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SparkEd Math offers 60+ practice questions for Areas of Parallelograms & Triangles across 3 difficulty levels (Easy, Medium, Hard) plus a timed test mode. Each question comes with visual step-by-step solutions and AI coaching hints.

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SparkEd Math uses visual step-by-step solutions for every Areas of Parallelograms & Triangles problem, making complex concepts easy to understand. The platform includes 3 difficulty levels for progressive learning, a Super Power Help AI coach for instant doubt-clearing, and interactive practice rather than passive reading.

Is SparkEd's Areas of Parallelograms & Triangles content aligned with the CBSE syllabus?

Yes, all Areas of Parallelograms & Triangles questions and lessons on SparkEd Math are fully aligned with the Central Board of Secondary Education (CBSE) syllabus for Class 9. The content follows the official curriculum structure and exam pattern.

Areas of Parallelograms & Triangles — Class 9 CBSE (2025-26)

About Areas of Parallelograms & Triangles — Class 9 CBSE

Prove theorems relating areas of figures on the same base and between the same parallels. This topic is part of the CBSE Class 9 mathematics syllabus (chapter: Chapter 9). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Areas of Parallelograms & Triangles

Introduction to Area and Parallelograms
Parallelograms on the Same Base and Between the Same Parallels
Linking Parallelograms and Triangles
The Core Proof: Triangles on the Same Base and Between the Same Parallels
Advanced Applications and Key Corollaries

Interactive lesson · about 15 minutes · checkpoint question after every unit

Areas of Parallelograms & Triangles — solved examples for Class 9 CBSE

Example 1easy

For two planar figures to be considered 'between the same parallels', what is the essential condition they must satisfy?

A)Their bases must lie on one parallel line, and their vertices opposite to the base must lie on the other parallel line.
B)They must have at least one common side.
C)Their areas must be equal.
D)They must be congruent.

Step-by-step solution

The definition states that if two figures have a common base (or equal bases) and the vertices opposite to the base of each figure lie on a line parallel to the base, then they are said to be on the same base and between the same parallels.
Option A directly captures this definition, explaining the geometric arrangement required.

Answer: A

Example 2medium

Which of the following conditions is necessary for two figures to be on the same base and between the same parallels?

A)A) They must be congruent.
B)B) They must have equal areas.
C)C) They must share a common base and their vertices opposite to the common base lie on a line parallel to the base.
D)D) They must be quadrilaterals.

Step-by-step solution

Step 1: Figures are said to be on the same base if they share a common side.
Step 2: They are said to be between the same parallels if the vertices opposite to the common base lie on a line parallel to the common base.
Step 3: Option C correctly combines these two conditions, which is the definition.

Answer: C) They must share a common base and their vertices opposite to the common base lie on a line parallel to the base.

Example 3hard

Two parallelograms, P1 and P2, have equal areas. Which of the following statements must always be true?

A)P1 and P2 share a common base and lie between the same pair of parallel lines.
B)P1 and P2 have the same base length and the same corresponding height.
C)P1 and P2 are congruent figures.
D)P1 and P2 have equal perimeters.

Step-by-step solution

The area of a parallelogram is given by the formula: Area = base × height. For two parallelograms to have equal areas, the product of their respective base and corresponding height must be equal.
If Area(P1) = Area(P2), then base_1 × height_1 = base_2 × height_2. This does not necessarily mean they share a common base and lie between the same parallels (which is a sufficient condition, but not necessary for *any* two parallelograms to have equal areas).
However, if they have the same base length and the same corresponding height, their areas will always be equal by definition. Options C and D are not generally true for parallelograms with equal areas.

Answer: P1 and P2 have the same base length and the same corresponding height.

Practice questions on Areas of Parallelograms & Triangles

Q1.easy
In a figure, parallelogram ABCD and triangle EBC are drawn such that point E lies on the side AD. For these two figures to be on the same base and between the same parallels, which condition is necessary?
1. A)E must be the midpoint of AD.
2. B)The line AD must be parallel to the line BC.
3. C)The line AB must be parallel to the line CD.
4. D)Area(ABCD) must be equal to Area(EBC).
Show answer
Answer: B
Hint: Consider BC as the common base. For the figures to be between the same parallels, the line containing the vertices opposite to BC must be parallel to BC.
Q2.easy
Parallelogram PQRS and parallelogram MNRS are on the same base SR and between the same parallels SR and PM. If Area(PQRS) = 45 cm², what is Area(MNRS)?
1. A)90 cm²
2. B)45 cm²
3. C)22.5 cm²
4. D)Cannot be determined without side lengths.
Show answer
Answer: B
Hint: Recall the theorem about the areas of parallelograms that share the same base and are situated between the same parallel lines.
Q3.easy
Which of the following conditions is *not necessarily true* for two triangles on the same base and between the same parallels?
1. A)Their heights corresponding to the common base are equal.
2. B)Their areas are equal.
3. C)They have the same perimeter.
4. D)Their vertices opposite to the common base lie on the same parallel line.
Show answer
Answer: C
Hint: While areas are equal, think about what determines the perimeter of a triangle. Does having the same base and height guarantee all side lengths are identical?
Q4.medium
A parallelogram has a base of 12 cm and the corresponding altitude (height) is 7 cm. What is its area?
1. A)A) 19 cm²
2. B)B) 42 cm²
3. C)C) 84 cm²
4. D)D) 168 cm²
Show answer
Answer: C) 84 cm²
Hint: Remember the formula for the area of a parallelogram involving its base and height.
Q5.medium
Consider two parallelograms ABCD and PQRS. If they are on the same base AB and between the same parallels AB and DE (where D and E are points on the parallel line), which of the following statements is TRUE?
1. A)A) Area(ABCD) < Area(PQRS)
2. B)B) Area(ABCD) > Area(PQRS)
3. C)C) Area(ABCD) = Area(PQRS)
4. D)D) Area(ABCD) cannot be compared to Area(PQRS) without more information.
Show answer
Answer: C) Area(ABCD) = Area(PQRS)
Hint: Think about the theorem concerning the areas of parallelograms that share the same base and lie between the same parallel lines.
Q6.medium
Parallelogram PQRS and parallelogram TURS share the same base RS and lie between the same parallel lines RS and PT. If Area(PQRS) is 60 cm², what is the area of parallelogram TURS?
1. A)A) 30 cm²
2. B)B) 60 cm²
3. C)C) 90 cm²
4. D)D) 120 cm²
Show answer
Answer: B) 60 cm²
Hint: Apply the theorem about parallelograms on the same base and between the same parallels.
Q7.hard
Let ΔABC and ΔDBC be two triangles sharing the same base BC. If Area(ΔABC) = Area(ΔDBC), which of the following statements must always be true?
1. A)AD is parallel to BC.
2. B)AB = DC.
3. C)AC = DB.
4. D)The triangles are congruent.
Show answer
Answer: AD is parallel to BC.
Hint: Consider the relationship between the areas of triangles, their bases, and their heights. If two triangles have the same base and equal areas, what must be true about their heights relative to that base?
Q8.hard
ABCD is a parallelogram. P is any point on the diagonal AC. If Area(ΔABP) = 15 cm², find Area(ΔADP).
1. A)10 cm²
2. B)20 cm²
3. C)15 cm²
4. D)30 cm²
Show answer
Answer: 15 cm²
Hint: Recall that a diagonal divides a parallelogram into two triangles of equal area. Also, consider the heights of triangles sharing a common base on a diagonal.
Q9.hard
In triangle ABC, AD is the median. E is the midpoint of AD. F is the midpoint of AB. What is the area of quadrilateral BDEF if Area(ΔABC) = 60 cm²?
1. A)10 cm²
2. B)15 cm²
3. C)20 cm²
4. D)30 cm²
Show answer
Answer: 30 cm²
Hint: Remember that a median divides a triangle into two triangles of equal area. Apply this property step-by-step to the smaller triangles formed.

These are 9 of the 60 questions available for Areas of Parallelograms & Triangles. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.