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All questions of Lines, Angles and Triangles for SAT Exam

In a triangle ABC, the internal bisector of the angle A meets BC at D. If AB = 4, AC = 3 and ∠  A = 600, then length of AD is :
  • a)
    2√3
  • b)
    (12√3) / 7
  • c)
    (15√3) / 8
  • d)
    (6√3) / 7
  • e)
    None of these
Correct answer is option 'B'. Can you explain this answer?

Lavanya Menon answered

Let BC = x and Ad = y, then as per bisector theorem,
BD /DC = AB /AC = 4 /3
Hence, BD = 4x/7 and DC = 3x/7
Now, in Δ ABD using cosine rule,
cos 300 = [{42 +y2 - (16x2/49)} /2*3*y]
Or, 4√3y =[{16 +y2 - (16x2/49)}] --------- (i)
Similarly in Δ ADC,
cos 300 = [{32 +y2 - (9x2/49)} /2*3*y]
Or, 3√3y =[{9 +y2 - (9x2/49)}] --------- (ii)
From equation (i) and (ii), we get
y = 12√3 /7.
Clear all your doubts with EduRev

In triangle PQR length of the side QR is less than twice the length of the side PQ by 2 cm. Length of the side PR exceeds the length of the side PQ by 10 cm. The perimeter is 40 cm. The length of the smallest side of the triangle PQR is :
  • a)
    6 cm
  • b)
    8 cm
  • c)
    7 cm
  • d)
    10 cm
  • e)
    9 cm
Correct answer is option 'B'. Can you explain this answer?

Nandita Yadav answered
In Δ PQR,
QR +2 = 2PQ 
QR = 2PQ - 2 ------- (2)
PR = PQ + 10 ----- (2)
Perimeter = 40 m
PQ + QR + Rp = 40
Putting the value of PQ and QR from equation (1) and (2),
Pq + 2PQ - 2 + PQ +10 = 40
4PQ = 32 
PQ = 8 cm which is the smallest side of the triangle.

The complement angle of 80° is
  • a)
    18 / π radian
  • b)
    5π / 9 radian
  • c)
    π / 18 radian
  • d)
    9 / 5π radian
Correct answer is option 'C'. Can you explain this answer?

Rutuja Banerjee answered
Complement Angle Calculation:
The complement angle of a given angle is the angle that, when added to the given angle, equals 90 degrees. In this case, we are given the angle of 80 degrees and we need to find its complement angle.

Calculation:
- The complement angle of 80 degrees can be calculated as follows:
- Complement angle = 90 degrees - 80 degrees
- Complement angle = 10 degrees

Conversion to Radians:
To convert the complement angle from degrees to radians, we use the conversion factor: 1 degree = π / 180 radians.
- Complement angle in radians = 10 degrees x (π / 180)
- Complement angle in radians = π / 18 radians
Therefore, the complement angle of 80 degrees is π / 18 radians, which corresponds to option 'c'.

In the given figure AB || CD, ∠A = 128°, ∠E = 144°. Then, ∠FCD is equal to :
  • a)
    72°
  • b)
    64°
  • c)
    136°
  • d)
    92°
Correct answer is option 'D'. Can you explain this answer?

Pioneer Academy answered
As per the given figure,
Through E draw EE’ || AB || CD.
Then, ∠AEE’ = 180° ‒ ∠BAE = (180° ‒ 128°) = 52°.
(Interior angles on the same side of the transversal are supplementary.)
Now, ∠E’EC = (144° ‒ 52°) = 92°.
∠FCD = ∠E’EC = 92° (Corr. ∠s).
Hence, option D is correct.

If one arm of an angle is respectively parallel to the arm of another angle formed with a common line between them, then the two angles are
  • a)
    Neither equal nor supplementary
  • b)
    not equal but supplementary
  • c)
    equal but not supplementary
  • d)
    Either equal or supplementary
Correct answer is option 'B'. Can you explain this answer?

Rahul Kapoor answered
If one arm of an angle is respectively parallel to the arm of another angle formed with a common line between them, then the two angles are not equal but supplementary.

If l1 || l2 ⇒ ∠1 + ∠2 = 180°
(Supplementary)
Hence, option B is correct.

ABC and BDC are two triangles as shown in the given figure. If BDC is an isosceles triangle in which BD = DC. Find the value of ∠BDC.
  • a)
    25o
  • b)
    50o
  • c)
    65o
  • d)
    130o
  • e)
    140o
Correct answer is option 'D'. Can you explain this answer?

Aditya Joshi answered
Since BD = CD,AngleDBC = AngleDCB (Let them be x)
Now three sides of triangles sums up to 180.AngleABC + AngleCBA + AngleCAB = 18040 +40+50+x+x = 180solving this, x = 25.
in triangle DBC,25+25+AngleDBC = 180angleDBC = 130

In the figure given below, ABC is a triangle. BC is parallel to AE. If BC = AC, then what is the value of ∠CAE?
  • a)
    20°
  • b)
    30°
  • c)
    40°
  • d)
    50°
Correct answer is option 'D'. Can you explain this answer?

Pioneer Academy answered
Given that, BC || AE
∠CBA + ∠EAB = 180°
⇒ ∠EAB = 180° – 65° = 115°
∵   BC = AC
Hence, ΔABC is an isosceles triangle. 

⇒ ∠CBA = ∠CAB = 65°
Now,  ∠EAB = ∠EAC + ∠CAB
⇒ 115° = x + 65° ⇒ x = 50°.
Hence, option D is correct.

The point of intersection of the altitudes of a triangle is called its:
  • a)
    Incentre
  • b)
    Excentre
  • c)
    Orthocentre
  • d)
    Centroid
  • e)
    none of these
Correct answer is option 'C'. Can you explain this answer?

Alok Verma answered
The three (possibly extended) altitudes intersect in a single point, called the orthocenter of the triangle, usually denoted by H. The orthocenter lies inside the triangle if and only if the triangle is acute (i.e. does not have an angle greater than or equal to a right angle).

The supplement of 154° 30’ is :
  • a)
    25° 30’
  • b)
    44° 45’
  • c)
    158° 45’
  • d)
    168° 30’
Correct answer is option 'A'. Can you explain this answer?

Pioneer Academy answered
Supplement of 154° 30’ is 180° ‒ (154° 30’)

= (179° 30’) ‒ (154° 30’) {since 1° = 60’}

= 25° 30’.

The straight lines AD and BC intersect one another at the point O. If ∠AOB + ∠BOD + ∠DOC = 274°, then ∠AOC is :
  • a)
    86°
  • b)
    90°
  • c)
    94°
  • d)
    137°
Correct answer is option 'A'. Can you explain this answer?

Rahul Kapoor answered
As we know that the sum of all the angles around a point is 360°.

(∠AOB + ∠BOD + ∠DOC) + ∠AOC = 360°
∴ 274° + ∠AOC = 360° or ∠AOC = 86°.
Hence, option A is correct.

In the trapezium PQRS, QR || PS, ∠Q = 90°, PQ = QR and ∠PRS = 20°. If ∠TSR = θ, then the value of θ is:
  • a)
    75°
  • b)
    55°
  • c)
    65°
  • d)
    45°
Correct answer is option 'C'. Can you explain this answer?

BT Educators answered
In the given figure, 
PQ = QR and ∠PQR = 90° ⇒ ∠QPR = ∠QRP = 45°.
∴ ∠QRS = (45° + 20°) = 65°.
∴ θ = ∠QRS = 65° (alt. ∠s)
Hence, option C is correct.

Two right angled triangles are congruent if :
I.The hypotenuse of one triangle is equal to the hypotenuse of the other.
II.a side for one triangle is equal to the corresponding side of the other.
III.Sides of the triangles are equal.
IV. An angle of the triangle are equal.
Of these statements, the correct ones are combination of:
  • a)
    I and II
  • b)
    II and III
  • c)
    I and III
  • d)
    IV only
  • e)
    I and IV
Correct answer is option 'A'. Can you explain this answer?

Hridoy Desai answered
Explanation:

To determine which statements are correct for two right-angled triangles to be congruent, let's analyze each statement one by one:

I. The hypotenuse of one triangle is equal to the hypotenuse of the other.
This statement is correct. The hypotenuse is the side opposite the right angle in a right-angled triangle. If the hypotenuse of one triangle is equal to the hypotenuse of another triangle, then the triangles are congruent by the Hypotenuse-Leg Congruence Theorem.

II. A side for one triangle is equal to the corresponding side of the other.
This statement is correct. If a side of one triangle is equal to the corresponding side of another triangle, then the triangles are congruent by the Side-Side-Side (SSS) Congruence Theorem.

III. Sides of the triangles are equal.
This statement is incorrect. While it is true that if all three sides of one triangle are equal to the corresponding sides of another triangle, then the triangles are congruent, this statement does not specify which sides are equal. It is possible for two triangles to have equal sides but still not be congruent because the angles may be different.

IV. An angle of the triangle is equal.
This statement is incorrect. While it is true that if all three angles of one triangle are equal to the corresponding angles of another triangle, then the triangles are congruent, this statement only mentions that "an angle" is equal. It does not specify which angle is equal, and therefore, it cannot be used to determine congruence.

Conclusion:
The correct statements for two right-angled triangles to be congruent are I. The hypotenuse of one triangle is equal to the hypotenuse of the other and II. A side for one triangle is equal to the corresponding side of the other. Therefore, the correct answer is option 'A': I and II.

In ΔLMN, LM = |x-7|, MN = |x-4| and NL = x + 1, where x is a number whose value is not known. Is ΔLMN an acute triangle (a triangle each of whose angles measures less than 90∘)?
(1) NM = 10
(2) LM = 7
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
  • d)
    EACH statement ALONE is sufficient to answer the question asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'D'. Can you explain this answer?

Shivam Ghoshal answered
Statement 1: NM = 10
Statement 2: LM = 7

To determine if LMN is an acute triangle, we need to compare the lengths of its sides.

Analysis of Statement 1:
NM = 10

Since we know the length of NM, we can find the difference between LM and NL by subtracting NM from MN.

MN - NM = |x-4| - 10 = |x-4|-10

Since we don't have any information about LM or NL, we cannot determine if LMN is an acute triangle based on this statement alone.

Analysis of Statement 2:
LM = 7

Since we know the length of LM, we can find the difference between LM and MN by subtracting LM from MN.

MN - LM = |x-4| - 7 = |x-4|-7

Again, since we don't have any information about NL or the value of x, we cannot determine if LMN is an acute triangle based on this statement alone.

Analysis of Statements 1 and 2 together:
Combining the information from both statements, we have:

NM = 10
LM = 7

Using these values, we can find the difference between LM and NL as well as the difference between LM and MN.

MN - NM = |x-4| - 10 = |x-4|-10

MN - LM = |x-4| - 7 = |x-4|-7

However, even with this additional information, we still cannot determine if LMN is an acute triangle because we do not know the value of x or have any constraints on its range.

Therefore, neither statement alone nor both statements together are sufficient to answer the question.

Hence, the correct answer is option D.

Triangle ABC is an isosceles triangle with AB = AC and AD is the perpendicular dropped from vertex A on the side BC. What is the perimeter of triangle ABC?
(1) ∠BAD = 2∠ACD
(2) The perimeter of triangle ADB is 15 + 5√3
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
  • d)
    EACH statement ALONE is sufficient to answer the question asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'C'. Can you explain this answer?

Arka Basu answered
Steps 1 & 2: Understand Question and Draw Inferences
Given:
  • Isosceles ΔABC
    • AB = AC
    • So, perpendicular AD will bisect side BC.
    • If we assume ∠ ABC to be xo, the different ∠s in the figure can be represented as below:
 
To find: Perimeter of ΔABC
  • =AB + BC + CA
  • = 2AB + BC
So, in order to find the perimeter, we need to know AC and BC
Knowing the ∠s may help us (because we may then employ trigonometric ratios)
 
Step 3: Analyze Statement 1 independently
 
(1) ∠BAD = 2∠ACD
90°- x° = 2x° 
⇒ 3x° = 90°  
⇒ x° = 30°
But we don’t know the magnitude of any side of the triangle. So, we cannot yet use trigonometric ratios to find the sides of the triangle.
Not sufficient.
Step 4: Analyze Statement 2 independently
(2) The perimeter of triangle ADB is 15 + 5√3
The value of 2AB + BC depends on the value of AD.
Since we do not know the value of AD, we cannot find the required value
Not sufficient.
  • From Statement 2 
 
  • Using Equation II, a unique value of AD will be obtained
  • Using Equation I, the value of 2AB + BC will be obtained
Sufficient.
Answer: Option C

If yo = 45o, is xo = 90o?
I.  AB is parallel to YZ
II.  AB = AY
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficien
  • c)
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. 
  • d)
    EACH statement ALONE is sufficient. 
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient.
Correct answer is option 'D'. Can you explain this answer?

Rhea Gupta answered
Steps 1 & 2: Understand Question and Draw Inferences
In ΔABY,
xo + yo + Angle ABY = 180o
  • xo + 45o + Angle ABY = 180o
  • xo + Angle ABY = 135o
We need to find if x is a right angle or not, i.e. if x = 90o.
For this we need the value of Angle ABY.
Step 3: Analyze Statement 1
Given that AB is parallel to YZ.
  • xo + 2yo = 180 (Since both of them are consecutive interior angles).
Given that y = 45o
  • x = 90o
Therefore, statement 1 is sufficient to arrive at a unique answer.
Step 4: Analyze Statement 2
Given AB = AY.
Therefore, Angle ABY = yo = 45o (given)
Therefore, xo + 45o = 135o
  • x = 90o
 Therefore, statement 2 is sufficient to arrive at a unique answer.
Step 5: Analyze Both Statements Together (if needed)
Since we arrived at a unique answer in at least one of step 3 and step 4, this step is not necessary.
Correct Answer: D

Consider the triangle shown in the figure where BC = 12 cm, Db = 9 cm, CD = 6 cm and What is the ratio of the perimeter of the triangle ADC to that of the triangle BDC ?
  • a)
    7:9
  • b)
    8:9
  • c)
    6:9
  • d)
    5:9
  • e)
    None of these
Correct answer is option 'A'. Can you explain this answer?

Nayanika Bajaj answered
Here, ∠ACB = c+180-(2c-b) = 180-(b+c)
So, We can say that Δ BCD and &delta ABC will be similar.
According to property of similarity,
AB/12 = 12/9 
Hence, 
AB = 16
AC/6 = 12/9 
AC = 8
Hence, AD = 7 and AC = 8
Now,
Perimeter of Delta; ADC / Perimeter of &delta BDC,
= (6+7+8)/(9+6+12) 
= 21/27 = 7/9.

 In the figure shown above, point E is the intersection point of the diagnoals AC and BD of rectangle ABCD. Is AB greater than BC?
  1.  AE = 5 cms and AB = 7.25 cm
  2. The perimeter of AEB is greater than the perimeter of BEC
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. 
  • c)
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. 
  • d)
    EACH statement ALONE is sufficient. 
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient.
Correct answer is 'D'. Can you explain this answer?

Aisha Gupta answered
Steps 1 & 2: Understand Question and Draw Inferences
In the given rectangle ABCD, we need to find if AB > BC, i.e., if the length is greater than the breadth.
Note that AC and BD are the diagonals of the rectangle and E is their point of intersection.
In right triangle ADC, by Pythagoras Theorem,
AD2 + DC2 = AC2  .  .  . (1)
 Similarly, in right triangle DCB,
DC2 + BC2 = DB2  .  .  . (2)
Equation (1)   – Equation (2):
AD2 – BC2 = AC2 – DB2
Being opposite sides of a rectangle, AD = BC
Therefore, the above equation becomes:
0 = AC2 – DB2Therefore, AC = DB.
Thus, we observe that the two diagonals of the rectangle are equal. 
As we know that diagonals of a rectangle bisect each other, we can write 
AE = BE = CE = DE = AC/2 = BD/2
(Now, it is also easy to notice that the rectangle is symmetrical about point E.
Therefore, point E will divide each diagonal in half. )
 
Step 3: Analyze Statement 1
Given AE = 5
  • AC = 10
Also, given that AB = 7.25
Notice that ABC is a right angled triangle. Applying Pythagoras theorem
AC2 = AB2 + BC2
  • BC2 = 102 – 7.252
Therefore, we can calculate the value of BC and so we can arrive at a unique answer on whether AB is greater than BC or not.
Statement 1 is sufficient to arrive at a unique answer.
Step 4: Analyze Statement 2
Perimeter of AEB > Perimeter of BEC
  • AE + BE + AB > BE + CE + BC
  • AB > BC (Since we already noted that AE = BE = CE = DE)
Statement 2 is sufficient to arrive at a unique answer.
Step 5: Analyze Both Statements Together (if needed)
We arrived at a unique answer in step 3 and step 4 above. Therefore, this step is not needed.
Correct Answer: D

What is the value of a?
(1) The area of an equilateral triangle of side length 3a units is 9√3 square units
(2) The area of an isosceles triangle whose sides are of length 4a, 4a and a units is 3√7 square units
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is
    not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is
    not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to
    answer the question asked, but NEITHER statement ALONE
    is sufficient to answer the question asked.
  • d)
    EACH statement ALONE is sufficient to answer the question
    asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to
    answer the question asked, and additional data specific to the
    problem are needed.
Correct answer is option 'D'. Can you explain this answer?

Pallabi Sengupta answered
Steps 1 & 2: Understand Question and Draw Inferences
To find: the value of a
Step 3: Analyze Statement 1 independently
  • Area of an equilateral Δ of side length 3a is 9√3
  •  
So, a = 2 (rejecting a = -2 because the side of a triangle, 3a, cannot be negative)
Thus, Statement 1 alone is sufficient to answer the question
Step 4: Analyze Statement 2 independently
  • The area of an isosceles triangle whose sides are of length 4a, 4a and a units is 3√7 square units
  • Let’s draw the given triangle:
  • A perpendicular on the side that measures a units from the opposite vertex will bisect this side (Property of Isosceles triangles)
    • Let the height of the perpendicular be h
  • So, by applying Pythagoras Theorem,
=>   (Rejecting the negative value of h since the length of a perpendicular cannot be negative)
so a2 =4
That is, a = 2 (rejecting a = -2 because the side of a triangle, 3a, cannot be
negative)
Thus, Statement 2 alone is sufficient to answer the question
Step 5: Analyze both Statements together (if needed)
Since we’ve already arrived at a unique answer in each of Steps 3 and 4, this
step is not required
Answer: Option D
 

In the given figure, AB || CD, m∠ABF = 45° and m∠CFC = 110°. Then, m∠FDC is:
  • a)
    25°
  • b)
    45°
  • c)
    35°
  • d)
    30°
Correct answer is option 'A'. Can you explain this answer?

Pioneer Academy answered
As in the given figure,
∠FCD = ∠FBA = 45° (alt. ∠s)
∴ ∠FDC = 180° ‒ (110° + 45°) = 25°.
Hence, option A is correct.

Consider the following statements
I. The locus of points which are equidistant from two parallel lines is a line parallel to both of them and drawn mid way between them.
II. The perpendicular distance of any point on this locus line from two original parallel lines are equal. Further, no point outside this locus line has this property.
Which of the above statements is/are correct?
  • a)
    Only I
  • b)
    Only II
  • c)
    Both I and II
  • d)
    Neither I nor II
Correct answer is option 'C'. Can you explain this answer?

Rahul Kapoor answered
Statements I and II are both true, because the locus of points which are equidistant from two parallel lines is a line parallel to both of them and draw mid way between them.

Also, it is true that the perpendicular distances of any point on this locus line from two original parallel lines are equal. Further, no point outside this locus line has this property.

Hence, option C is correct.


In the figure given above LOM is a straight line. What is the value of x°?
  • a)
    45°
  • b)
    60°
  • c)
    70°
  • d)
    80°
Correct answer is option 'B'. Can you explain this answer?

Pioneer Academy answered
From the given figure,
∠LOQ + ∠QOP + ∠POM = 180°
(straight line)
∴   (x° + 20°) + 50° + (x° – 10°) = 180°
⇒   2x° + 60° = 180° ⇒ 2x° = 120°
∴   x° = 60°
Hence, optuion B is correct.

In the given figure, line CE is drawn parallel to DB. If ∠BAD = 110°, ∠ABD = 30°, ∠ADC = 75° and ∠BCD = 60°, then the value of x° is:
  • a)
    45°
  • b)
    75°
  • c)
    85°
  • d)
    120°
Correct answer is option 'C'. Can you explain this answer?

EduRev GMAT answered
As in the given figure,
∠ADB = 180° ‒ (110° + 30°) = 40°.
So, ∠BDC = (75° ‒ 40°) = 35°.
∴ ∠DBC = 180° ‒ (60° + 35°) = 85°.
∴ ∠BCE = ∠DBC = 85° (alt. ∠s).
So, x = 85°.
Hence, option C is correct.

Is the perimeter of triangle ABC greater than 6 cm?
 (1) One side of triangle ABC measures 3 cm
 (2) The lengths of the three sides of triangle ABC are consecutive integers 
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. 
  • c)
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. 
  • d)
    EACH statement ALONE is sufficient. 
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient.
Correct answer is option 'D'. Can you explain this answer?

Ananya Iyer answered
Steps 1 & 2: Understand Question and Draw Inferences
Let’s say the sides of the triangle are a, b, and c.
Let the perimeter of the triangle be X.
Thus, X = a + b + c
We need to find whether X > 6         
Since we don’t have any other information, let’s move on to the analysis of the statement 1.
Step 3: Analyse Statement 1
Statement 1 says: One side of triangle ABC measures 3 cm. 
Let’s assume a = 3 cm
Now, we know that the sum of two sides of a triangle is always greater than the third side.
So, we can write:
b + c > a
Adding a to both sides of this inequality, we get:
a + b + c > 2a
That is, X > 2a
Substituting a = 3 cm in this inequality, we get:
X > 6 cm
Thus, Statement 1 alone is sufficient to answer the question.
Step 4: Analyse Statement 2
Statement 2 says: The lengths of the three sides of triangle ABC are consecutive integers.    
So, the lengths of the sides are n, (n+1), (n+2) cm, where n is a positive integer.
Perimeter X = n + (n+1) + (n+2)
X = 3n + 3
X = 3(n+1)
Now, we know that the sum of two sides of a triangle is always greater than the third side.
This means, we can write:
n + (n+1) > n + 2
2n + 1 > n + 2
n > 1
Adding 1 to both sides of this inequality:
n + 1 > 2
We know that multiplying both sides of an inequality with a positive number doesn’t change the inequality. Therefore, multiplying both sides of the above inequality with 3:
3(n+1) > 6
This means, X > 6
Thus, Statement 2 alone is sufficient to determine if X > 6
Step 5: Analyse Both Statements Together (if needed)
Since we got the answer to the question in Steps 3 and 4, we don’t need to perform this step.
Answer: Option (D)  

In the given figure AB || CD, ∠ALC = 60° and EC is the bisector of ∠LCD. If EF || AB then the value of ∠CEF is
  • a)
    120°
  • b)
    140°
  • c)
    150°
  • d)
    None of these
Correct answer is option 'C'. Can you explain this answer?

Notes Wala answered

∠ALC = ∠LCD = 60°
[∵  Alternate angles]
EC is the bisector of ∠LCD
∴ ∠ ECD = (1 / 2) × ∠ LCD
= (1 / 2) × 60º = 30º
∠CEF + ∠ECD = 180°
[∵  Pair of interior angles]
∠CEF + 30° = 180°
∠CEF = 180° – 30° = 150°
Hence, option C is correct.

In the adjoining figure, ∠ABC = 100°, ∠EDC = 120° and AB || DE. Then, ∠BCD is equal to:
  • a)
    80°
  • b)
    60°
  • c)
    40°
  • d)
    20°
Correct answer is option 'C'. Can you explain this answer?

Pioneer Academy answered
In the given figure,
Produce AB to meet CD at F.
∠BFD = ∠EDF = 120° (alt. ∠s)
∠BFC = (180° ‒ 120°) = 60°.
∠CBF = (180° ‒ 100°) = 80°.
∴ ∠BCF = 180° ‒ (60° + 80°) = 40°.
Hence, option C is correct.

If two supplementary angles differ by 44°, then one of the angle is:
  • a)
    72°
  • b)
    102°
  • c)
    65°
  • d)
    68°
Correct answer is option 'D'. Can you explain this answer?

Hridoy Desai answered
If two supplementary angles differ by 44 degrees, we can set up the equation:

x + (x + 44) = 180

Simplifying this equation, we have:

2x + 44 = 180

Subtracting 44 from both sides, we get:

2x = 136

Dividing both sides by 2, we find:

x = 68

So, one angle is 68 degrees, and the other angle (which differs by 44 degrees) is 112 degrees.

In Δ PQR, PS is the bisector of ∠ P and PT ⊥ OR, then ∠ TPS is equal to:
  • a)
    ∠Q + ∠ R
  • b)
    900 + 1/2 ∠Q
  • c)
    900 - 1/2 ∠R
  • d)
    1/2 (∠ Q - ∠ R)
  • e)
    None of These
Correct answer is option 'D'. Can you explain this answer?

Abhishek Kapoor answered
∠1 + ∠2 = ∠3 [PS is bisector.] ------ (1)
∠Q = 900 - ∠1
∠R = 900 -∠2 - ∠3
So,
∠Q - ∠R = (900 - ∠1) - (900 - ∠2 - ∠3)
∠Q - ∠R = ∠2 + ∠3 - ∠1
∠Q - ∠ R = ∠2 + (∠1 + ∠2) -∠1[using equation 1]
∠Q - ∠R = 2 ∠2
1/2 * (∠Q - ∠R) = ∠TPS.

In ΔABC, point D lies on the side AB between vertices A and B.
If CD = CB, is AD > BC?
(1) ∠ADC = 110
(2) ∠ACD = 30
 
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is
    not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is
    not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to
    answer the question asked, but NEITHER statement ALONE
    is sufficient to answer the question asked.
  • d)
    EACH statement ALONE is sufficient to answer the question
    asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to
    answer the question asked, and additional data specific to the
    problem are needed.
Correct answer is option 'C'. Can you explain this answer?

Mrinalini Dasgupta answered
step 1 & 2: Understand Question and Draw Inference
Given:
Representing the given information visually as below:
  • In triangle BCD, CD = CB
    • So, angles opposite to these sides will be equal. Let these angles measure Xo
  • Let ∠ACD = y∘
  • So, the different angles in this figure will be as under:
  •  
To find: If AD > BC
  • That is, if AD > CD (since BC = CD)
  • That is, if in triangle ADC, 
    • ∠ACD > ∠DAC (since sides opposite to greater angles are greater)
    • That is, y∘ > x∘ − y∘
    • That is, 2y∘ > x∘
  • So, the answer to the question will be YES if
  • 2y∘ > x∘
​Step 3 : Analyze Statement 1 independent
Step 4 : Analyze Statement 2 independent
Step 5: Analyze Both Statements Together (if needed)
From Statement 1:x = 70∘

From Statement 2: y = 30∘
Since we now know the values of xo and yo  we can determine if 2yo > Xo
Thus, the 2 statements together are sufficient to answer the question.
Answer: Option C

The complement of 72° 40’ is :
  • a)
    107° 20’
  • b)
    27° 20’
  • c)
    17° 20’
  • d)
    12° 40’
Correct answer is option 'C'. Can you explain this answer?

Rahul Kapoor answered
Complement of 72° 40’ is 90° ‒ (72° 40’)
= (89° 60’) ‒ (72° 40’) {since 1° = 60’}
= 17° 20’

In the given figure, if AB || CD, then ∠FXE is equal to:
  • a)
    30°
  • b)
    50°
  • c)
    70°
  • d)
    80°
Correct answer is option 'D'. Can you explain this answer?

Rahul Kapoor answered
As per the given figure,
∠BFE = ∠CEF = 110° (alt. ∠s).
So, ∠XFE = ∠BFE ‒ ∠BFX = (110° ‒ 50°) = 60°.
And on straight line CD,
110° + ∠FEX + 30° = 180° ⇒ ∠FEX = 40°.
Now, ∠XFE + ∠FEX + ∠FXE = 180° ⇒ 60° + 40° + ∠FXE = 180°.
∴ ∠FXE = 80°.
Hence, option D is correct.

In the figure given below, AB is parallel to CD. ∠ABC = 65°, ∠CDE = 15° and AB = AE. What is the value of ∠AEF?
  • a)
    30°
  • b)
    35°
  • c)
    40°
  • d)
    45°
Correct answer is option 'B'. Can you explain this answer?

Rahul Kapoor answered
Given that,
∠ABC = 65° and ∠CDE = 15°
Here, ∠ABC + ∠TCB = 180°
(∵   AB || CD)
∠TCB = 180° – ∠ABC
∴ ∠TCB = 180° – 65° = 115°
∵ ∠TCB + ∠DCB = 180°
(Linear pair)
∴ ∠DCB = 65°
Now, in ΔCDE
∠CED = 180° – (∠ECD + ∠EDC)
(∵ ∠ECD = ∠BCD)
= 180° – (– 65° + 15° ) = 100°

∵ ∠DEC + ∠FEC = 180°
⇒ ∠FEC = 180° – 100° = 80°
Given that, AB = AE.
i.e. ΔABE an isosceles triangle.
∴ ∠ABE = ∠AEB = 65°
∵ ∠AEB + ∠AEF + ∠FEC = 180°
(straight line)
⇒ 65° + x° + 80° = 180°
∴ x° = 180° – 145° = 35°.
Hence, option B is correct.

In the given figure, AB || CD, ∠ABO = 40° and ∠CDO = 30°. If ∠DOB = x°, then the value of x is:
  • a)
    35°
  • b)
    110°
  • c)
    70°
  • d)
    140°
Correct answer is option 'C'. Can you explain this answer?

Notes Wala answered
In the given figure,

Through O draw EOF parallel to AB & so to CD.
∴ ∠BOF = ∠ABO = 40° (alt. ∠s)
Similarly, ∠FOD = ∠CDO = 30° (alt. ∠s)
∴ ∠BOD = (40° + 30°) = 70°.
So, x = 70°.
Hence, option C is correct.

Two straight lines AB and CD cut each other at O. If ∠BOD = 63°, then ∠BOC is:
  • a)
    63°
  • b)
    117°
  • c)
    17°
  • d)
    153°
Correct answer is option 'B'. Can you explain this answer?

Rahul Kapoor answered
As given ∠BOD = 63° 
Since COD is a straight line, we have:

∠BOC + ∠BOD = 180°. So, ∠BOC = (180° ‒ 63°) = 117°.

The figure above shows the position, at two instants of time, of a 10-meter-long rod as it was falling down to the ground. If one end of the rod was pinned to the ground, by what vertical distance did the rod fall between the two instants of time?
  • a)
  • b)
  • c)
  • d)
  • e)
Correct answer is option 'A'. Can you explain this answer?

Aarav Sharma answered
Given:
  • Length of rod = 10 meters
  • Angle with the ground at 1st instant = 60o
  • Angle with the ground at 2nd instant = 30o
To find:
  • Vertical distance covered between the 2 instants
Approach:
  • We will first try to understand the question
  • To do so, let us:
    • Drop perpendiculars from the free end of the rod at both instants of time
    • Label the triangles thus formed
 
  • From the diagram, it is clear that we need to find AB – PQ
    • (Note: if you are not sure how AB – PQ is equal to the vertical distance covered by the ladder, drop a perpendicular PX on AB. You’ll see that BQPX is a rectangle and so, PQ = BX. So, AB – PQ = AB – BX = AX. The ladder was earlier at a height of A and it is now at a height of X. So, the distance AX clearly denotes the vertical distance covered by the ladder.)
  1. Thus, we need to find AB – PQ
    1. AB is a side of right ΔOBA, which is 30-60- 90 Triangle and angle AOB = 60o
    2. PQ is a side of right ΔOQP, which is also a 30-60-90 Triangle with angle POQ = 30o
  2. In each of the 2 right triangles, we know a side and two angles.
    1. So, by using side ratio property of 30-60-90 triangle, we can find the required values
 
Working Out:
  • Finding AB
    • In right ΔOBA
      • OB: AB: OA = 1: √3: 2 
      • We know OA = 10
      • AB: OA = √3: 2
      • Thus AB = 5√3
  • Finding PQ
    • In right ΔOQP
      • PQ: OQ: PO = 1: √3: 2 
      • We know PO = 10
      • PQ: PO = 1: 2
      • Thus PQ = 5
  • Thus, AB – PQ = 5√3 – 5 = 5 (√3 -1) meters
  • Therefore, the correct answer is Option A.

In the figure given above, EC is parallel to AB, ∠ECD = 70° and ∠BDO = 20°. What is the value of ∠OBD?
  • a)
    20°
  • b)
    30°
  • c)
    40°
  • d)
    50°
Correct answer is option 'D'. Can you explain this answer?

Notes Wala answered
Given that, EC || AB
∴    ∠ECO + ∠AOC = 180°

⇒   ∠AOC = 180° – 70° = 110°
∴    ∠BOD = ∠AOC = 110°
(alternate angle)
Now, in ΔOBD
∠BOD + ∠ODB + ∠DBO = 180°
∴   110° + 20° + x° = 180° ⇒ x° = 50°.
Hence, option D is correct.

The angles x°, a°, c° and (π – b)° are indicated in the figure given below. Which one of the following is correct?
  • a)
    x° = a° + c° – b°
  • b)
    x° = b° – a° – c°
  • c)
    x° = a° + b° + c°
  • d)
    x° = a° – b° + c°
Correct answer is option 'C'. Can you explain this answer?

Rahul Kapoor answered
∠PCT + ∠PCB = π
(Linear pair)
∠PCB = π – (π – b°) = b° ..... (i)

In ΔBPC,
∠PCB + ∠BPC + ∠PBC = π
∠PBC = π – ∠PCB – ∠BPC  =  π – b° – a° ..... (ii)
∵ ∠ABE + ∠EBC = π
(∵ ∠PBC = ∠EBC)      (linear pair)
∠ABE = π – ∠PBC = π – (π – b° – a°)  =  a° + b° ...(iii)
Now, in ΔABE
Sum of two interior angles = Exterior angle
∠EAB + ∠ABE = ∠BES ⇒ c° + b° + a° = x°
∴   x° = a° + b° + c°.
Hence, option C is correct.

A wheel makes 12 revolutions per min. The angle in radian described by a spoke of the wheel in 1 s is:
  • a)
    5π/2
  • b)
    2π/5
  • c)
    3π/5
  • d)
    4π/5
Correct answer is option 'B'. Can you explain this answer?

Advait Roy answered
To find the angle described by a spoke of the wheel in 1 second, we need to find the angle described by the wheel in 1 second and then divide it by the number of spokes on the wheel.

The wheel makes 12 revolutions per minute, so it makes 12 * 2π radians per minute.

To find the angle described by the wheel in 1 second, we divide by 60 (since there are 60 seconds in a minute): (12 * 2π) / 60 = 2π/5 radians per second.

Since there are typically 4 spokes on a wheel, we divide the angle described by the wheel in 1 second by 4 to find the angle described by a spoke in 1 second: (2π/5) / 4 = π/10 radians per second.

Therefore, the angle described by a spoke of the wheel in 1 second is π/10 radians.

In the given figure, AOB is a straight line. If ∠AOC + ∠BOD = 85°, then ∠COD is:
  • a)
    85°
  • b)
    90°
  • c)
    95°
  • d)
    100°
Correct answer is option 'C'. Can you explain this answer?

Pioneer Academy answered
Clearly,
∠AOC + ∠COD + ∠BOD = 180°
∴ 85° + ∠COD = 180°. So, ∠COD = (180° ‒ 85°) = 95°.
Hence, option C is correct.

In the given figure lines AP and OQ intersect at G. If ∠AGO + ∠PGF = 70° and ∠PGQ = 40°. Find the angle value of ∠PGF.
  • a)
    31°
  • b)
    35°
  • c)
    30°
  • d)
    20°
Correct answer is option 'C'. Can you explain this answer?

Notes Wala answered
As AP is a straight line and rays GO and GF stands on it.
∴ ∠AGO + ∠OGF + ∠PGF = 180°
⇒ (∠AGO + ∠PGF) + ∠OGF = 180°
⇒ 70° + ∠OGF = 180°
⇒ ∠OGF = 180° – 70° 
⇒ ∠OGF = 110°
As, OQ is a straight line, rays GF and GP stands on it.
∠OGF + ∠PGF + ∠PGQ = 180°
Putting the value of ∠OGF & ∠PGQ 
110° + ∠PGF + 40° = 180°
∠PGF = 180° – 150° = 30°
Hence, option C is correct.

A right angled triangle, right angled at point B has angles such that a<c<b where a, b, c correspond to the angles at points A, B, C. If the area of the triangle is 2 which of the following inequality indicates the range of values of side BC.
  • a)
    0 < BC < 2
  • b)
    0 < BC < √2
  • c)
    0 < BC < 4
  • d)
    0<BC < 2√2
  • e)
    0 < BC < 8
Correct answer is option 'A'. Can you explain this answer?

Moumita Sen answered
Since it is given that ΔABC is right angled at B, Angle ABC = 90∘
Also, area of ΔABC = ½ * AB * BC = 2
AB*BC = 4
But it is given that a < c
Therefore, BC < AB
In the extreme case of BC = AB (which is not possible according to the given condition),
BC = AB = 2
However, since BC < AB
BC should be less than 2 and AB should be greater than 2.
Therefore, BC < 2
Also, since BC is a side of a triangle, it is positive and greater than zero
Therefore, BC > 0
Combining the above two inequalities, we get:
0 < BC < 2
Correct Answer: A

The figure above shows a triangle ABC in which the measure of ∠ABC, b∘is greater than 90∘
 If AB = 3 units and BC = 4 units, which of the following can be the measure of side AC?
  1. 6
  2. 7
  3. 8
 
  • a)
    I only
  • b)
    II only
  • c)
    III only
  • d)
    I, II and III
  • e)
    None of the above
Correct answer is option 'A'. Can you explain this answer?

Mrinalini Dasgupta answered
Given:
  • An obtuse triangle, 2 of whose sides are AB = 3 and BC = 4 units
To find:
  • The possible measure of the side AC that is opposite to the obtuse angle
Approach & Working:
  • To answer the question, we need to think of the constraints on AC. From the theory of triangles, we can figure that there are 2 constraints on AC:
    • Constraint 1: Since triangle ABC is obtuse-angled at B, AC2 > AB2 + BC2
    • Constraint 2: The sum of any 2 sides of a triangle must be greater than the 3rd side. So, AB + BC > AC
  • So, we will work out the range of possible values of AC by applying these 2 constraints, and then see, which of the 3 given values fit this range.
  • Applying Constraint 1
    • AC2 > 32 + 42
    • AC2 > 9 + 16
    • AC2 > 25
    • So, AC > 5
  • Applying Constraint 2
    • AB + BC > AC
    • So, 3 + 4 > AC
    • 7 > AC
    • Rewritten as: AC < 7
  • Getting to the answer
    • Thus, we see that 5 < AC < 7
    • The only value, out of the 3 options, that satisfies this range is Option I
 
Thus, the correct answer is Option A

Is AD > BC?
I.  Angle CAB = Angle BCA
II.  CD > AB
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. 
  • c)
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. 
  • d)
    EACH statement ALONE is sufficient. 
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient.
Correct answer is option 'B'. Can you explain this answer?

Athira Choudhury answered
Steps 1 & 2: Understand Question and Draw Inferences
Given a quadrilateral ABCD with right angles at B and D.
We need to find if AD > BC is true. 
Step 3: Analyze Statement 1
Given that Angle CAB = Angle BCA
Therefore, in ΔABC,
Angle CAB = Angle BCA = 45o
Therefore, BC = AB
This doesn’t give us any relation between AD and BC.
So statement 1 is not sufficient to arrive at a unique answer.
 Step 4: Analyze Statement 2
Given that CD > AB.
In the right angled triangle ABC, we have
AC2 = AB2 + BC2
Similarly, in the right angled triangle ADC, we have
AC2 = AD2 + CD2
Therefore,
AD2 + CD2 = AB2 + BC2……………. (I)
 Given that CD > AB
  • CD2 > AB2
In other words, a term in the LHS of (I) is greater than another term in the RHS of (I)
So for (I) to be true (the equality to hold true), the remaining terms on either sides of LHS and RHS should compensate for the imbalance created by CD2 and AB2.
From (I),
CD2 -AB2 = BC2 -AD2
Let’s say CD2 – AB2 = k (a positive number since CD2 > AB2)
Then
BC2 – AD2 = k (the same positive number).
Therefore, BC2 > AD2
  • BC > AD
Therefore, statement 2 is sufficient to arrive at a unique answer.
Step 5: Analyze Both Statements Together (if needed)
We arrived at a unique answer in step 4 above. Therefore, this step is not needed.
Correct Answer: B

In a triangle ABC,∠ A = 900, AL is drawn perpendicular to BC, Then ∠ BAL is equal to:
  • a)
    ∠ALC
  • b)
    ∠ACB
  • c)
    ∠BAC
  • d)
    ∠B - ∠BAL
  • e)
    None of these
Correct answer is option 'B'. Can you explain this answer?

Nayanika Bajaj answered
∠ BAL + ∠ B + 900 = 1800
or, ∠ BAL + ∠ B = 900
or, ∠ BAL = 900 - ∠ B ----------- (1)
Now, in Δ ABC,
∠ ACB + ∠ B + ∠ A = 1800
∠ ACB = 900
-∠ B ----- (2)
From, (1) and (2),
∠ BAL = ∠ ACB.

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