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

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

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

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

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

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.

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