Case study (4 Marks)

Question 14 Marks

The figure below consists of a square and an equilateral triangle connected together with a
common side.

In the design, $DF$ and $IG$ are two iron rods perpendicular to $BC.$ The measure of $\angle BAC = 120^\circ .$
$8.$ Which type of triangle is $ABC?$ Why$?$
$9.$ The central triangle AFG is equilateral. What is the measure of $\angle FDA?$
$A. 30^\circ $
$B. 60^\circ $
$C. 90^\circ $
$D. 120^\circ $
$10.$ The length of $IG$ is half of the length of $GC$. Write a proof for the statement.

Answer

$8.$ Writes either isosceles or obtuse or both. Reasoning involves symmetry or measure of angle or both.
● Isosceles, as the design is symmetrical.
● Obtuse, as one of the angle is greater than $90^\circ .$
$9. D. 120^\circ $
$10.$ Valid mathematical proof involving properties of triangles.
● $IG$ is perpendicular to $BC,$ thus triangle $IGC$ is a right-angled triangle.
Measure of $\angle ICG = 30^\circ .$
Hence, $\angle CIG = 60^\circ .$
The sides of the triangle $IGC$ are in the ratio $2:1.$

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MCQ 24 Marks

In the igure below, $BC = AC.$

What is the measure of $∠BAD?$

A
$ 30^\circ$
B
$ 60^\circ$
C
$ 75^\circ$
✓
$ 90^\circ$

Answer

Correct option: D.

$ 90^\circ$

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MCQ 34 Marks

The igure below consists of a square and an equilateral triangle connected together with a
common side.

What is the measure of $‘x’?$

A
$15$
B
$ 30$
✓
$ 45$
D
$ 60$

Answer

Correct option: C.

$ 45$

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MCQ 44 Marks

The figure below shows an equilateral triangle bounded by two straight lines.

What is the sum of the four marked angles$?$

A
$ 180^\circ$
✓
$ 240^\circ$
C
$ 270^\circ$
D
$ 360^\circ$

Answer

Correct option: B.

$ 240^\circ$

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MCQ 54 Marks

Two equilateral triangles on a straight line are shown below.What is the measure of $ 'x\ ’?$

A
$ 30$
✓
$ 40$
C
$ 60$
D
$ 65$

Answer

Correct option: B.

$ 40$

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Question 64 Marks

A parking lot for a city mall is shown below. The painted lines that separate the parking spaces are parallel.

$3.$ Parking space number $378$ is inclined at $60^\circ $ to the horizon line. At what angle is parking space $380$ inclined to the horizontal line$?$ Why$?$

Answer

$ 60^\circ ,$ reasoning includes properties of parallel lines.
$ 60^\circ ,$ as the lines are parallel, thus corresponding angles will be equal.

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Question 74 Marks

The game of billiards is played with balls placed on a rectangular table. One ball is struck with the
end of a stick, called a cue. The ball bounces into other balls and relects off the sides of the table. In a real game, the ball may spin, but for mathematical purposes, it is considered that the ball travels in a straight line with the same relection and incidence angles.

On a billiard table $ABCD,$ the ball placed at $O$ is struck with the cue.
$1.$ What is the value of $\angle a + \angle d?$
$2.$ Why is the line $OM$ parallel to $PN?$

Answer

$1. 90$
$90^\circ $
$2.$ Mathematically valid proof
● Let angles on line $AMB$ be $a, x$ and $b$ and angles on line $BNC$ be $c, y$ and $d.$
$x = 180 – (a + b) …..1$
$y = 180 – (c + d) …...2$
Adding $1$ and $2,$
$x + y = 360 – (a + b + c + d)$
$= 360 – (2a + 2c)$
$= 360 – 2 \times 90 = 180$
Thus, lines $OM$ and $NP$ are parallel.

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Question 84 Marks

Read the Source Text given below and answer any four questions:
Maths teacher draws a straight line $AB$ shown on the blackboard as per the following figure.

$i.$ Now he told Raju to draw another line $CD$ as in the figure.
$ii.$ The teacher told Ajay to mark $\angle\text{AOD}$ as $2z.$
$iii.$ Suraj was told to mark $\angle\text{AOC}$ as 4y.
$iv.$ Clive Made and angle $\angle\text{COE}=60^\circ.$
$v.$ Peter marked $\angle\text{BOE}$ and $\angle\text{BOD}$ as $y$ and $x$ respectively.
Now answer the following questions:
$i.$ What is the value of $ x?$
$a. 48^\circ $
$b. 96^\circ $
$c. 100^\circ $
$d. 120^\circ $
$ii.$ What is the value of $y?$
$a. 48^\circ $
$b. 96^\circ $
$c. 100^\circ $
$d. 24^\circ $
$iii.$ What is the value of $z?$
$a. 48^\circ $
$b. 96^\circ $
$c. 42^\circ $
$d. 120^\circ $
$iv.$ What should be the value of $x + 2z?$
$a. 148^\circ $
$b. 360^\circ $
$c. 180^\circ $
$d. 120^\circ$
$v.$ What is the relation between $y$ and $z?$
$a. 2y + z = 90^\circ $
$b. 2y + z = 180^\circ $
$c. 4y + 2z = 120^\circ $
$d. y = 2z$

Answer

$(i)$	$(b)$	$96^\circ $
$(ii)$	$(d)$	$24^\circ $
$(iii)$	$(c)$	$42^\circ $
$(iv)$	$(c)$	$180^\circ $
$(v)$	$(a)$	$2y + z = 90^\circ $

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Question 94 Marks

Read the Source/ Text given below and answer these questions: Reeta and Rohan were playing a game on parallel lines and the angles formed with the transverse line$($ie alternate angles, corresponding angle and interior angles$)$. First Reeta drew a straight line $AB,$ then Rohan drew another straight line $CD \| AB$. Further, a transverse line $PQ $ was drawn which intersects lines $AB$ and $CD$ at points $X$ and $Y$ respectively.

Now they did toss with a coin and Rohan won the toss. Following were the rules of the game:
$i.$ Toss winner will ask a question and others will answer.
$ii.$ If the answer is correct then person answering will ask question else questioner will ask next question.
$iii.$ Who wins the last question he/ she will be the winner.
$iv.$ Total of $5$ questions will be asked.
Based on the above paragraph please answer these questions:
$i.$ Which is the alternate angle to $\angle\text{6}?$
$a. \angle1$
$b. \angle2$
$c. \angle3$
$d. \angle4$
$ii.$ Which is the corresponding angle to $\angle1?$
$a. \angle4$
$b. \angle5$
$c. \angle6$
$d. \angle7$
$iii.$ If $\angle4=120^\circ$ then what is measure of $\angle6?$
$a. 80^\circ $
$b. 120^\circ $
$c. 100^\circ $
$d. 60^\circ $
$iv.$ What is the sum of $\angle3$ and $\angle5?$
$a. 180^\circ $
$b. 160^\circ $
$c. 100^\circ $
$d. 60^\circ $
$v. \angle5$ is equal to which of the following pair of angles$?$
$a. \angle7\text{ and }\angle\text{8}$
$b. \angle6\text{ and }\angle\text{7}$
$c. \angle4\text{ and }\angle\text{8}$
$d. \angle7\text{ and }\angle\text{8}$

Answer

$(i)$	$(c)$	$\angle3$
$(ii)$	$(b)$	$\angle5$
$(iii)$	$(d)$	$60^\circ $
$(iv)$	$(a)$	$180^\circ $
$(v)$	$(c)$	$\angle4$ and $\angle\text{8}$

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Question 104 Marks

Read the Source/ Text given below and answer any four questions: There were two parallel roads $AM$ and $XY$ in New Delhi. Due to increasing pollution, $\text{MCD}$ planned to get planted trees on these roads. On the road $AM$, plants of Ashoka were planted by one company. While on the road $XY$ mango trees were planted by another company.

Between these roads three streets $St_1, St_2$ and $St_3$ were situated. During the survey, $\angle\text{BPQ}$ was measured to be $70^\circ$ and other angles $p, q, r, s$ and $t$ were also measured. Now answer the following questions:
$i.$ What is the measure of $\angle\text{p}?$
$a. 60^\circ$
$b. 70^\circ$
$c. 160^\circ$
$d. 100^\circ$
$ii.$ What is the measure of $\angle\text{q}?$
$a.60^\circ$
$b.70^\circ$
$c.160^\circ$
$d.110^\circ$
$iii.$ What is the value of $\frac{\text{p}+\text{q}+\text{t}}{5}?$
$a.50^\circ$
$b.70^\circ$
$c.160^\circ$
$d.100^\circ$
$iv.$ What is the measure of $\angle\text{EsY}?$
$a.60^\circ$
$b.140^\circ$
$c.70^\circ$
$d.110^\circ$
$v.$ What is value of ${4p - (q + r) - (r - s)}?$
$a.50^\circ$
$b.100^\circ$
$c.160^\circ$
$d.180^\circ$

Answer

$(i)$	$(b)$	$70^\circ$
$(ii)$	$(d)$	$110^\circ$
$(iii)$	$(a)$	$50^\circ$
$(iv)$	$(c)$	$70^\circ$
$(v)$	$(b)$	$100^\circ$

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Question 114 Marks

Read the Source/ Text given below and answer these questions:
Once the Maths teacher of class $IX\ D$ told students that today we will prove that the sum of all three angles is $180^\circ .$ As shown in the figure, he told to draw any $\triangle ABC$ in the notebook.
Further side $BC$ was extended to $D.$

Now the teacher said to draw $CE \| BA.$ Further angles were named $1$ to $5$ as shown in the figure.
Now answer the following questions:
$i. BA \| CE$ and $AC$ is the transverse line, So $\angle1$ is equal to which angle$?$
$a. \angle2$
$b. \angle3$
$c. \angle4$
$d. \angle5$
$ii. \angle2$ is equal to which angle$?$
$a.\angle2$
$b.\angle3$
$c.\angle4$
$d. \angle5$
$iii.$ What is value of $\angle3+\angle4+\angle5?$
$a.180^\circ $
$b.120^\circ $
$c. 200^\circ $
$d. 360^\circ $
$iv.$ What is value of $\angle\text{ECD}=\angle4+\angle5?$
$a.\angle3+\angle5$
$b.\angle1+\angle2$
$c. \angle2+\angle3$
$d. \angle3+\angle4$
$v.$ What is value of $\angle1+\angle2+\angle3?$
$a.\angle3+\angle4+\angle5=180^\circ$
$b.360^\circ$
$c. \angle3+\angle4=100^\circ$
$d. 280^\circ$

Answer

$(i)$	$(c)$	$\angle4$
$(ii)$	$(d)$	$\angle5$
$(iii)$	$(a)$	$180^\circ $
$(iv)$	$(b)$	$\angle1+\angle2$
$(v)$	$(a)$	$\angle3+\angle4+\angle5=180^\circ$

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Question 124 Marks

Read the Source/ Text given below and answer these questions:

Ashok is studying in $9^{th}$ class in Govt School, Chhatarpur. Once he was at his home and was doing his geometry homework. He was trying to measure three angles of a triangle using the Dee, but his dee was old and his Dee's numbers were erased and the lines on the dee were visible. Let us help Ashok to find the angles of the triangle. He found that the second angle of the triangle was three times as large as the first. The measure of the third angle is double of the first angle. Now answer the following questions:
$i.$ What was the value of the first angle$?$
$a. 30^\circ $
$b. 45^\circ $
$c. 60^\circ $
$d. 90^\circ $
$ii.$ What was the value of the third angle$?$
$a.30^\circ $
$b.45^\circ $
$c.60^\circ $
$d. 90^\circ $
$iii.$ What was the value of the second angle$?$
$a.30^\circ $
$45^\circ $
$c.60^\circ $
$d. 90^\circ $
$iv.$ What was the value of $\angle4$ as shown the figure$?$
$a.120^\circ $
$b.45^\circ $
$c.60^\circ $
$d. 90^\circ $
$v.$ What was the sum of all three angles measured by Ashok using Dee$?$
$a.270^\circ $
$b.180^\circ $
$c.100^\circ $
$d. 90^\circ $

Answer

$(i)$	$(a)$	$30^\circ $
$(ii)$	$(c)$	$60^\circ$
$(iii)$	$(d)$	$90^\circ $
$(iv)$	$(a)$	$120^\circ $
$(v)$	$(b)$	$180^\circ $

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Maths Case study (4 Marks)

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