5 Marks Questions

Question 15 Marks

The drawings below $Fig.$ show angles formed by the goalposts at different positions of a football player. The greater the angle, the better chance the player has of scoring a goal. For example, the player has a better chance of scoring a goal from Position $A$ than from Position $B.$
$i.$

$ii.$

$iii.$

In Parts $(a)$ and $(b)$ given below it may help to trace the diagrams and draw and measure angles.
$a.$ Seven football players are practicing their kicks. They are lined up in a straight line in front of the goalpost $[Fig. (ii)].$ Which player has the best $($thegreatest$)$ kicking angle?
$b.$ Now the players are lined up as shown in $Fig. (iii)$. Which player has the best kicking angle?
$c.$ Estimate atleast two situations such that the angles formed by different positions of two players are complement to each other.

Answer

$a.$

$b.$

$c.$ From the above figure, we can say that player $4$ has the best kicking angle, as it is greatest.
$d.$ Since, the angles are complementary. Hence, two situations are $45^\circ , 45^\circ $ and $30^\circ , 60^\circ .$

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Question 25 Marks

Iron rods $a, b, c, d, e$ and $f$ are making a design in a bridge as shown in $Fig.$ in which $a \| b, c \| d, e \| f.$ Find the marked angles between.
$i. b$ and $c$
$ii. d$ and $e$
$iii. d$ and $f$
$iv. c$ and $f$

Answer

Since, $l, m$ are two parallel lince and $PQ, RS$ and $TU$ are transversal.
Then, $\angle4=\angle\text{QPS} [$Alternate interior angles$] \Rightarrow\angle4=75^\circ$
$[\because\angle\text{QPS}=75^\circ]$ Again, $\angle1 = \angle\text{QOR} [$Vertically opposite angles$]$ $\Rightarrow\angle1=30^\circ$
$[\because\angle\text{QOR}=30^\circ]$ Also, $PQ$ and $TU$ are parallel and $m$ and $l$ are transversal.
Therefore, $\angle2+\angle\text{QPT}=180^\circ [$Consecutive interior angles$]$
$\Rightarrow\angle2=180^\circ-75^\circ$
$[\because\angle\text{QPT}=75^\circ,\text{given}]$
$\Rightarrow\angle2=105^\circ$ Also, $\angle2+\angle3=180^\circ$
$\Rightarrow105^\circ+\angle3=180^\circ$
$\Rightarrow\angle3=180^\circ-105^\circ$
$\Rightarrow\angle3=75^\circ$ Hence,
$i. 30^\circ$
$ii. 105^\circ$
$iii. 75^\circ$
$iv. 75^\circ$

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Question 35 Marks

In Fig. $PQ, RS$ and $UT$ are parallel lines.
$i.$ If $c = 570$ and $\text{a}=\frac{\text{c}}{3}$, find the value of $d$.
$ii.$ If $c = 750$ and $\text{a} = \frac{2}{5}\text{c}$, find $b.$

Answer

$i.$ Since, $PQ \| UT$ and $PT$ is transversal,
Therefore, $\angle\text{QPT}=\angle\text{UTP} [$Alternate Interior angles$]$
$\Rightarrow\text{a}+\text{b}=\text{c}$ $\big[\because\text{a}=\frac{\text{c}}{3},\text{given}\big]$
$\Rightarrow\frac{\text{c}}{3}+\text{b}=\text{c}$
$\Rightarrow\text{b}=\text{c}-\frac{\text{c}}{3}$
$\Rightarrow\text{b}=\frac{3\text{c-}\text{c}}{3}$
$\Rightarrow\text{b}=\frac{2\text{c}}{3}=\frac{2}{3}\times57^\circ [\because\text{c}=57^\circ,\text{given}]$
$ \therefore \text{b}=38^\circ$
Again, $PQ \| RS$ and $PR$ is transversal.
Therefore, $\angle\text{QPR}+\angle\text{PRS}=180^\circ [$Consecutive interior angles$]$
$\Rightarrow\text{b}+\text{d}=180^\circ$
$\Rightarrow\text{d}=180^\circ-\text{b}$
$\Rightarrow\text{d}=180^\circ-38^\circ [\because\text{b}=38^\circ]$
$\Rightarrow\text{d}=142^\circ$
$ii.$ Since, $PQ \| UT$ and $PT$ is transversal.
Therefore, $\angle\text{QPT}=\angle\text{UTP} [$Alternate interior angles$]$
$\Rightarrow\text{a}+\text{b}=\text{c}$
$\Rightarrow\ \text{b}=\text{c}-\text{a}$
$\Rightarrow\ \text{b}=\text{c}-\frac{2}{5}\text{c} \big[\because\text{a}=\frac{\text{2}}{5}\text{c},\text{given}\big]$
$\Rightarrow\text{b}=\frac{5\text{c-}2\text{c}}{5}$
$\Rightarrow\text{b}=\frac{3\text{c}}{5}$
$\Rightarrow\text{b}=\frac{3\times75^\circ}{5}$ $[\because\text{c}=75^\circ,\text{given}]$
$\Rightarrow\text{b}=45^\circ$

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