This MCQ module is based on: Congruence of Triangles and SSS Condition
TOPIC 1 OF 23
Congruence of Triangles and SSS Condition
🎓 Class 7
Mathematics
CBSE
Theory
Ch 1 — Triangles
⏱ ~35 min
🌐 Language: [gtranslate]
🧠 AI-Powered MCQ Assessment
▲
📐 Maths Assessment
▲
This mathematics assessment will be based on: Congruence of Triangles and SSS Condition
Targeting Class 7 level in General Mathematics, with Basic difficulty.
Upload images, PDFs, or Word documents to include their content in assessment generation.
1.1 Geometric Twins
Imagine a symbol on a signboard that needs to be recreated on another board. The symbol looks like an open angle — two arms meeting at a point. Let us name the corner points of this symbol.
The signboard symbol with corner points A (vertex), B and C (arm endpoints)
How do we recreate this figure? One way is to trace the outline on tracing paper. But this is difficult for big symbols. What else can we do?
Can we take some measurements that would allow us to exactly recreate this figure? If yes, what measurements should we take?
Suppose the arm lengths are AB = 4 cm and BC = 8 cm. We observe that several such symbols can be constructed with the same lengths.
Different symbols with AB = 4 cm, BC = 8 cm but different angles at A
To get the exact replica, would it help to take any other measurement? Yes — the measure of \(\angle ABC\), along with the two arm lengths AB and BC, fixes the shape and size of this figure.
Can you draw the symbol if it is known that AB = 4 cm, BC = 8 cm, and \(\angle ABC = 80°\)?
These three measurements can help us create an exact replica of the symbol on the signboard. Figures that are exact copies of each other or have the same shape and size are said to be congruent?. Congruent figures can be superimposed exactly, one over the other.
Definition
Congruent figures are figures that have the same shape and size. They can be superimposed exactly, one over the other. A figure can be rotated or flipped before superimposing it on the other figure.
Two congruent crescent shapes — they can be superimposed exactly
Two congruent quadrilaterals (signboard shapes) — identical in shape and size
Congruent D-shapes and rotated L-shapes — rotation and flipping preserve congruence
Note
While checking for congruence, a figure can be rotated or flipped before superimposing it on the other figure. So the following pairs of figures are also congruent to each other.
Figure it Out (Section 1.1)
Q1. Check if the two figures are congruent.
Answer: Yes, the two L-shaped figures are congruent. They have the same shape and dimensions. One can be superimposed on the other exactly by rotating or flipping.
Q2. Circle the pairs that appear congruent.
Answer: The leaf pair and the star pair appear congruent — they have the same shape and size. The cloud shapes may or may not be congruent depending on exact measurements.
Q3. What measurements would you take to make a figure congruent to a given: (a) Circle (b) Rectangle
Answer:
(a) Circle: We need to measure the radius. Two circles with the same radius are congruent.
(b) Rectangle: We need to measure both the length and the width. Two rectangles with the same length and width are congruent.
(a) Circle: We need to measure the radius. Two circles with the same radius are congruent.
(b) Rectangle: We need to measure both the length and the width. Two rectangles with the same length and width are congruent.
Q4. How would we check if two Y-shaped figures like the ones below are congruent?
Answer: To check if two Y-shaped figures are congruent, we need to measure: (1) the length of the stem, (2) the lengths of both arms, and (3) the angle between the two arms. If all these measurements match, the two Y-shapes are congruent. Alternatively, we can try to superimpose one on the other using tracing paper.
1.2 Congruence of Triangles
Meera and Rabia have been asked to make a cardboard cutout identical to a triangular frame they have in school. They see that the frame is too big to be traced on a paper and replicated.
Meera and Rabia looking at the large triangular frame — too big to trace!
What do you think they can do?
Measuring the Sidelengths
Can certain measurements of the triangle be used for this? Using a measuring tape, the girls measure the sides of the triangle to be 40 cm, 60 cm, and 80 cm.
Meera: The angles of the triangle are not required! With the side lengths we have measured, we can create a triangle congruent to this one.
Do you agree with Meera?
Instead of 40 cm, 60 cm, and 80 cm, suppose the sidelengths had been 4 cm, 6 cm, 8 cm (fitting on our page). Is this information sufficient to replicate the triangle with the same size and shape? If yes, can you do so?
Rabia: If I were to construct this triangle, I would first draw a line segment having one of the given lengths, say 6 cm, and then draw circles from each of its end points with radii 4 cm and 8 cm. But the circles would intersect at two points, forming two triangles: \(\triangle ABE\) and \(\triangle ABF\).
Construction: circles from A (radius 4 cm) and B (radius 8 cm) intersect at E and F, forming \(\triangle ABE\) and \(\triangle ABF\)
Examine whether \(\triangle ABE\) and \(\triangle ABF\) are congruent.
We see that \(\triangle ABE\) and \(\triangle ABF\) are congruent. From this general construction, we can see that all triangles with the same sidelengths are congruent. Hence, Meera was right — the sidelengths are sufficient to construct a congruent triangle.
SSS Condition
If two triangles have the same sidelengths, then they are congruent. We call this the SSS (Side Side Side)? condition for congruence.
Conventions to Express Congruence
The two triangles given below are congruent. How can these two triangles be superimposed? Which vertices of \(\triangle XYZ\) and \(\triangle ABC\) should we overlap?
\(\triangle ABC\) and \(\triangle XYZ\) — tick marks show corresponding equal sides: AB = XY (single), BC = YZ (double), AC = XZ (triple)
Overlapping Vertex A over Vertex X, Vertex B over Vertex Y, and Vertex C over Vertex Z will ensure that equal sides overlap, making the triangles fit exactly over each other.
The fact that these triangles are congruent shows that their respective angles are equal:
\(\angle A = \angle X, \quad \angle B = \angle Y, \quad \angle C = \angle Z\)
When two triangles are congruent, there are corresponding? vertices, sides, and angles that fit exactly over each other when the triangles are superimposed:
Corresponding Vertices
A and X, B and Y, C and Z
Corresponding Sides
AB and XY, BC and YZ, AC and XZ
Corresponding Angles
\(\angle A\) and \(\angle X\), \(\angle B\) and \(\angle Y\), \(\angle C\) and \(\angle Z\)
To capture this relation, the congruence is written as:
\(\triangle ABC \cong \triangle XYZ\)
Congruence notation: the 1st vertex in ΔABC corresponds to the 1st vertex in ΔXYZ, and so on
Important — Vertex Order Matters
By writing \(\triangle ABC \cong \triangle XYZ\), we mean that the first vertex in the name of \(\triangle ABC\) corresponds to the first vertex in the name of \(\triangle XYZ\), the second to the second, and the third to the third.By this convention, it is incorrect to write for these two triangles that \(\triangle ACB \cong \triangle XYZ\).
However, another correct way of saying it is \(\triangle ACB \cong \triangle XZY\).
Example: Rectangle with a Diagonal
Can you identify a pair of congruent triangles below? Why are they congruent?
Fig 1.1 — Rectangle ABCD with diagonal BD
Consider \(\triangle ABD\) and \(\triangle CDB\). Since ABCD is a rectangle, we have:
- AB = CD (opposite sides of a rectangle are equal)
- AD = CB (opposite sides of a rectangle are equal)
- BD is a common side
The remaining side BD is common, so the SSS condition is satisfied, confirming the congruence of the two triangles.
Identify the correct correspondence of vertices and express the congruence between the two triangles.
Verifying congruence by superimposing paper cutouts — the triangles fit exactly over each other
Verify this by superimposing paper cutouts of the triangles obtained from the rectangle ABCD (Fig. 1.1).
Are there other ways of overlapping the vertices so that the triangles fit exactly over each other?
We need corresponding sides to match: AB = CD, AD = CB, BD = BD. So the correspondence is A↔C, B↔D, D↔B. But wait — B is common to both triangles, and D is common. The correct congruence is:
\(\triangle ABD \cong \triangle CDB\)
Figure it Out (Section 1.2 — SSS)
Q1. Suppose \(\triangle HEN \cong \triangle BIG\). List all the other correct ways of expressing this congruence.
Answer: The correspondence is H↔B, E↔I, N↔G. All valid expressions must preserve this correspondence:
\(\triangle HEN \cong \triangle BIG\) ✓
\(\triangle HNE \cong \triangle BGI\) ✓
\(\triangle EHN \cong \triangle IBG\) ✓
\(\triangle ENH \cong \triangle IGB\) ✓
\(\triangle NHE \cong \triangle GBI\) ✓
\(\triangle NEH \cong \triangle GIB\) ✓
\(\triangle HEN \cong \triangle BIG\) ✓
\(\triangle HNE \cong \triangle BGI\) ✓
\(\triangle EHN \cong \triangle IBG\) ✓
\(\triangle ENH \cong \triangle IGB\) ✓
\(\triangle NHE \cong \triangle GBI\) ✓
\(\triangle NEH \cong \triangle GIB\) ✓
Q2. Determine whether the triangles are congruent. If yes, express the congruence.
Answer: Let us match the sides:
RE = 3.5 cm = IA = 3.5 cm ✓
ED = 5 cm = MA = 5 cm ✓
RD = 6 cm = MI = 6 cm ✓
All three pairs of sides are equal (SSS condition). The triangles are congruent.
Correspondence: R↔A (not I!), let's check: RE = 3.5 matches AI = 3.5? Actually IA = 3.5. So R↔I? RE=3.5, and IA=3.5, so R↔I. ED=5 and AM=5, so E↔A, D↔M.
\(\triangle RED \cong \triangle IAM\)
RE = 3.5 cm = IA = 3.5 cm ✓
ED = 5 cm = MA = 5 cm ✓
RD = 6 cm = MI = 6 cm ✓
All three pairs of sides are equal (SSS condition). The triangles are congruent.
Correspondence: R↔A (not I!), let's check: RE = 3.5 matches AI = 3.5? Actually IA = 3.5. So R↔I? RE=3.5, and IA=3.5, so R↔I. ED=5 and AM=5, so E↔A, D↔M.
\(\triangle RED \cong \triangle IAM\)
Q3. In the figure below, AB = AD and CB = CD. Can you identify any pair of congruent triangles? If yes, explain why they are congruent. Does AC divide \(\angle BAD\) and \(\angle BCD\) into two equal parts?
Answer: In \(\triangle ABC\) and \(\triangle ADC\):
AB = AD (given, single tick marks) ✓
CB = CD (given, double tick marks) ✓
AC = AC (common side) ✓
By SSS condition, \(\triangle ABC \cong \triangle ADC\).
Since corresponding angles are equal: \(\angle BAC = \angle DAC\), so AC bisects \(\angle BAD\). Similarly, \(\angle BCA = \angle DCA\), so AC bisects \(\angle BCD\). Yes, AC divides both angles into two equal parts.
AB = AD (given, single tick marks) ✓
CB = CD (given, double tick marks) ✓
AC = AC (common side) ✓
By SSS condition, \(\triangle ABC \cong \triangle ADC\).
Since corresponding angles are equal: \(\angle BAC = \angle DAC\), so AC bisects \(\angle BAD\). Similarly, \(\angle BCA = \angle DCA\), so AC bisects \(\angle BCD\). Yes, AC divides both angles into two equal parts.
Q4. In the figure below, are \(\triangle DFE\) and \(\triangle GED\) congruent to each other? It is given that DF = DG and FE = GE.
Answer: In \(\triangle DFE\) and \(\triangle DGE\):
DF = DG (given, single tick marks) ✓
FE = GE (given, double tick marks) ✓
DE = DE (common side) ✓
By SSS condition, \(\triangle DFE \cong \triangle DGE\).
Yes, the triangles are congruent.
DF = DG (given, single tick marks) ✓
FE = GE (given, double tick marks) ✓
DE = DE (common side) ✓
By SSS condition, \(\triangle DFE \cong \triangle DGE\).
Yes, the triangles are congruent.
Activity: Exploring Congruence with Paper Cutouts
L3 Apply
Materials needed: Tracing paper, ruler, protractor, scissors
Predict: If two triangles have all three sides equal, will they always be congruent? Can you think of a case where SSS might fail?
- Draw a triangle with sides 5 cm, 7 cm, and 9 cm on a sheet of paper.
- On another sheet, draw another triangle with the same three side lengths.
- Cut out both triangles carefully.
- Try to superimpose one triangle on the other. You may need to flip one triangle.
- Observe: Do they fit exactly? Measure the angles of both triangles.
Observe: The two triangles fit exactly, one over the other (possibly after flipping). All corresponding angles are equal.
Explain: The SSS condition guarantees congruence. When all three sidelengths match, there is only one possible triangle shape (up to reflection). This is because the three sides uniquely determine the triangle — there is no flexibility in the angles once all sides are fixed.
Competency-Based Questions
Scenario: Rahul is designing a kite. He has two sticks of lengths 30 cm and 50 cm that cross at right angles. He joins the endpoints to form a quadrilateral ABCD where A is the top, C is the bottom, B is the left, and D is the right. The horizontal stick BD = 50 cm crosses the vertical stick AC at point O, where AO = 15 cm and OC = 35 cm. The sticks are perpendicular.
Q1. Are the triangles \(\triangle AOB\) and \(\triangle AOD\) congruent? Which condition applies?
L3 Apply
Answer: (a) Yes, by SSS condition. AO is common, OB = OD (each is half of 50 cm = 25 cm since the sticks cross at the centre), and AB = AD (both can be found using Pythagoras: \(\sqrt{15^2 + 25^2}\)). So all three sides match.
Q2. Rahul says \(\triangle BOC \cong \triangle DOC\). Analyse whether this is correct and identify the corresponding vertices.
L4 Analyse
Answer: Yes, this is correct. OC is common, OB = OD = 25 cm, and BC = DC (by Pythagoras: \(\sqrt{35^2 + 25^2}\)). The correspondence is B↔D, O↔O, C↔C. So \(\triangle BOC \cong \triangle DOC\) by SSS.
Q3. Is it correct to write \(\triangle ABC \cong \triangle ADC\)? Justify your answer by listing corresponding sides.
L5 Evaluate
Answer: Yes, it is correct. Corresponding sides: AB = AD, BC = DC, AC = AC (common). The correspondence A↔A, B↔D, C↔C is valid. All three pairs of sides are equal, so \(\triangle ABC \cong \triangle ADC\) by SSS.
Q4. If Rahul changes the design so that the sticks no longer cross at right angles (but still bisect each other), would \(\triangle AOB\) and \(\triangle COD\) still be congruent? Create a logical argument.
L6 Create
Answer: If the sticks bisect each other (AO = OC and BO = OD), then in \(\triangle AOB\) and \(\triangle COD\): AO = OC, BO = OD, and \(\angle AOB = \angle COD\) (vertically opposite angles). This gives us SAS congruence (which we will study next). However, using only SSS: we need AB = CD. Since the sticks bisect each other and the angle between them is the same for both triangles, AB will indeed equal CD. So yes, the congruence holds even without the right angle, as long as the sticks bisect each other.
Assertion–Reason Questions
Assertion (A): If \(\triangle PQR \cong \triangle XYZ\), then PQ = XY, QR = YZ, and PR = XZ.
Reason (R): Corresponding parts of congruent triangles are equal (CPCT).
Reason (R): Corresponding parts of congruent triangles are equal (CPCT).
Answer: (a) — Both A and R are true. The reason (CPCT) directly explains why the corresponding sides are equal when triangles are congruent.
Assertion (A): Two triangles with the same perimeter are always congruent.
Reason (R): The SSS condition states that if three sides of one triangle are equal to three sides of another, the triangles are congruent.
Reason (R): The SSS condition states that if three sides of one triangle are equal to three sides of another, the triangles are congruent.
Answer: (d) — A is false: two triangles can have the same perimeter (e.g., sides 3, 4, 5 and sides 2, 5, 5 — both have perimeter 12) but different shapes and sizes, so they are not congruent. R is true: the SSS condition is a valid criterion for congruence.
Assertion (A): In rectangle ABCD, \(\triangle ABD \cong \triangle CDB\).
Reason (R): In a rectangle, opposite sides are equal and the diagonal is common to both triangles.
Reason (R): In a rectangle, opposite sides are equal and the diagonal is common to both triangles.
Answer: (a) — Both are true. In rectangle ABCD: AB = CD, AD = CB, and BD is common. So by SSS, \(\triangle ABD \cong \triangle CDB\). The reason correctly explains why (opposite sides equal + common diagonal gives three matching sides).
Frequently Asked Questions
What is the SSS congruence condition?
The SSS (Side-Side-Side) congruence condition states that two triangles are congruent if all three pairs of corresponding sides are equal. If AB equals PQ, BC equals QR and CA equals RP, then triangle ABC is congruent to triangle PQR. No angle measurement is needed. NCERT Class 7 Part II Chapter 1 covers this criterion.
What does congruence mean in geometry?
Congruence means two figures have exactly the same shape and size. When placed on top of each other, they match perfectly. Congruent triangles have equal corresponding sides and equal corresponding angles. The symbol for congruence is a special equals sign. NCERT Class 7 Ganita Prakash Part II Chapter 1 introduces this concept.
How do you identify congruent triangles using SSS?
To use SSS, measure or calculate all three sides of both triangles. If you can match each side of one triangle with an equal side of the other such that all three pairs match, the triangles are congruent by SSS. Write the correspondence correctly: if AB=PQ, BC=QR, CA=RP, then triangle ABC is congruent to triangle PQR.
What is the correspondence in congruent triangles?
Correspondence identifies which vertices of one triangle match which vertices of the other. In triangle ABC congruent to triangle PQR, A corresponds to P, B to Q and C to R. This means angle A equals angle P, angle B equals angle Q, and angle C equals angle R. Correct correspondence is essential in NCERT Class 7 problems.
Can two triangles with equal perimeters be congruent?
Not necessarily. Two triangles can have equal perimeters but different side lengths and shapes. For example, a triangle with sides 3, 4, 5 and another with sides 4, 4, 4 both have perimeter 12 but are not congruent. The SSS condition requires each individual side to match, not just the total.
Frequently Asked Questions — Chapter 1
What is Congruence of Triangles and SSS Condition in NCERT Class 7 Mathematics?
Congruence of Triangles and SSS Condition is a key concept covered in NCERT Class 7 Mathematics, Chapter 1: Chapter 1. This lesson builds the student's foundation in the chapter by explaining the core ideas with worked examples, definitions, and step-by-step methods aligned to the CBSE curriculum.
How do I solve problems on Congruence of Triangles and SSS Condition step by step?
To solve problems on Congruence of Triangles and SSS Condition, follow the NCERT method: identify the given quantities, choose the relevant formula or theorem, substitute values carefully, and simplify. Class 7 exercises gradually increase in difficulty — start with solved NCERT examples before attempting exercise questions, and always verify your answer by substitution or diagram.
What are the most important formulas for Chapter 1: Chapter 1?
The essential formulas of Chapter 1 (Chapter 1) are listed in the chapter summary and highlighted throughout the lesson in formula boxes. Memorise them and practise at least 2–3 problems per formula. CBSE board exams frequently test direct application as well as combined use of multiple formulas from this chapter.
Is Congruence of Triangles and SSS Condition important for the Class 7 board exam?
Congruence of Triangles and SSS Condition is part of the NCERT Class 7 Mathematics syllabus and appears in CBSE board exams. Questions typically include short-answer, long-answer, and competency-based items. Review the NCERT examples, exercise questions, and previous-year board problems on this topic to prepare confidently.
What mistakes should students avoid in Congruence of Triangles and SSS Condition?
Common mistakes in Congruence of Triangles and SSS Condition include skipping steps, misapplying formulas, sign errors, and losing track of units. Write each step clearly, double-check algebraic manipulations, and re-read the question after solving to verify that your answer matches what was asked.
Where can I find more NCERT practice questions on Congruence of Triangles and SSS Condition?
End-of-chapter NCERT exercises for Congruence of Triangles and SSS Condition cover all difficulty levels tested in CBSE exams. After completing them, try the examples again without looking at the solutions, attempt the NCERT Exemplar questions for Chapter 1, and solve at least one previous-year board paper to consolidate your understanding.
AI Tutor
Mathematics Class 7 — Ganita Prakash-II
Ready