\angle \mathrm{d}=125°\left(\text{Vertically\hspace{0.17em}opposite\hspace{0.17em}angles}\right)\end{array}$

Q.18 Find the value of x in each of the following figures if

$\mathrm{l}\parallel \mathrm{m}$

.

Ans.

$\begin{array}{l}\text{(i)}\\ \angle \mathrm{y}=110°\text{(corresponding angles)}\\ \angle \mathrm{x}+\angle \mathrm{y}=180°\left(\text{linear pair}\right)\\ \mathrm{So},\angle \mathrm{x}=180°-110°=70°\\ \\ \text{(ii)}\\ \angle \mathrm{x}=100°\left(\text{corresponding\hspace{0.17em}angles}\right)\end{array}$

Q.19

$$\begin{gathered} \mathrm{In}\mathrm{the}\mathrm{given}\mathrm{figure},\mathrm{the}\mathrm{arms}\mathrm{of}\mathrm{two}\mathrm{angles}\mathrm{are}\mathrm{parallel}. \\ \mathrm{If}\angle \mathrm{ABC}=70°,\mathrm{then}\mathrm{find} \\ \left(\mathrm{i}\right)\angle \mathrm{DGC} \\ \left(\mathrm{ii}\right)\angle \mathrm{DEF} \end{gathered}$$

Ans.

$\begin{array}{l}\text{(i) Consdier that AB}\parallel \text{DG and a transerval line BC is}\\ \text{intersecting them.}\\ \angle \mathrm{DGC}=\angle \mathrm{ABC}\left(\text{Corresponding angles}\right)\\ \mathrm{SO},\boxed{\angle \mathrm{DGC}=70°}\\ \\ \text{(ii) Consdier that BC}\parallel \text{EF and a transerval line DE is}\\ \text{intersecting them.}\\ \angle \mathrm{DEF}=\angle \mathrm{DGC}\left(\text{Corresponding angles}\right)\\ \boxed{\angle \mathrm{DEF}=70°}\end{array}$

Q.20 In the given figures below, decide whether l is parallel to m.

Ans.

$\begin{array}{l}\text{(i)}\\ \text{Consider two lines, l and m, and a transversal line n which is}\\ \text{intersecting them. Sum of the interior angles on the same side}\\ \text{of transersal}=126°+44°=170°\text{.}\\ \text{As the sum of interior angles on the same side of transervsal is}\\ \text{not 180}°\text{, Therefore}\mathrm{l}\text{is not parallel to}\mathrm{m}\text{.}\\ \\ \text{(ii)}\mathrm{x}+75°=180°\text{(linear pair)}\\ \mathrm{x}=180°-75°=105°\\ \mathrm{For}\mathrm{l}\text{and}\mathrm{m}\text{to be parallel to each other, corresponding}\\ \text{angles (}\angle \text{ABC and}\angle \mathrm{x})\text{\hspace{0.17em}should be equal. However, here they}\\ \text{are 75}°\text{and 105}°\text{.}\\ \begin{array}{l}\text{Hence these lines are not parallel to each other.}\\ \\ \text{(iii)}\mathrm{x}+123°=180°\\ \mathrm{x}=\text{180}°-123°=57°\\ \mathrm{For}\text{l and m to be parallel to each other, corresponding}\\ \text{angles (}\angle \text{ABC and}\angle \mathrm{x})\text{\hspace{0.17em}should be equal. However, here they}\\ \text{are 57}°\text{and 57}°\text{.}\\ \text{Hence these lines are parallel to each other.}\\ \\ \text{(iv) 98}°+\mathrm{x}=180°\\ \mathrm{x}=180°-98°=82°\\ \text{For}\mathrm{l}\text{and}\mathrm{m}\text{to be parallel to each other, corresponding}\\ \text{angles (}\angle \text{ABC and}\angle \mathrm{x})\text{\hspace{0.17em}should be equal. However, here they}\\ \text{are 72}°\text{and 82}°\text{.}\\ \text{Hence these lines are not parallel to each other}\end{array}\end{array}$