Unit: 5: Lines and Angles

Find the complement of each of the following angles:

(i)              

(ii)           

(iii)       

Complementary angle $=90°-$ given angle

(i)             Complement of $20°=90°-20°=70°$

(ii)          Complement of $63°=90°-63°=27°$

(iii)      Complement of $57°=90°-57°=33°$

Find the supplement of each of the following angles:

         i.        

       ii.       

 

 

      iii.     

Supplementary angle $=180°-$ given angle

(i)             Supplement of $105°=180°-105°=75°$

(ii)          Supplement of $87°=180°-87°=93°$

(iii)      Supplement of $154°=180°-154°=26°$

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

(i)             $65°,115°$

(ii)          $63°,27°$

(iii)      $112°,68°$

(iv)       $130°,50°$

(v)          $45°,45°$

(vi)       $80°,10°$

If sum of two angles is $180°$, then they are called supplementary angles.

If sum of two angles is $90°$, then they are called complementary angles.

(i)             $65°+115°=180°$    These are supplementary angles.

(ii)          $63°+27°=90°$ These are complementary angles.

(iii)      $112°+68°=180°$    These are supplementary angles.

(iv)       $130°+50°=180°$    These are supplementary angles.

(v)          $45°+45°=90°$ These are complementary angles.

(vi)       $80°+10°=90°$ These are complementary angles.

Find the angle which is equal to its complement.

Let one of the two equal complementary angles be $x$.

$\therefore x+x=90°$

$\Rightarrow 2x=90°$

$\Rightarrow x=\frac{90°}{2}=45°$

Thus, $45°$ is equal to its complement.

Find the angle which is equal to its supplement.

Let $x$ be two equal angles of its supplement.

Therefore, $x+x=180°$  [Supplementary angles]

$\Rightarrow 2x=180°$

$\Rightarrow x=\frac{180°}{2}=90°$

Thus, $90°$ is equal to its supplement.

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.

If $\angle 1$ is decreased then, $\angle 2$ will increase with the same measure, so that both the angles still remain supplementary.

Can two angles be supplementary if both of them are:

(i)             acute?

(ii)          obtuse?

(iii)      right?

(i)             No, because sum of two acute angles is less than $180°.$

(ii)          No, because sum of two obtuse angles is more than $180°.$

(iii)      Yes, because sum of two right angles is $180°.$

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

Let the complementary angles be $x$ and $y,$ $\text{i}\text{.e}\text{.},x+y=90°$.

It is given that $x>45°$

Adding $y$ both sides, $x+y>45°+y$

$\Rightarrow 90°>45°+y$

$\Rightarrow 90°-45°>y$

$\Rightarrow y<45°$

Thus, its complementary angle is less than $45°.$

In the adjoining figure:

(i)             Is $\angle 1$ adjacent to $\angle 2$?

(ii)          Is $\angle \text{AOC}$ adjacent to $\angle \text{AOE}$?

(iii)      Do $\angle \text{COE}$ and $\angle \text{EOD}$ form a linear pair?

(iv)       Are $\angle \text{BOD}$ and $\angle \text{DOA}$ supplementary?

(v)          Is $\angle 1$ vertically opposite to $\angle 4$?

(vi)       What is the vertically opposite angle of $\angle 5$?

(i)             Yes, in $\angle \text{AOE}$, OC is common arm.

(ii)          No, they have no non-common arms on opposite side of common arm.

(iii)      Yes, they form linear pair.

(iv)       Yes, they are supplementary.

(v)          Yes, they are vertically opposite angles.

(vi)       Vertically opposite angles of $\angle 5$ is $\angle \text{COB}$.

Indicate which pairs of angles are:

(i)             Vertically opposite angles.

(ii)          Linear pairs.

(i)             Vertically opposite angles, $\angle 1$ and $\angle 4$; $\angle 5$ and $\angle 2$ + $\angle 3$.

(ii)          Linear pairs $\angle 1$ and $\angle 5$; $\angle 5$ and $\angle 4$.

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

$\angle 1$ and $\angle 2$ are not adjacent angles because their vertex is not common.

Find the values of the angles $x$, $y$, and $z$ in each of the following:

(i)              

(ii)           

(i)             $x=55°$                                     [Vertically opposite angles]

Now $55°+y=180°$                 [Linear pair]

$\Rightarrow y=180°-55°=125°$

Also $y=z=125°$                    [Vertically opposite angles]

Thus, $x=55°,y=125°\text{ and}z=125°.$ 

(ii)          $40°+x+25°=180°$                [Angles on straight line]

$\Rightarrow 65°+x=180°$

$\Rightarrow x=180°-65°=115°$

Now $40°+y=180°$                 [Linear pair]

$\Rightarrow y=180°-40°=140°$       ……….(i)

Also $y+z=180°$                     [Linear pair]

$\Rightarrow 140°+z=180°$                [From equation (i)]

$\Rightarrow z=180°-140°=40°$

Thus, $x=115°,y=140°\text{ and }z=40°.$

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 ______________.

(i)             If two angles are complementary, then the sum of their measures is $90°$.

(ii)          If two angles are supplementary, then the sum of their measures is $180°$.

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

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

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

(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 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

(i)             $\angle \text{AOD}=\angle \text{BOC}\text{.}$

(ii)           $\angle \text{AOB}=\angle \text{AOE}\text{.}$

(iii)       $\angle \text{BOE}=\angle \text{DOE}\text{.}$

(iv)       $\angle \text{AOE,}\angle \text{EOC; }\angle \text{AOD, }\angle \text{DOC and }\angle \text{AOB,}\angle \text{BOC}$

(v)          $\angle \text{AOB,}\angle \text{AOE; }\angle \text{AOE, }\angle \text{EOD and }\angle \text{EOD,}\angle \text{COD}$

 

 

State the property that is used in each of the following statements?

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

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

(iii)      If $\angle 4+\angle 5=180°$, then $a\parallel b.$

(i)             Given, $a\parallel b,$ then $\angle 1=\angle 5$      [Corresponding angles]

If two parallel lines are cut by a transversal, each pair of corresponding angles are equal in measure.

(ii)          Given, $\angle 4=\angle 6$, then $a\parallel b,$     [Alternate interior angles]

When a transversal cuts two lines such that pairs of alternate interior angles are equal, the lines have to be parallel.

(iii)      Given, $\angle 4+\angle 5=180°$, then $a\parallel b,$  [Co-interior Angles]

When a transversal cuts two lines, such that pairs of interior angles on the same side of transversal are supplementary, the lines have to be parallel.

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.

(i)             The pairs of corresponding angles:

$\angle 1\text{and}\angle 5;\angle 2\text{and}\angle 6;\angle 4\text{and}\angle 8;\angle 3\text{and}\angle 7$

(ii)          The pairs of alternate interior angles are:

$\angle 3\text{and}\angle 5;\angle 2\text{and}\angle 8$

(iii)      The pair of interior angles on the same side of the transversal:

$\angle 3\text{and}\angle 8;\angle 2\text{and}\angle 5$

(iv)       The vertically opposite angles are:

$\angle 1\text{and}\angle 3;\angle 2\text{and}\angle 4;\angle 6\text{and}\angle 8;\angle 5\text{and}\angle 7$

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

Given, $p\parallel q.$ and cut by a transversal line.

$\because 125°+e=180°$                          [Linear pair]

$\therefore e=180°-125°=55°$         ……….(i)

Now $e=f=55°$                              [Vertically opposite angles]

Also $a=f=55°$                              [Alternate interior angles]

$a+b=180°$                             [Linear pair]

$\Rightarrow 55°+b=180°$                          [From equation (i)]

$\Rightarrow b=180°-55°=125°$

Now $a=c=55°\text{ and }b=d=125°$   [Vertically opposite angles]

Thus, $a=55°,b=125°,c=55°,d=125°,e=55°\text{ and }f=55°.$

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

(i)              

(ii)           

(iii)       

(i)             Given, $l\parallel m$ and $t$ is transversal line.

$\therefore$     Interior vertically opposite angle between lines $l$ and $t=110°.$

$\therefore 110°+x=180°$  [Supplementary angles]

$\Rightarrow x=180°-110°=70°$

(ii)          Given, $l\parallel m$ and $t$ is transversal line.

$x+2x=180$                    [Interior angles on same side of transversal]

$\Rightarrow 3x=180°$

$\Rightarrow x=\frac{180°}{3}=60°$

(iii)      Given, $l\parallel m$ and $a\parallel b$.

$x=100°$                           [Corresponding angles]

In the given figure, the arms of two angles are parallel. If $\angle \text{ABC}=70°,$ then find

(i)             $\angle \text{DGC}$

(ii)          $\angle \text{DEF}$

(i)             Given, $\text{AB}\parallel \text{DE}$ and BC is a transversal line and $\angle \text{ABC}=70°$

$\because \angle \text{ABC}=\angle \text{DGC}$        [Corresponding angles]

$\therefore \angle \text{DGC}=70°$              ……….(i)

(ii)          Given, $\text{BC}\parallel \text{EF}$ and DE is a transversal line and $\angle \text{DGC}=70°$

$\because \angle \text{DGC}=\angle \text{DEF}$        [Corresponding angles]

$\therefore \angle \text{DEF}=70°$              [From equation (i)]

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

(i)              

(ii)           

(iii)       

(iv)        

(i)             $126°+44°=170°$

 l is not parallel to m because sum of interior angles on same side of transversal is not $180°$.

(ii)          $75°+75°=150°$

 because sum of angles on same side of transversal does not obey the property of parallel lines.

(iii)      $57°+123°=180°$

$l\parallel m$ due to supplementary angles property of parallel lines.

(iv)       $98°+72°=170°$

$l$ is not parallel to $m$ because sum of angles does not obey the property of parallel lines.