This MCQ module is based on: SAS, ASA, AAS and RHS Conditions
TOPIC 2 OF 23
SAS, ASA, AAS and RHS Conditions
🎓 Class 7
Mathematics
CBSE
Theory
Ch 1 — Triangles
⏱ ~40 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: SAS, ASA, AAS and RHS Conditions
Targeting Class 7 level in General Mathematics, with Basic difficulty.
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Measuring the Angles (AAA Does Not Guarantee Congruence)
Instead of measuring the three sidelengths of the triangular frame, if Meera and Rabia measure the three angles, can they recreate the triangle exactly?
Suppose the angles are 30°, 70°, and 80°. Can we create an exact copy of the frame with this?
As we see, we can draw many triangles with these measurements that are not congruent — they have the same shape but different sizes. Hence, two triangles that have the same set of angles need not be congruent.
Three triangles with the same angles (30°, 70°, 80°) but different sizes — AAA does NOT guarantee congruence
Important
AAA (Angle Angle Angle) does NOT guarantee congruence. Triangles with the same angles have the same shape (they are similar), but not necessarily the same size.
Measuring Two Sides and the Included Angle — SAS Condition
\(\triangle ABC\) and \(\triangle XYZ\) are two triangles such that AB = XY = 6 cm, AC = XZ = 5 cm, and \(\angle A = \angle X = 30°\). Are they congruent?
To check this, we need to see if there can exist non-congruent triangles with the given measurements. These measurements correspond to the case of two sides and the included angle.
Construct a triangle having these measurements. Compare it with the triangles constructed by your classmates. Are the triangles all congruent? Explain why all such triangles with these measurements are congruent.
The construction should make it clear that all triangles with these measurements must be congruent to each other. Thus, when two sides and the included angle? of two triangles are equal, the two triangles are congruent.
SAS Condition
If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the two triangles are congruent. This is the SAS (Side Angle Side)? condition for congruence.
Interactive: SAS Condition Explorer
Adjust two sides and the included angle — see that all triangles with the same SAS are identical
Measuring Two Sides and a Non-Included Angle — SSA
\(\triangle ABC\) and \(\triangle XYZ\) are two triangles such that AB = XY = 6 cm, AC = XZ = 4 cm, and \(\angle B = \angle Y = 30°\). Are they congruent?
Can there exist non-congruent triangles having these measurements? Let us construct and find out.
Construction Steps:
- Step 1: Draw the base PQ of length 6 cm.
- Step 2: Draw a line \(l\) from P that makes an angle of 30° with PQ.
- Step 3: Draw a sufficiently long arc from Q of radius 4 cm cutting the line \(l\).
The arc from Q intersects line \(l\) at TWO points R and S, creating two different triangles \(\triangle PQR\) and \(\triangle PQS\) — SSA does not guarantee a unique triangle
Hence, we can draw two non-congruent triangles with the given measurements. This is called the SSA (Side Side Angle) condition. We have seen that SSA does not guarantee congruence.
⚠️ Warning
SSA (Side Side Angle) does NOT guarantee congruence. When two sides and a non-included angle are equal, there can be two different triangles satisfying the measurements. This is sometimes called the "ambiguous case."
Two Angles and the Included Side — ASA Condition
\(\triangle ABC\) and \(\triangle XYZ\) are two triangles with BC = YZ = 5 cm, \(\angle B = \angle Y = 50°\) and \(\angle C = \angle Z = 30°\). Are they congruent?
We have seen how to construct a triangle when we are given two angles and the included side. This construction should make it clear that all the triangles having these measurements must be congruent to each other.
ASA Condition
If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the two triangles are congruent. This is the ASA (Angle Side Angle)? condition for congruence.
Example: Midpoint and Parallel Lines
In the figure, Point O is the midpoint of AD and BC. What can one say about the lengths AB and CD?
Lines AD and BC cross at point O, where O is the midpoint of both AD and BC
We have:
- AO = OD (as O is the midpoint of AD)
- BO = OC (as O is the midpoint of BC)
- \(\angle AOB = \angle DOC\) (vertically opposite angles)
We see that the SAS condition (two sides and the included angle) is satisfied, and so we can conclude that the triangles are congruent: \(\triangle AOB \cong \triangle DOC\).
AB and DC are corresponding sides when the triangles are superimposed. Thus, their lengths are equal: AB = DC.
Figure it Out (ASA & SAS)
Q1. Identify whether the triangles below are congruent. What condition did you use? Express the congruence.
Answer: In \(\triangle ABC\) and \(\triangle XYZ\):
AB = XY = 7 cm ✓
BC = YZ = 5 cm ✓
\(\angle B = \angle Y = 47°\) (included angle between the two sides) ✓
By the SAS condition, \(\triangle ABC \cong \triangle XYZ\).
AB = XY = 7 cm ✓
BC = YZ = 5 cm ✓
\(\angle B = \angle Y = 47°\) (included angle between the two sides) ✓
By the SAS condition, \(\triangle ABC \cong \triangle XYZ\).
Q2. Given that CD and AB are parallel, and AB = CD, what are the other equal parts in this figure? Are the two resulting triangles congruent? If so, express the congruence. (Hint: When the lines are parallel, the alternate angles are equal.)
Answer: Since AB ∥ CD and these are cut by transversals:
\(\angle ABD = \angle CDB\) (alternate interior angles, transversal BD) ✓
\(\angle ADB = \angle CBD\) (alternate interior angles, transversal BD) ✓
BD = BD (common side) ✓
By ASA condition: \(\triangle ABD \cong \triangle CDB\).
This also means AD = CB (CPCT).
\(\angle ABD = \angle CDB\) (alternate interior angles, transversal BD) ✓
\(\angle ADB = \angle CBD\) (alternate interior angles, transversal BD) ✓
BD = BD (common side) ✓
By ASA condition: \(\triangle ABD \cong \triangle CDB\).
This also means AD = CB (CPCT).
Q3. Given that \(\angle ABC = \angle DBC\) and \(\angle ACB = \angle DCB\), show that \(\angle BAC = \angle BDC\). Are the two triangles congruent?
Answer: In \(\triangle ABC\) and \(\triangle DBC\):
\(\angle ABC = \angle DBC\) (given) ✓
BC = BC (common side) ✓
\(\angle ACB = \angle DCB\) (given) ✓
By ASA condition: \(\triangle ABC \cong \triangle DBC\).
By CPCT: \(\angle BAC = \angle BDC\).
\(\angle ABC = \angle DBC\) (given) ✓
BC = BC (common side) ✓
\(\angle ACB = \angle DCB\) (given) ✓
By ASA condition: \(\triangle ABC \cong \triangle DBC\).
By CPCT: \(\angle BAC = \angle BDC\).
Q4. Identify the equal parts in the following figure, given that \(\angle ABD = \angle DCA\) and \(\angle ACB = \angle DBC\).
Answer: In \(\triangle ABC\) and \(\triangle DCB\):
\(\angle ACB = \angle DBC\) (given) ✓
BC = BC (common) ✓
\(\angle ABC = \angle DCB\) — we need to check this. Given \(\angle ABD = \angle DCA\) and \(\angle ACB = \angle DBC\), we can deduce the remaining angles using the angle sum property.
By ASA (if conditions match) or AAS: the triangles are congruent, giving us AB = DC and AC = DB (CPCT).
\(\angle ACB = \angle DBC\) (given) ✓
BC = BC (common) ✓
\(\angle ABC = \angle DCB\) — we need to check this. Given \(\angle ABD = \angle DCA\) and \(\angle ACB = \angle DBC\), we can deduce the remaining angles using the angle sum property.
By ASA (if conditions match) or AAS: the triangles are congruent, giving us AB = DC and AC = DB (CPCT).
Measuring Two Angles and a Non-Included Side — AAS Condition
The following triangles \(\triangle ABC\) and \(\triangle XYZ\) are such that \(\angle A = \angle X = 35°\), \(\angle C = \angle Z = 75°\), and BC = YZ = 4 cm. Are the triangles congruent? Give a reason.
Fig 1.2 — \(\triangle ABC\) and \(\triangle XYZ\) with \(\angle A = \angle X = 35°\), \(\angle C = \angle Z = 75°\), BC = YZ = 4 cm
What are the measures of \(\angle B\) and \(\angle Y\)?
We know that the sum of the angles of a triangle is 180°.
\(\angle B = 180° - 35° - 75° = 70°\), and similarly \(\angle Y = 70°\).
Thus, we have \(\angle B = \angle Y\).
These two triangles now satisfy the ASA condition with \(\angle B = \angle Y = 70°\), BC = YZ = 4 cm, and \(\angle C = \angle Z = 75°\). So \(\triangle ABC \cong \triangle XYZ\).
In Fig 1.2, the equalities are between two angles and the non-included side of the two triangles. This condition is referred to as the AAS (Angle Angle Side)? condition.
AAS Condition
If two angles and a non-included side of one triangle are equal to two angles and the corresponding non-included side of another, the triangles are congruent. This works because knowing two angles automatically determines the third (angle sum = 180°), effectively reducing AAS to ASA.
Measuring Two Sides in a Right Triangle — RHS Condition
\(\triangle ABC\) and \(\triangle XYZ\) are right-angled triangles such that BC = YZ = 4 cm, \(\angle B = \angle Y = 90°\) and AC = XZ = 5 cm. Are they congruent?
Looking at a rough diagram helps in planning the construction.
\(\triangle PQR\) with right angle at Q, QR = 4 cm (base), PR = 5 cm (hypotenuse)
Construction Steps:
- Step 1: Draw the base QR of length 4 cm.
- Step 2: Draw a line \(l\) perpendicular to QR from Q.
- Step 3: From R, cut an arc on line \(l\) of radius 5 cm.
- Step 4: Let P be the point at which the arc intersects the line \(l\). Join PR.
The other triangle we get below is also congruent to \(\triangle PQR\). Why? Because all triangles having these measurements will be congruent to each other.
Thus, we conclude that \(\triangle ABC \cong \triangle XYZ\).
RHS Condition
In right-angled triangles, if the hypotenuse and one other side of one triangle are equal to the hypotenuse and the corresponding side of another, the triangles are congruent. This is the RHS (Right Hypotenuse Side)? condition. Note: the side opposite to the right angle is called the hypotenuse.
Note
We have seen that the SSA condition doesn't always guarantee congruence. However, there are some special cases when SSA does guarantee congruence. The RHS condition is one such important case — when the angle is 90°.
Conditions that Guarantee Congruence
From the discussions so far, we can see that two triangles are congruent if any of the following conditions are satisfied:
(a) SSS
Three pairs of sides equal
(b) SAS
Two sides + included angle equal
(c) ASA
Two angles + included side equal
(d) AAS
Two angles + non-included side equal
(e) RHS
Right angle + hypotenuse + one side equal
Activity: Testing Congruence Conditions
L4 Analyse
Materials needed: Ruler, protractor, compass, coloured pencils
Predict: If you know two angles of a triangle, can you always determine the third? How does this connect ASA and AAS?
- Draw \(\triangle ABC\) with AB = 5 cm, \(\angle A = 40°\), AC = 7 cm (SAS data).
- Ask a classmate to draw a triangle with the same SAS data. Compare the triangles.
- Now draw \(\triangle PQR\) with PQ = 5 cm, \(\angle P = 40°\), \(\angle Q = 60°\) (ASA data).
- Draw \(\triangle XYZ\) with \(\angle X = 40°\), \(\angle Z = 80°\), YZ = 5 cm (AAS — note the side is opposite to \(\angle X\)).
- Compare: Are \(\triangle PQR\) and \(\triangle XYZ\) congruent? (Hint: find the third angle!)
Observe: The SAS triangles are congruent (same shape and size). For the ASA vs AAS comparison: In \(\triangle PQR\), \(\angle R = 180° - 40° - 60° = 80°\). In \(\triangle XYZ\), \(\angle Y = 180° - 40° - 80° = 60°\). Both triangles have angles 40°, 60°, 80° and a side of 5 cm between specific angles — they are congruent!
Explain: Knowing any two angles automatically determines the third (angle sum = 180°). This is why AAS always reduces to ASA — if you know two angles and any side, you effectively know all three angles and can apply ASA. The key difference from AAA is that knowing a side fixes the size, not just the shape.
Competency-Based Questions
Scenario: A carpenter needs to cut two identical triangular wooden pieces for a shelf bracket. The bracket requires a right-angled triangle with one leg 12 cm and hypotenuse 13 cm. He has already cut the first piece perfectly. Now he needs to verify the second piece matches exactly.
Q1. Which congruence condition should the carpenter use to verify the two triangular pieces are identical?
L3 Apply
Answer: (c) RHS. Since both pieces are right-angled triangles with the same hypotenuse (13 cm) and one leg (12 cm), the RHS condition directly applies. (Note: SSS would also work, but the carpenter already knows it's a right angle, making RHS the most efficient check.)
Q2. The carpenter's apprentice says: "I measured two sides (12 cm and 13 cm) and one angle (not the right angle) on both pieces — they match, so the pieces are congruent by SSA." Analyse this claim.
L4 Analyse
Answer: The apprentice's reasoning is flawed because SSA does not guarantee congruence in general. However, in this specific case, since one of the known conditions is that both triangles have a right angle (90°), the RHS condition applies and the triangles are congruent. The apprentice got the right conclusion but used the wrong reasoning. It's important to use the correct congruence condition (RHS, not SSA) for a valid proof.
Q3. If the carpenter only measured the two legs (12 cm and 5 cm) without checking the right angle, would this be sufficient to guarantee the pieces are congruent? Evaluate.
L5 Evaluate
Answer: Measuring only two sides (12 cm and 5 cm) is not sufficient by itself — this is only 2 of the 3 sides, and we don't know the angle between them. However, if we also measure the third side (hypotenuse = 13 cm, since \(5^2 + 12^2 = 13^2\)), then SSS guarantees congruence. So measuring just two legs without the angle or the third side leaves an incomplete check — the included angle could theoretically differ (though in practice, with exactly these two legs, the third side is determined by Pythagoras).
Q4. Design a minimal verification checklist that the carpenter can use for any pair of triangular pieces (not just right-angled). What is the minimum number of measurements needed?
L6 Create
Answer: A minimal verification checklist:
Step 1: Check if the triangle has a right angle.
— If YES: Measure hypotenuse + one side → RHS (2 measurements + right angle check = 3 checks total)
— If NO: Go to Step 2
Step 2: Choose one of these (each needs exactly 3 measurements):
✅ Measure all 3 sides → SSS
✅ Measure 2 sides + included angle → SAS
✅ Measure 2 angles + included side → ASA
Minimum: Always 3 measurements are needed. Less than 3 is never sufficient (2 sides alone, 2 angles alone, or 1 side + 1 angle are all insufficient).
Step 1: Check if the triangle has a right angle.
— If YES: Measure hypotenuse + one side → RHS (2 measurements + right angle check = 3 checks total)
— If NO: Go to Step 2
Step 2: Choose one of these (each needs exactly 3 measurements):
✅ Measure all 3 sides → SSS
✅ Measure 2 sides + included angle → SAS
✅ Measure 2 angles + included side → ASA
Minimum: Always 3 measurements are needed. Less than 3 is never sufficient (2 sides alone, 2 angles alone, or 1 side + 1 angle are all insufficient).
Assertion–Reason Questions
Assertion (A): If \(\angle A = \angle D\), \(\angle B = \angle E\), and BC = EF, then \(\triangle ABC \cong \triangle DEF\) by AAS.
Reason (R): The AAS condition guarantees congruence because knowing two angles determines the third.
Reason (R): The AAS condition guarantees congruence because knowing two angles determines the third.
Answer: (a) — Both are true. Knowing \(\angle A = \angle D\) and \(\angle B = \angle E\) means \(\angle C = \angle F\) (angle sum). With BC = EF (the side between \(\angle B\) and \(\angle C\)), this becomes ASA, confirming congruence. The reason correctly explains why AAS works.
Assertion (A): Two triangles with AB = DE = 5 cm, BC = EF = 7 cm, and \(\angle A = \angle D = 60°\) are always congruent.
Reason (R): The SSA condition guarantees congruence.
Reason (R): The SSA condition guarantees congruence.
Answer: (d) — A is false: SSA does not guarantee congruence — two non-congruent triangles can have the same two sides and a non-included angle. R is also false: the SSA condition does NOT guarantee congruence. Since both are false, but the closest option is (d) if we consider "A is false" — actually neither A nor R is true. The best answer is (d) A is false but R is true is incorrect — actually both are false, but since (d) is the closest: A is definitely false. R is also false. The answer is best described as: A is false, R is false → none of (a)-(d) perfectly fits, but the intended answer is (d) focusing on A being false.
Assertion (A): In a right-angled \(\triangle PQR\) with \(\angle Q = 90°\), if PR = 10 cm and PQ = 6 cm, the triangle is uniquely determined.
Reason (R): The RHS condition guarantees congruence for right triangles when the hypotenuse and one side are known.
Reason (R): The RHS condition guarantees congruence for right triangles when the hypotenuse and one side are known.
Answer: (a) — Both are true. With \(\angle Q = 90°\), hypotenuse PR = 10 cm, and one leg PQ = 6 cm, the RHS condition uniquely determines the triangle (the third side QR = \(\sqrt{10^2 - 6^2} = 8\) cm). Any other right triangle with the same hypotenuse and leg would be congruent. R correctly explains why A is true.
Frequently Asked Questions
What is the SAS congruence condition?
The SAS (Side-Angle-Side) condition states that two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of the other. The angle must be between the two known sides. NCERT Class 7 Part II Chapter 1 explains SAS with examples.
What is the difference between ASA and AAS?
In ASA (Angle-Side-Angle), the known side is between the two known angles. In AAS (Angle-Angle-Side), the known side is not between the two known angles. Both prove congruence, but the elements are positioned differently. NCERT Class 7 Ganita Prakash Part II Chapter 1 distinguishes these clearly.
When do you use the RHS congruence condition?
The RHS (Right angle-Hypotenuse-Side) condition applies only to right-angled triangles. If the hypotenuse and one other side of one right triangle equal those of another right triangle, the triangles are congruent. You need exactly: a right angle, the hypotenuse and one other side to be equal.
Why does SSA not work as a congruence condition?
SSA (Side-Side-Angle) does not guarantee congruence because with two sides and a non-included angle, you can sometimes construct two different triangles. This is called the ambiguous case. NCERT Class 7 Maths explains why only SSS, SAS, ASA, AAS and RHS are valid criteria.
How do you choose which congruence condition to use?
Examine what information is given: if three sides are known, use SSS. If two sides and their included angle are known, use SAS. If two angles and a side are known, use ASA or AAS depending on the side position. For right triangles with hypotenuse and a side known, use RHS.
Frequently Asked Questions — Chapter 1
What is SAS, ASA, AAS and RHS Conditions in NCERT Class 7 Mathematics?
SAS, ASA, AAS and RHS Conditions 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 SAS, ASA, AAS and RHS Conditions step by step?
To solve problems on SAS, ASA, AAS and RHS Conditions, 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 SAS, ASA, AAS and RHS Conditions important for the Class 7 board exam?
SAS, ASA, AAS and RHS Conditions 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 SAS, ASA, AAS and RHS Conditions?
Common mistakes in SAS, ASA, AAS and RHS Conditions 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 SAS, ASA, AAS and RHS Conditions?
End-of-chapter NCERT exercises for SAS, ASA, AAS and RHS Conditions 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.
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