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QUESTION: 1

AB, CD and EF are three line segments intersecting at a common point G as shown in the given figure. Find the value of ∠CGB.

Solution:

QUESTION: 2

In the figure above, what is the value of angle ∠ABC?

I. BD and BE are the angle bisectors of ∠CBE and ∠ABD respectively.

II. ∠CBD = 45^{o}

Solution:

QUESTION: 3

In the given figure, BC and EF are line segments intersecting at F. FD and AF are the angle bisectors of ∠EFC and ∠EFB respectively. Find the value ∠AFE + ∠DFE.

Solution:

QUESTION: 4

In the given diagram, line segment EF and BC are parallel to each other. Find the value of (x+y).

Solution:

QUESTION: 5

Three parallel lines l_{1}, l_{2} and l_{3 }are intersected by two parallel transversals m_{1} and m_{2}. What is the measure of angle y?

Solution:

__Step 1: Question statement and Inferences__

We are given that three parallel lines l_{1}, l_{2} and l_{3} are intersected by two parallel transversals m_{1} and m_{2}. Also, two angles 3x-100 and 3x+10 are formed as shown in the figure.

We have to find the value of angle y.

We know that a line which intersects two or more parallel lines is called a **Transversal** line.

Let us name the intersecting points and understand angles associated them.

To find the value of angle y, we must build a relationship between ∠B,?E,and ∠F

with the use of properties discussed above.

__Step 2: Finding required values__

Two parallel lines l1,&l2

are intersected by transversal m_{1,} so

∠F=y=∠E=3x+10

(Since F and E are corresponding angles.)

So, if we get the value of x we can get the value of y.

To get the value of x, we must make a relationship between ∠Band?E.

We find that

∠B=∠D(Correspondingangles)

∠D=∠FED(Correspondingangles) and ∠FED=180−∠E (Pair of linear angles) Thus, we can say: ∠D = ∠FED = 180^{o}−∠E

∠B=180o−∠E

__Step 3: Calculating the final answer__

Now, ∠B=180^{o}−∠E

3x−100^{o}=180^{o}−(3x+10)^{o }

3x−100^{o}=180^{o}−3x−10^{o} 6x=270^{o}

x=45^{o }

or, y^{o}=∠E=3×45^{o}+10^{o}=145^{o}

**Answer: Option (E)**

QUESTION: 6

If the lines l_{1} and l_{2} are parallel to each other, and the transversal line m intersects it such that ∠PRS= 3a and ∠RYX= 6b, then what is the value of “a + 2b” ?

Solution:

__Step 1: Question statement and Inferences__

We are given that two parallel lines l_{1} and l_{2} are intersected by a transversal line m. Also, two angles 3a and 6b are formed as shown in the figure.

We have to find the value of "a + 2b".

We know that,

∠RYX=∠PRQ (Corresponding angles) So, ∠PRQ = 6b……………(1) Also,

∠PRS = ∠QRY=3a (Vertically opposite angles)

And ,

∠PRQ = ∠SRY= 6b (Vertically opposite angles)

__Step 2: Finding required values__

Now, we know that the sum of all the angles formed around a point is 360^{o}. So,

∠PRS + ∠SRY + ∠QRY + ∠PRQ=360^{∘}

Thus,

3a + 6b + 3a + 6b =360^{∘}

6a+12b=360^{∘}

a+2b=60^{∘}

__Step 3: Calculating the final answer__

So, the sum a + 2b = 60

**Answer: Option (B)**

QUESTION: 7

In the given figure, three angles X, Y, and Z are formed at the centre O of the circle. If the angles X, Y, and Z are in the ratio 1:3:5 respectively, then find the value of Y.

Solution:

__Step 1: Question statement and Inferences __

We are given a circle with centre O. Three lines originate from the point O and form three angles X, Y, and Z at the centre.

These angles are in the ratio 1:3:5 i.e.

⇒ X : Y : Z = 1 : 3 : 5

We have to find the value of Y.

__Step 2: Finding required values__

Let’s say:

X = p

This means,

Y = 3p

Z = 5p

Now, we know that the sum of all the angles formed around a point is 360^{o}. So,

X + Y + Z = 360^{o}

p + 3p + 5p = 360^{o}

p = 40^{o}

__Step 3: Calculating the final answer __

So, the measure of the angle Y is 3p = 3*40^{o} =120^{o}

**Answer: Option (D)**

QUESTION: 8

In the figure above, AOD is a straight line. Find the measure of ∠AOB

Solution:

__Step 1: Question statement and Inferences __

We are given a straight line AOD. ∠AOB, ∠BOC and ∠COD

, are the angles formed on the same side of the straight line. Also, the values of the angles are as follows:

∠AOB = 4x

∠BOC = 3x +5

∠COD = 5x -5

Now, we have to find the value of ∠AOB

. For that, first we need to find the value of x.

__Step 2: Finding required values__

Now, we know that the sum of all the angles formed on a straight line is equal to 180^{o}.

Thus,

∠AOB + ∠BOC+ ∠COD = 180^{0}

4x +3x+5+5x -5 = 180^{0}

12x =180^{0}

Thus , x =15^{o}

So,

∠AOB = 4 x 15^{o }= 60^{o}

__Step 3: Calculating the final answer __

So, the measure of the angle AOB is 60 degrees.

**Answer: Option (D)**

QUESTION: 9

In the given figure, if the lines *l _{1}* and

Solution:

__Step 1: Question statement and Inferences__

We are given that two parallel lines l_{1} and l_{2} are intersected by a transversal line *m*. Also, the angles formed are shown as 2x, 3x + 50, a, b, and y.

We have to find the value of angle y.

Now, to find the value of y, we need to establish the relation between x and y. Since the lines *l _{1}* and

2x = a (Corresponding angles)

Also, since

a = b (Vertically opposite angles)

Hence,

b = 2x

Also,

(3x + 50) + a = 180º

(3x + 50) + 2x = 180º

5x + 50 = 180º

5x = 130º

x = 26º ……………………… (1)

__Step 2: Finding required values__

Now, we know that the sum of all the angles formed around a point is 360^{o} . So,

a + b + (3x + 50) + y = 360º

Thus,

2x + 2x + y + y = 360º

7x + y + 50 = 360º

7x + y = 310º …… (2)

By putting the value of x from equation (1) to equation (2), we get:

7*26 + y = 310º

182 + y = 310º

y = 128º

__Step 3: Calculating the final answer__

So, the value of y = 128º

**Answer: Option (C)**

QUESTION: 10

In the given figure, what is the value of a + b, if both a and b are positive integers?

Solution:

__Step 1: Question statement and Inferences __

We are given that two lines XY and PQ intersect each other at point O. The measures of the four angles thus formed are given in terms of the positive integers a and b. These angles are:

∠POX=2a+20

∠XOQ=3b+20

∠QOY=4b–50

∠YOP=4a+10

We need to find the values of a and b to get the value of “a + b”.

Now, we know that the sum of all the angles formed around a point is 360. So,

∠POX+∠XOQ+∠QOY+∠YOP=360∘

By putting the values of these angles in the above equation, we get:

(2a + 20) + (3b + 20) + (4b – 50) + (4a + 10) = 360^{o}

6a + 7b = 360^{o} …………….. (1)

Also, we know that the vertically opposite angles formed between straight lines are equal, so:

∠POX=∠YOQ

2a + 20 = 4b – 50

2a = 4b – 70

a = 2b – 35 ……………… (2)

And,

∠POY=∠XOQ

4a + 10 = 3b + 20

4a = 3b + 10 …………………. (3)

__Step 2: Finding required values__

By putting the value of a from equation (2) in equation (1),

6a + 7b = 360^{o}

6(2b – 35) + 7b = 360^{o}

12b – 210 + 7b = 360^{o}

19b = 570^{o}

b = 30^{o}

Now, let’s plug the value of b in equation (2):

a = 2b – 35

a = 2(30^{o}) – 35

a = 25^{o}

Thus,

a + b = 30^{o }+ 25^{o}

a + b = 55^{o}

__Step 3: Calculating the final answer __

So, the sum of the values of a and b is 55^{o }.

**Answer: Option (B)**

QUESTION: 11

In the figure above, AD bisects angle BAE and AE bisects angle CAD. Also, the line XY and AC are parallel to each other. Find the value of ∠DAE

.

(1) ∠BAC = 60^{o}

(2) ∠XOB = 120

Solution:

__Steps 1 & 2: Understand Question and Draw Inferences__

We are given that in the figure line AD bisects and the line AE bisects angle . We have to find the value of the .

now, since AD bisects ∠BAE

⇒∠BAD=∠DAE.......(1)

and, AE bisects∠CAD⇒∠CAE=∠DAE.......(2) ∠BAC+∠BAD+∠DAE+∠CAE=360.......(3)(sincethesumofallanglesformedaroundapointis360o)

Now, from equation (1), (2), and (3)

∠DAE + ∠DAE + ∠DAE + ∠BAC=360^{∘}

⇒3∠DAE+∠BAC=360∘.......(4) Also, since AC and XY are parallel to each other, ∠BAC =∠BOY (Corresponding angles).......(5)

However,wedon′tknowthevalueof∠BOY.

So,tofindthevalueof∠DAE,weneedtofindthevalueofeither∠BACor∠BOY.

__Step 3: Analyse Statement 1__

Statement 1 says: ∠BAC= 60^{o}

This statement directly gives us the value of ∠BAC. By putting this value in equation (4), we get:

3∠DAE+∠BAC=360∘3∠DAE+60=360∘3∠DAE=300∘⇒∠DAE=100∘

Hence, statement (2) alone is sufficient to answer the question: What is the value of ∠BAE?

__Step 4: Analyse Statement 2__

Statement 2 says: ∠XOB= 120

Now, we know that the sum of all the angles formed around a point is 360. So,

∠AOX+∠XOB+∠BOY+∠YOA=360∘.......(6)Also,∠AOX=∠BOY∠XOB=∠YOA(Verticallyoppositeangles)Thus,equation(6)becomes:2∠XOB+2∠BOY=360∘2×120∘+2∠BOY=360∘2∠BOY=120∘∠BOY=60∘Fromequation(5),∠BAC=∠BOYThus,∠BAC=60∘Byputtingthisvalueinequation(4),weget:3∠DAE+∠BAC=360∘3∠DAE+60=360∘3∠DAE=300∘⇒∠DAE=100∘

Hence, statement (2) alone is sufficient to answer the question: What is the value of ∠BAE?

__Step 5: Analyse Both Statements Together (if needed)__

Since statement (1) and (2) alone are sufficient to answer the question, we don’t need to perform this step.

**Correct Choice : D**

QUESTION: 12

Is the segment CD a perpendicular bisector of the segment AB?

(1) Half of CD = CO

(2) AO = DO

Solution:

__Steps 1 & 2: Understand Question and Draw Inferences__

We are given that segments AB & CD are perpendicular to each other, and we have to find: Does egment CD bisect segment AB? (The perpendicular part is omitted as we already know from the square symbol used at the point O that the segments are perpendicular to each other.)

So, the question becomes, is AO = OB?

Since we are not given any other information, let’s move on to the analysis of the statements.

__Step 3: Analyse Statement 1__

Statement I says: Half of CD = CO

It means that point O is the mid-point of CD or we can say that segment AB bisects segment CD.

Did we get the answer to the question? Well, no!

Pay attention to the question, *Is segment CD, a perpendicular bi-sector of segment AB?* It is not answered. We seek for an answer whether CD bi-sects AB and not vice-versa. Had you hurriedly done this, you would have been trapped laid by the test-maker!

**Take Away**: Pay attention to the verbiage!

Hence, statement I is not sufficient to answer the question: is AO = OB?

__Step 4: Analyse Statement 2__

Statement II says: AO = DO

It does not help in any way. It simply builds a relationship between sub-parts of two different segments. We are interested in whether AO = OB. This question is still not answered.

Hence, statement II is not sufficient to answer the question: is AO = OB?

__Step 5: Analyse Both Statements Together (if needed)__

Since step 3 and step 4 were not able to provide the answer to the question, let’s analyse both the statements together:

Statement I:

Segment AB bisects segment CD.

Statement II:

AO = DO.

Even the combination of both the statements does not help us to find the answer to our question.

Hence, both statements combined together are not sufficient to answer the question.

**Answer: Option (E)**

QUESTION: 13

Refer to the above diagram. Evaluate w.

Statement 1: x=2v

Statement 2: y=2x

Solution:

Statement 1 alone gives insufficient information. It only gives a relationship between x and v, but no further clues about the measures of any angle are given.

Statement 2 alone gives insufficient information; w=y=2x, since the angles with those measures are vertical; since no measures are known, w cannot be calculated.

Now assume both statements are true. Again, from Statement 1, x=2v; from Statement 2, y=2x=2(2v)=4v. Again, from the diagram, w=y=4v. Three angles with measures x,v,w together form a straight angle, so

x+v+w=180

2v+v+4v=180

7v=180

Therefore, both statements together are sufficient to answer the question.

QUESTION: 14

Refer to the above diagram. Evaluate v.

Statement 1: x+y=150

Statement 2: w+v=130

Solution:

Assume Statement 1 alone. From the diagram, the three angles of measure x,y,v together form a straight angle, so

x+y+v=180

From Statement 1,

x+y=150,

so by the subtraction property of equality,

x+y+v−(x+y)=180−150

v=30

Assume Statement 2 alone. w+v=130, but there is no clue about the value of w or any other angle measure, so the value of v cannot be computed.

QUESTION: 15

Evaluate m∠1.

Statement 1: m∠4=88∘

Statement 2: m∠8=88∘

Solution:

Assume Statement 1 alone. ∠1 and ∠4 are vertical angles, so they must have the same measure; m∠1=m∠4=88∘.

Assume Statement 2 alone. ∠1 and ∠8 are alternating exterior angles, which are congruent if and only if l||m; however, we do not know whether l||m, so no conclusions can be made about m∠1.

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