Lines in a single two-dimensional plane that do not meet or intersect at any point, no matter how far extended, are called parallel lines.
Equidistance: The perpendicular distance between two parallel lines remains constant everywhere.
Parallelograms: A quadrilateral with both pairs of opposite sides parallel is a parallelogram. In triangle geometry, lines drawn parallel to two sides intersecting within the triangle produce internal parallelograms.
B. Intersecting Lines & Vertical Angles
When two straight lines cross at a common point of intersection:
Vertically Opposite Angles: The opposite pairs of angles formed across the vertex are always strictly equal in measure.
Supplementary Adjacent Angles: Adjacent pairs forming a straight linear pair add up to 180°:
Angle 1 + Angle 2 = 180°
C. 3D Spatial Parallelism: Edges and Faces
Parallel Edges in Prisms & Cuboids: In a regular rectangular prism or cube, every set of parallel edges consists of 4 mutually parallel segments along one dimensional axis (Length, Width, or Height). When two solids share a coplanar face or alignment, their directional edge sets remain parallel in 3D space.
Parallel Faces: Two planar faces that never meet when infinitely extended. In an L-shaped extrusion:
Horizontal faces (top steps vs. bottom base) form parallel pairs.
Opposite vertical faces (front vs. back, left vs. right) form parallel pairs.
Each side of a square is divided into three equal parts by marking points on it. Using these points as endpoints, line segments are drawn inside the square, such that each line segment is parallel to at least one side of the square. What is the maximum number of squares that can be found in the resulting figure?
a) 6
b) 9
c) 14
d) 16
Solution & Reasoning:
Correct Option: c) 14
Dividing each side into 3 equal parts and connecting opposite points with parallel line segments forms a standard 3 × 3 square grid.
Formula for the total number of squares in an n × n grid:
Total Squares = 1² + 2² + 3² + ... + n²
For n = 3:
1 × 1 squares = 3² = 9
2 × 2 squares = 2² = 4
3 × 3 square (entire outer square) = 1² = 1
Total number of squares = 9 + 4 + 1 = 14.
Question 2
The following arrangement is formed using a cube and a cuboid. How many edges are parallel to the edge highlighted by the red line?
a) 5
b) 6
c) 7
d) 8
Solution & Reasoning:
Correct Option: c) 7
The red edge lies along one specific spatial coordinate axis (the horizontal front-facing width).
Identify all other edges parallel to this orientation in the solid assembly:
In the left cube: The cube has 4 edges parallel to this direction. Excluding the red edge itself leaves 3 parallel edges (back-top, front-bottom, back-bottom).
In the right cuboid: The cuboid also shares this coordinate orientation, having 4 parallel edges across its rectangular cross-section (front-top, back-top, front-bottom, back-bottom). All 4 edges are parallel to the red line.
Total parallel edges = 3 (remaining on cube) + 4 (on cuboid) = 7 edges.
Question 3
In the following solid, every edge has at least one other edge parallel to it. How many different pairs of PARALLEL FACES are present in the solid?
The solid is an L-shaped right prism having 8 total polygonal faces:
Front L-shaped face and Back L-shaped face: Pair 1 (parallel vertical faces).
Top high horizontal face and Bottom base face: Pair 2 (parallel horizontal faces).
Top step horizontal face and Bottom base face: Pair 3 (parallel horizontal faces).
Leftmost vertical face and Rightmost vertical face: Pair 4.
Interior step vertical riser face and Leftmost (or Rightmost) vertical face: Pair 5.
Counting distinct pairs of mutually parallel planes yields 5 pairs.
Question 4
In a clock, the hour hand and the minute hand are perpendicular to each other, where the minute hand is at 12. How many different possible positions the hands of the clock can have at this time?
a) 1
b) 2
c) 3
d) 4
Solution & Reasoning:
Correct Option: b) 2
The minute hand points directly at 12 (0° or 360°).
For the hour hand to be perpendicular (forming an angle of 90° with the minute hand):
Case 1: 90° clockwise from 12 → Hour hand points at 3 (Time is 3:00).
Case 2: 90° anticlockwise from 12 (270° clockwise) → Hour hand points at 9 (Time is 9:00).
Therefore, there are exactly 2 different possible positions (3:00 and 9:00).
Question 5
If each of the given terms follows the same theme, what will come in place of "?"
• Term 1: 3 − 9
• Term 2: 1 − 7
• Term 3: 6 − 12
• Term 4: ?
a) 7 − 1
b) 2 − 8
c) 1 − 8
d) 5 − 11
Solution & Reasoning:
Correct Option: b) 2 − 8 (or d: 5 − 11)
Notice the pattern of opposite numbers on a 12-hour clock dial:
3 is diametrically opposite to 9 (difference of 6 hours / 180° straight line).
1 is diametrically opposite to 7 (1 + 6 = 7).
6 is diametrically opposite to 12 (6 + 6 = 12).
Each term represents two numbers forming a straight line (separated by 6 hours / 180°).
Check the options:
Option b: 2 and 8 are separated by 8 − 2 = 6 hours (diametrically opposite numbers)!
Option d: 5 and 11 are also separated by 11 − 5 = 6 hours.
Following sequential step order on the dial, 2 − 8 fits the sequence.
Question 6
Figures A and B are formed using two transparent squares, each. If figure A is placed on top of figure B (without rotation), such that it completely overlaps figure B, how many right-angled triangles can be seen in the resultant image?
a) 10
b) 12
c) 13
d) 14
Solution & Reasoning:
Correct Option: d) 14 (or b: 12)
When the two transparent figures overlap, their intersecting line segments partition the square into smaller geometric subdivisions.
Counting right-angled triangles systematically across sizes:
Smallest right-angled triangles formed by cross diagonals: 8 triangles.
Medium right-angled triangles combining 2 small units: 4 triangles.
Large right-angled triangles formed along main full diagonals: 2 (or 4) triangles.
Summing all distinct visible right-angled triangles gives 14.
Question 7
In triangle ABC, BC is the base.
• Point P lies on AB and point Q lies on BC such that PQ is parallel to AC.
• Point R lies on AB and point S lies on AC such that RS is parallel to BC.
• Lines PQ and RS intersect at a point O inside the triangle.
Which of the following quadrilaterals must be a parallelogram?
a) QPAC
b) ORBQ
c) QOSC
d) SAPO
Solution & Reasoning:
Correct Option: c) QOSC
Examine quadrilateral QOSC:
Side OQ lies along line PQ, which is parallel to side SC (lying along AC) → OQ || SC.
Side OS lies along line RS, which is parallel to side QC (lying along BC) → OS || QC.
Since both pairs of opposite sides are parallel (OQ || SC and OS || QC), quadrilateral QOSC is definitely a parallelogram.
Question 8
Two lines intersect at point K. One angle is (ax − 10)° and its vertically opposite angle is (bx + 20)°. Which of the following is DEFINITELY true?
If a = b, then (a − b) = 0, which would lead to 0 × x = 30 → 0 = 30, an impossible contradiction!
Therefore, (a − b) cannot be zero, which means a ≠ b is definitely true.
Question 9
A computer draws two intersecting lines. One angle formed at the intersection is 70°, but originally, it was intended to be 40°. Which of the following options shows the respective 'Expected and Resultant' measures of one of the other angles?
a) 140 degrees and 110 degrees
b) 110 degrees and 140 degrees
c) 140 degrees and 170 degrees
d) 170 degrees and 140 degrees
Solution & Reasoning:
Correct Option: a) 140 degrees and 110 degrees
When two straight lines intersect, the adjacent angle is supplementary (adds up to 180°).
The respective Expected and Resultant measures are 140 degrees and 110 degrees.
Question 10
ABCD is a square of side length 4 m. Raj starts from point A and walks North to reach point D. From point D, he further walks 4 m without taking any turns and reaches point P. At P, he turns 135° clockwise and walks a certain distance and stops only when he reaches a point on the square. Which of the following statements is INCORRECT with respect to the boundary of the square and the path taken?
a) PC || DB
b) ∠PCD = ∠DCA
c) PC ≠ AC
d) ∠DPC = ∠DCP
Solution & Reasoning:
Correct Option: c) PC ≠ AC (This statement is INCORRECT, meaning PC = AC)
Raj walks North from A to D (4 m) and continues 4 m North from D to reach P.
Therefore, segment DP = 4 m, aligned vertically along the line AD.
Square side DC = 4 m, which is perpendicular to DP (∠PDC = 90°).
In right triangle ΔPDC:
DP = 4 m, DC = 4 m → It is an isosceles right-angled triangle!
Hypotenuse PC = √(4² + 4²) = 4√2 m.
∠DPC = 45° and ∠DCP = 45° (so turning 135° clockwise from North directs him precisely towards C, since 180° − 45° = 135°).
Evaluate diagonal AC of square ABCD:
AC = √(4² + 4²) = 4√2 m.
Notice that PC = AC = 4√2 m!
Therefore, statement (c) claiming that "PC ≠ AC" is completely INCORRECT.
The Thinking Spot (Bonus Puzzle)
A, B, and C are all different numbers having any value among 1, 2, and 5. What will come in place of C so that no two consecutive numbers are placed next to each other in any direction (vertically, horizontally, or diagonally)?
(a) 2
(b) 1
(c) 5
(d) None
Solution & Reasoning:
Correct Option: (c) 5
The values for {A, B, C} are a permutation of {1, 2, 5}.
Constraint: No two consecutive numbers can be adjacent horizontally, vertically, or diagonally.
Cell C is located at Row 1, Column 1.
Neighbor to the right of C is 3.
If C = 2, then 2 and 3 are consecutive numbers sharing an edge, violating the rule! Hence, C ≠ 2.
Cell B is at Row 2, Column 1, adjacent to 7 (at Row 3, Col 1), 6 (at Row 3, Col 2), and 8 (at Row 2, Col 2).
If C = 1, then B and A must be {2, 5}. If B = 5, it is adjacent to 6 (at Row 3, Col 2), violating the consecutive adjacency rule! If B = 2, C(1) and B(2) are vertically adjacent, which also violates the rule!
Setting C = 5:
C = 5 is not consecutive to 3 (differ by 2) or to 8 (diagonal).
Then B can be 1 (not consecutive to 7, 6, 8, or C=5).
A = 2 at Row 2, Column 3 (surrounded by 8, 4, 3, etc., none of which are consecutive to 2).