1
Congruent triangles have the same —
💡 Hint:
One covers the other.
Explanation:
Both shape and size.
Class 9 · Maths · Chapter 7 · बिहार बोर्ड (BSEB)CBSE · NCERT 2026-27
Triangles
Triangles
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1. Read — Congruence · SAS · ASA · AAS · Exercise 7.1 · isosceles · Exercise 7.2 · SSS · RHS · Exercise 7.3 · longer side · Exercise 7.4 · Exercise 7.5, diagram, worked example, board tip
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- 1 Congruent triangles and corresponding parts
- 2 SAS, ASA and AAS — Exercise 7.1
- 3 Isosceles triangles — Exercise 7.2
- 4 SSS and RHS — Exercise 7.3
- 5 The longer side and the larger angle
- 6 The triangle inequality — Exercise 7.4
- 7 Optional Exercise 7.5
- Chapter winner — every lesson at mastery ★
🎬 Shorts
This chapter has no Short on the channel yet, so ten empty cards are not shown. The notes, diagrams and checks are complete.
📄 PDF download — notes + answer key
Congruent triangles and corresponding parts
सर्वांगसम त्रिभुज और संगत भाग · Textbook 7.2 · correspondence · CPCT
The correspondence decides which part matchesNotes
Two triangles are congruent when one can be placed on the other and cover it fully. Shape and size are both the same. Equal angles alone are not enough. A side length must match as well.
The correspondence ΔABC ↔ ΔPQR means A with P, B with Q, and C with R. Then AB = PQ, BC = QR, CA = RP, and the angles match in that same order. Writing this after congruence is CPCT: corresponding parts are equal.
The book’s cut-out — placing one triangle on anotherActivity
The book asks you to cut copies of a triangle and place them on another. Some copies cover it after a turn, and one does not. This is not a numbered activity. A copy that covers the triangle is congruent. A copy that matches only the angles, and not a side, is not congruent.
Worked exampleExample
Question: ΔABC ≅ ΔDEF with A↔D, B↔E, C↔F. AB = 6 cm. Write the match of DE and of ∠B.
Rule: CPCT, corresponding parts are equal.
Substitution: DE = AB = 6 cm. The match of ∠B is ∠E, so ∠B = ∠E.
10-second revision
- Congruent: both shape and size
- Correspondence fixes the corresponding parts
- Write CPCT after the congruence
Board tip · BSEBBoard tip
Keep the order of the correspondence that is written in the congruence sign. If ABC ≅ DEF, the match of AB is DE, not DF.
Board tip · CBSEBoard tip
Do not write only “they are equal”. Say in one line which part corresponds to which.
Check your understandingall correct = mastery ★
2
If all three angles are equal, the triangles must be congruent.
Explanation:
False — equal angles can leave a side longer. One side must match as well.
3
If ΔABC ≅ ΔPQR, the part corresponding to BC is ______.
💡 Hint:
B matches Q and C matches R.
Explanation:
QR.
4
CPCT means —
💡 Hint:
After the correspondence.
Explanation:
Corresponding parts are equal.
5
ΔABC ≅ ΔLMN, with A↔L, B↔M, C↔N. Write three corresponding pairs.
Model answer (board format)
AB = LM, BC = MN, and ∠A = ∠L. This is CPCT.
Check your answer:
SAS, ASA and AAS — Exercise 7.1
SAS, ASA और AAS — अभ्यास 7.1 · Textbook 7.3 · Axiom 7.1 · Theorem 7.1 · Exercise 7.1
One of the equal parts must be a sideNotes
Axiom 7.1 (SAS): if two sides and the included angle are equal, the triangles are congruent. Theorem 7.1 (ASA): two angles and the included side also give congruence.
AAS looks different. Two angles and one corresponding side are equal, even if the side is not between the angles. The third angle is what remains from 180°, so the pair becomes ASA. Three equal angles with no side do not give congruence. The book shows many triangles with 40°, 50° and 90°.
Exercise 7.1 — name the rule, then CPCTActivity
Exercise 7.1 asks you to prove two triangles congruent from the given equal parts. First mark the equal angles and sides. Then write one rule from SAS, ASA or AAS. After that, use CPCT for the side or angle the question asks. Do not write SAS before you check that the angle is the included one.
Worked exampleExample
Question: In ΔABC and ΔDEF, ∠B = ∠E = 50°, ∠C = ∠F = 60°, and BC = EF = 8 cm. Name the rule.
Formula: third angle = 180° − (the two given angles). Side BC lies between the two given angles, so ASA applies.
Substitution: ∠A = 180° − (50° + 60°) = 70°. The unit is the degree. BC is the included side. ΔABC ≅ ΔDEF by ASA.
10-second revision
- SAS: included angle, Axiom 7.1
- ASA: included side, Theorem 7.1
- AAA does not give congruence
Board tip · BSEBBoard tip
In a BSEB answer give both the name and the number of the rule. The sign ≅ alone is half a reason.
Board tip · CBSEBoard tip
Use triangles of 40°, 50° and 90° with different sizes to show that three angles are not enough without a side.
Check your understandingall correct = mastery ★
1
In SAS the equal angle is —
💡 Hint:
The included angle.
Explanation:
Between the two sides.
2
ASA congruence is Theorem 7.1.
Explanation:
True. SAS is Axiom 7.1, not a theorem.
3
If the angles are 50° and 60°, the third angle is ______ degrees.
💡 Hint:
180 − 110.
Explanation:
70°.
4
Two angles of 45° and 60° match, and it is not the sides opposite them but another corresponding side that is equal. Which rule will you use, and why?
Model answer (board format)
AAS. The third angle 180° − 105° = 75° becomes equal too, so congruence follows with one side. Not SAS, because the angle is not given as the included angle.
Check your answer:
Isosceles triangles — Exercise 7.2
समद्विबाहु त्रिभुज — अभ्यास 7.2 · Textbook 7.4 · Theorem 7.2 · Theorem 7.3 · Exercise 7.2
The angle opposite an equal side is equalNotes
An isosceles triangle has two equal sides. Theorem 7.2: angles opposite the equal sides are equal. Theorem 7.3 is the converse: sides opposite equal angles are equal. The proof draws the angle bisector and uses SAS or ASA.
The vertex angle and the two base angles add to 180°. If the base angles are equal, each one is (180° − vertex) / 2. In an equilateral triangle all three sides are equal, so each angle is 60°.
Exercise 7.2 — bisectors and equal anglesActivity
Exercise 7.2 gives angle bisectors in an isosceles triangle, equal angles, and two isosceles triangles on the same base. First write the equal parts from Theorem 7.2 or 7.3. Then use SAS or ASA on the smaller triangles. Finish with CPCT.
Worked exampleExample
Question: AB = AC and the vertex ∠A = 80°. Find the base angles.
Formula: each base angle = (180° − vertex angle) / 2. Theorem 7.2 makes them equal.
Substitution: (180° − 80°) / 2 = 100° / 2 = 50°. The unit is the degree. ∠B = ∠C = 50°.
10-second revision
- Theorem 7.2: equal sides, equal angles
- Theorem 7.3 is the converse
- Base angle = (180° − vertex) / 2
Board tip · BSEBBoard tip
Do not halve 80° and write 40°. First 180° − 80° = 100°, then half is 50°.
Board tip · CBSEBoard tip
If the converse is asked, write Theorem 7.3. Equal angles prove the sides, not the other way.
Check your understandingall correct = mastery ★
1
If AB = AC, then —
💡 Hint:
The opposite angles.
Explanation:
∠B = ∠C, Theorem 7.2.
2
Theorem 7.3 is the converse of Theorem 7.2.
Explanation:
True — sides opposite equal angles are equal.
3
If the vertex angle is 80°, each base angle is ______ degrees.
💡 Hint:
(180 − 80) / 2.
Explanation:
50°.
4
Each angle of an equilateral triangle is —
💡 Hint:
180 / 3.
Explanation:
60°.
5
∠B = ∠C. Which sides will be equal? Give the theorem number.
Model answer (board format)
The side opposite ∠B is AC and the side opposite ∠C is AB, so AC = AB. This is Theorem 7.3.
Check your answer:
6
In an isosceles triangle all three angles are always 60°.
Explanation:
False — 60° belongs to the equilateral case. In an isosceles triangle only the two base angles are equal.
SSS and RHS — Exercise 7.3
SSS और RHS — अभ्यास 7.3 · Textbook 7.5 · Theorem 7.4 · Theorem 7.5 · Exercise 7.3
All three sides, or the hypotenuse and one sideNotes
Theorem 7.4 (SSS): if all three sides are equal, the triangles are congruent. The rule works without being given an angle. Theorem 7.5 (RHS): in two right triangles, if the hypotenuse and one side are equal, the triangles are congruent. The right angle faces the hypotenuse.
In RHS the right angle is not the included angle. So do not write it as SAS. If one acute angle and the hypotenuse match, the third angle follows in a right triangle, but the named rules of this section are RHS and SSS.
Exercise 7.3 — choose SSS or RHSActivity
Exercise 7.3 often gives two isosceles triangles on the same base, or a right angle and a hypotenuse. If all three pairs are sides, use SSS. If one angle is 90° and the hypotenuse and one side are equal, use RHS. Do not turn the right angle into an ordinary side and force SSS.
Worked exampleExample
Question: In two right triangles the hypotenuse 13 cm and one side 5 cm are equal. Name the rule.
Rule: RHS, Theorem 7.5. The hypotenuse is the side opposite the right angle.
Check: 5 cm is a side and 13 cm is the hypotenuse. This pair matches in both triangles, so the triangles are congruent.
10-second revision
- SSS is the three-side rule
- RHS uses the hypotenuse and one side
- Do not call RHS by the name SAS
Board tip · BSEBBoard tip
The hypotenuse is the side opposite 90°. Do not call any long side the hypotenuse unless there is a right angle.
Board tip · CBSEBoard tip
In SSS show three pairs. Do not write two sides and leave the third as “obvious”.
Check your understandingall correct = mastery ★
1
SSS congruence is —
💡 Hint:
Three sides.
Explanation:
Theorem 7.4.
2
The RHS rule works on any triangle, and a right angle is not needed.
Explanation:
False — both triangles must be right-angled.
3
The hypotenuse is the side ______ the right angle.
💡 Hint:
Opposite 90°.
Explanation:
Opposite.
4
If the hypotenuse 13 cm and one side 5 cm are equal in two right triangles, the rule is —
💡 Hint:
Theorem 7.5.
Explanation:
RHS.
5
AB = DE = 7 cm, BC = EF = 8 cm, CA = FD = 9 cm. Which theorem gives the congruence?
Model answer (board format)
All three sides are equal, so ΔABC ≅ ΔDEF by Theorem 7.4, SSS.
Check your answer:
The longer side and the larger angle
बड़ी भुजा और बड़ा कोण · Textbook 7.6 · Theorem 7.6 · Theorem 7.7
Unequal sides, unequal opposite anglesNotes
Theorem 7.6: if two sides are unequal, the angle opposite the longer side is larger. Theorem 7.7 is the converse: the side opposite the larger angle is longer. It is proved by contradiction.
The isosceles result is different. There the sides were equal, so the angles were equal. Here the sides are unequal, so the angles are unequal too. The larger angle still cannot pass 180°.
The book’s three measures — thread, scalene, arcActivity
The book first ties side BC with pins and a thread and moves a pencil. Then it asks you to measure a scalene triangle: the longest side faces the largest angle. Then it joins several points on an arc drawn with AB as radius. All three checks are headed Activity. Do not number them 7.1 or 7.2. A measurement does not replace the theorem.
Worked exampleExample
Question: The sides are BC = 7 cm, CA = 5 cm, AB = 6 cm. Which angle is the largest?
Rule: Theorem 7.6, the longer side faces the larger angle.
Substitution: the longest side is BC = 7 cm. Opposite it is ∠A. So the largest angle is ∠A.
10-second revision
- Theorem 7.6: longer side, larger angle
- Theorem 7.7 is the converse
- The measuring checks are not numbered activities
Board tip · BSEBBoard tip
Read the angle opposite the side. Opposite BC is ∠A, not ∠B.
Board tip · CBSEBoard tip
Do not write the scalene measurement as the proof. Write that the measurement agrees with Theorem 7.6.
Check your understandingall correct = mastery ★
1
If BC is the longest side, the largest angle is —
💡 Hint:
The vertex opposite it.
Explanation:
∠A.
2
Theorem 7.7 says the side opposite the larger angle is longer.
Explanation:
True — it is the converse of 7.6.
3
Among sides 5 cm, 6 cm and 7 cm, the side opposite the largest angle is ______ cm.
💡 Hint:
The longest.
Explanation:
7 cm.
4
If ∠B > ∠C, which side is longer? Name the theorem.
Model answer (board format)
Opposite ∠B is AC and opposite ∠C is AB. AC > AB, by Theorem 7.7.
Check your answer:
The triangle inequality — Exercise 7.4
त्रिभुज असमिका — अभ्यास 7.4 · Textbook 7.6 · Theorem 7.8 · Exercise 7.4
Two sides together beat the thirdNotes
Theorem 7.8: the sum of any two sides is greater than the third side. Make three checks: a + b > c, b + c > a, and c + a > b. If one check fails, the lengths do not make a triangle.
Exercise 7.4 speaks in this inequality and in the language of the larger angle. In a right triangle the hypotenuse is the longest side, because the 90° opposite it is the largest angle. If a sum is only equal, the points fall on one line and no triangle is formed.
Exercise 7.4 — three sums and the larger angleActivity
Exercise 7.4 brings the longest side of a right triangle, a comparison of angles in two triangles, and an inequality with an angle bisector. If numbers are given, write all three sums and keep the unit cm. If there is only a figure, name Theorem 7.6 or 7.7. The optional set is not yet. That is 7.5 in the next lesson.
Worked exampleExample
Question: Do 3 cm, 4 cm and 8 cm make a triangle?
Formula: the sum of every two sides is greater than the third. Theorem 7.8.
Substitution: 3 + 4 = 7, and 7 is not greater than 8. 3 + 8 = 11 > 4, and 4 + 8 = 12 > 3. One check fails, so there is no triangle. The unit is cm.
10-second revision
- Three sums, each greater than the third
- An equal sum does not make a triangle
- The hypotenuse is the longest side, opposite the right angle
Board tip · BSEBBoard tip
The sum of the two shorter sides is the check that usually fails, but write all three checks in the answer.
Board tip · CBSEBoard tip
Do not call 7 cm “almost greater” than 8 cm. 7 > 8 is false, so there is no triangle.
Check your understandingall correct = mastery ★
1
With 3 cm, 4 cm, 8 cm a triangle —
💡 Hint:
3 + 4 = 7.
Explanation:
It is not formed, because 7 is not greater than 8.
2
A triangle is still formed if the sum of two sides equals the third.
Explanation:
False — the sum must be greater, not equal.
3
The sum of 3 cm and 4 cm is ______ cm.
💡 Hint:
Add.
Explanation:
7 cm.
4
In a right triangle the longest side is —
💡 Hint:
Opposite 90°.
Explanation:
The hypotenuse.
5
Write all three inequality checks for 5 cm, 6 cm and 7 cm.
Model answer (board format)
5 + 6 = 11 > 7, 6 + 7 = 13 > 5, and 7 + 5 = 12 > 6. All three are true, so a triangle is formed. The unit is cm.
Check your answer:
6
Theorem 7.8 is about the sum of two sides.
Explanation:
True — the sum is greater than the third side.
Optional Exercise 7.5
वैकल्पिक अभ्यास 7.5 · Textbook · Exercise 7.5 optional
Equal distance from three vertices, equal distance from three sidesNotes
The book marks Exercise 7.5 optional. The number is not dropped. The first point is equidistant from the three vertices. The second point is equidistant from the three sides.
The perpendicular bisector of AB is the path of points equidistant from A and B. The perpendicular bisector of BC does the same for B and C. Where they meet, all three vertices are at the same distance. For a point equidistant from the sides, the angle bisectors meet. The rangoli question fills matching pieces. It is not a new number.
Exercise 7.5 — the point and the rangoliActivity
First find the interior point equidistant from the three vertices. Two perpendicular bisectors are enough. Then find the point equidistant from the three sides. Use the angle bisectors. The park question uses the same language of distance. In the rangoli, fill the empty matching parts by the same rule.
Worked exampleExample
Question: The perpendicular bisectors of AB and BC meet at O. OA = 5 cm. Find OB and OC.
Rule: a point on the perpendicular bisector is equidistant from the endpoints.
Substitution: OA = OB = 5 cm, because O is on the perpendicular bisector of AB. OB = OC = 5 cm, because O is on the perpendicular bisector of BC. So OC = 5 cm. The unit is cm.
10-second revision
- Exercise 7.5 is optional, and the number stays
- Equidistant from vertices: perpendicular bisectors
- Equidistant from sides: angle bisectors
Board tip · BSEBBoard tip
Do not write the two points as one. The point for the vertices and the point for the sides are different questions.
Board tip · CBSEBoard tip
Only OA is given as 5 cm. OB and OC are also 5 cm because O lies on both perpendicular bisectors. Write that reason.
Check your understandingall correct = mastery ★
1
The point equidistant from all three vertices is found by —
💡 Hint:
Distance from the endpoints.
Explanation:
The perpendicular bisectors.
2
Exercise 7.5 is optional, so its number should be changed to 7.4.
Explanation:
False — the number stays 7.5. The book calls it optional.
3
If OA = 5 cm and O lies on both perpendicular bisectors, OC is ______ cm.
💡 Hint:
Equal distance.
Explanation:
5 cm.
4
The point equidistant from all three sides is linked to —
💡 Hint:
The angles are halved.
Explanation:
The angle bisectors.
5
Write the difference between the two interior points of Exercise 7.5 in one line each.
Model answer (board format)
The first point is equidistant from the three vertices and is the meeting of the perpendicular bisectors. The second point is equidistant from the three sides and is the meeting of the angle bisectors.
Check your answer:
❓ Full question bank — with answers and explanations — 66 questions
Pick a type. The 35 lesson checks are separate — each lesson has as many as its topic needs. All correct earns mastery ★.
BSEB modelpractice · not annual60 tried · 0 correctMultiple choiceMCQ130 tried · 0 correctTrue / falseTrue / false80 tried · 0 correctFill in the blanksFill80 tried · 0 correctMatchMatch20 tried · 0 correctAssertion–reasonAssertion–reason50 tried · 0 correctBalancingBalancing40 tried · 0 correctIdentifyIdentify20 tried · 0 correctShort and longVSA, SA, LA120 tried · 0 correctCBSE-stylenot a PYQ60 tried · 0 correct
No question is marked as a verified past paper. The BSEB set is a model for practice. CBSE items are CBSE-style, not a copy of any year’s paper.
Multiple choice
0/13Pick one option. A wrong try brings a hint.
1
Congruent triangles are —
💡 Hint:
They cover fully.
Explanation:
The same in both shape and size.
2
The angle in SAS is —
💡 Hint:
Axiom 7.1.
Explanation:
The angle between the two sides.
3
ASA is —
💡 Hint:
The included side.
Explanation:
Theorem 7.1.
4
If all three angles are equal, then —
💡 Hint:
AAA is not enough.
Explanation:
A side is also needed.
5
If AB = AC, the equal angles are —
💡 Hint:
Theorem 7.2.
Explanation:
∠B = ∠C.
6
With vertex 80°, each base angle is —
💡 Hint:
(180 − 80) / 2.
Explanation:
50°.
7
SSS is —
💡 Hint:
Three sides.
Explanation:
Theorem 7.4.
8
RHS applies to which triangles?
💡 Hint:
The hypotenuse faces the right angle.
Explanation:
Right triangles.
9
If the longest side is BC, the largest angle is —
💡 Hint:
Theorem 7.6.
Explanation:
∠A.
10
3 cm, 4 cm, 8 cm —
💡 Hint:
3 + 4 = 7.
Explanation:
No triangle.
11
If ΔABC ≅ ΔPQR, the match of AB is —
💡 Hint:
A matches P, B matches Q.
Explanation:
PQ.
12
The point equidistant from the three vertices lies on —
💡 Hint:
Exercise 7.5.
Explanation:
The perpendicular bisectors.
13
In Theorem 7.8 the sum of two sides is —
💡 Hint:
The inequality.
Explanation:
Greater than the third.
True or false
0/81
CPCT is the name of a construction done before congruence.
Explanation:
False — CPCT is the equality of corresponding parts, after congruence.
2
SAS is Axiom 7.1.
Explanation:
True.
3
In AAS the third angle becomes equal from the 180° sum.
Explanation:
True — so congruence follows with one side.
4
Theorem 7.3 proves the angles from the equal sides.
Explanation:
False — 7.3 is the converse. Equal angles give equal sides.
5
RHS can also be called SAS because 90° is the included angle.
Explanation:
False — the right angle is not treated as the included angle between the hypotenuse and the side. The name is RHS.
6
The side opposite the larger angle is longer.
Explanation:
True — Theorem 7.7.
7
5 cm, 6 cm and 7 cm do not make a triangle.
Explanation:
False — 5 + 6 = 11 > 7, and the other checks are true as well.
8
Exercise 7.5 stays numbered 7.5 in the book even though it is marked optional.
Explanation:
True.
Fill in the blanks
0/81
If ΔABC ≅ ΔDEF, the match of BC is ______.
💡 Hint:
B matches E, C matches F.
Explanation:
EF.
2
After angles 50° and 60°, the third is ______ degrees.
💡 Hint:
180 − 110.
Explanation:
70.
3
With vertex 80°, a base angle is ______ degrees.
💡 Hint:
Half of 100.
Explanation:
50.
4
The SSS theorem number is ______.
💡 Hint:
RHS is 7.5.
Explanation:
7.4.
5
The RHS theorem number is ______.
💡 Hint:
SSS is 7.4.
Explanation:
7.5.
6
Among sides 5, 6, 7 cm the largest angle is opposite the side of ______ cm.
💡 Hint:
The longest side.
Explanation:
7.
7
3 + 4 = ______ , which is not greater than 8.
💡 Hint:
The sum.
Explanation:
7.
8
If OA = 5 cm, OC is ______ cm when O is on both perpendicular bisectors.
💡 Hint:
Equal distance.
Explanation:
5.
Match
0/21
Match the rule with its number.
Column B: A. Theorem 7.5 · B. Axiom 7.1 · C. Theorem 7.1 · D. Theorem 7.4
1. SAS
2. ASA
3. SSS
4. RHS
Explanation:
SAS is Axiom 7.1, ASA is Theorem 7.1, SSS is Theorem 7.4, and RHS is Theorem 7.5.
2
Match the statement with the theorem.
Column B: A. 7.8 · B. 7.2 · C. 7.7 · D. 7.6
1. Equal sides, equal angles
2. Larger angle, longer side
3. Sum of two sides
4. Longer side, larger angle
Explanation:
7.2 is the isosceles angle, 7.7 the converse side, 7.8 the sum, and 7.6 the larger angle.
Assertion–reason
0/5Check both statements, then see whether the reason explains the assertion.
1
Assertion (A): In SAS the angle lies between the two sides.
Reason (R): The included angle is the one formed by the two equal sides.
Explanation:
Both are true and R explains A.
2
Assertion (A): Angles opposite equal sides are equal.
Reason (R): The sum of two sides is greater than the third.
Explanation:
Both are true, but R does not explain the isosceles angles.
3
Assertion (A): RHS is a rule for two right triangles.
Reason (R): RHS does not need a right angle.
Explanation:
A is true. R is false.
4
Assertion (A): 3 cm, 4 cm and 8 cm make a triangle.
Reason (R): The sum of two sides must be greater than the third.
Explanation:
A is false because 7 is not greater than 8. R is true, Theorem 7.8.
5
Assertion (A): The larger angle is opposite the longer side.
Reason (R): With unequal sides the opposite angles are unequal, and the larger angle faces the longer side.
Explanation:
Both are true and R explains Theorem 7.6.
Coefficient practice
0/4These are not results of this maths chapter. They are only coefficient practice for H2 + O2 → H2O.
1
This is coefficient practice, not a result of this maths chapter. Fill coefficients for H2 + O2 making H2O (blank = 1). This is not a triangle result.
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
2
This is coefficient practice, not a result of this maths chapter. Fill coefficients for H2 + O2 making H2O (blank = 1). Match the hydrogen count.
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
3
This is coefficient practice, not a result of this maths chapter. Fill coefficients for H2 + O2 making H2O (blank = 1). Match the oxygen count.
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
4
This is coefficient practice, not a result of this maths chapter. Fill coefficients for H2 + O2 making H2O (blank = 1). A blank coefficient is 1.
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
Classify
0/21
Place each rule in its type.
SAS
ASA
SSS
RHS
Explanation:
Only SAS is Axiom 7.1. ASA, SSS and RHS are theorems.
2
Place each statement on the right theorem.
Equal sides, equal angles
Longer side, larger angle
Sum of two sides is greater
Two equal sides, two equal angles
Explanation:
The isosceles fact belongs with 7.2. The longer side is 7.6. The sum is 7.8.
Very short answer
0/51
Write CPCT in one line.
Model answer (board format)
Corresponding parts of congruent triangles are equal.
Check your answer:
2
Write Axiom 7.1.
Model answer (board format)
If two sides and the included angle are equal, the triangles are congruent.
Check your answer:
3
Find the base angles of an isosceles triangle with vertex angle 40°.
Model answer (board format)
(180° − 40°) / 2 = 70°. Both base angles are 70°.
Check your answer:
4
Which triangles does Theorem 7.5 cover?
Model answer (board format)
Two right triangles, when the hypotenuse and one side are equal.
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5
Write Theorem 7.8 in one line.
Model answer (board format)
The sum of any two sides is greater than the third side.
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Short answer
0/41
∠B = ∠E = 40°, ∠C = ∠F = 70°, BC = EF. Write the rule and the third angle.
Model answer (board format)
The third angle = 180° − 110° = 70°. BC is the included side, so ASA, Theorem 7.1.
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2
The sides are 4 cm, 5 cm and 6 cm. Which side is opposite the largest angle? Also write one inequality check.
Model answer (board format)
The longest side is 6 cm, so the largest angle is opposite it. 4 + 5 = 9 > 6, so a triangle is formed.
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3
OA = 6 cm and O is the meeting of the perpendicular bisectors of AB and AC. Write OB and OC.
Model answer (board format)
OB = OA = 6 cm and OC = OA = 6 cm, because distance is equal on a perpendicular bisector.
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4
In two right triangles the hypotenuse 10 cm and one side 6 cm are equal. Give the name and number of the rule. Why not SAS?
Model answer (board format)
RHS, Theorem 7.5. Not SAS, because 90° is not taken as the included angle of that side and the hypotenuse.
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Long answer
0/31
Write SAS, ASA and AAS in your own words. Then find the third angle in the example with 50°, 60° and included side 8 cm.
Model answer (board format)
SAS is two sides and the angle between them. ASA is two angles and the side between them. AAS is two angles and one corresponding side, and the third angle matches from 180°. The third angle = 180° − 110° = 70°.
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2
Write Theorems 7.2, 7.6 and 7.8. Find the base angles of an isosceles triangle with vertex 100°. Write one sentence for 2 cm, 3 cm and 6 cm.
Model answer (board format)
Angles opposite equal sides are equal. The larger angle faces the longer side. The sum of two sides is greater than the third. Each base angle = (180° − 100°) / 2 = 40°. 2 + 3 = 5, which is not greater than 6, so there is no triangle.
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3
Write the order of Exercises 7.1 to 7.5 in one line. Then write the difference between SSS and RHS, and the difference between the two points of Exercise 7.5.
Model answer (board format)
The order is 7.1, 7.2, 7.3, 7.4, then optional 7.5. SSS needs three sides. RHS needs a right angle, the hypotenuse and one side. One point in 7.5 is equidistant from the vertices, and the other from the sides.
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BSEB model paper · practice
0/6This model set is for practice. It is not a question from any year’s annual examination. Annual questions will be added only when a source page is available.
1
In an isosceles triangle AB = AC and ∠A = 70°. ∠B is —
💡 Hint:
(180 − 70) / 2.
Explanation:
55°.
2
If the hypotenuse and one side are equal, two right triangles are —
💡 Hint:
Theorem 7.5.
Explanation:
RHS.
3
For 4 cm, 5 cm and 10 cm, the sum of the two shorter sides is ______ cm.
💡 Hint:
4 + 5.
Explanation:
9.
4
Theorem 7.1 is ASA.
Explanation:
True.
5
The sides are AB = 8 cm, BC = 6 cm and CA = 7 cm. Which angles are opposite the largest and the smallest?
Model answer (board format)
Opposite the longest side AB = 8 cm is ∠C. Opposite the shortest side BC = 6 cm is ∠A. Theorem 7.6.
Check your answer:
6
This is coefficient practice, not a result of the triangles chapter. Balance H2 + O2 making H2O by filling coefficients (blank = 1).
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
CBSE-style questions
0/6These are competency-based practice questions. They are not copies of a CBSE paper.
1
A student treats 3 cm, 4 cm and 8 cm as a triangle. The correction is —
💡 Hint:
Theorem 7.8.
Explanation:
7 is not greater than 8, so there is no triangle.
2
A park has three poles and a tap must be equidistant from them. The tap sits —
💡 Hint:
The language of Exercise 7.5.
Explanation:
The meeting of the perpendicular bisectors is equidistant from the three vertices.
3
Assertion (A): In an isosceles triangle the base angles are equal.
Reason (R): The three sides of a triangle add to 180°.
Explanation:
A is true. R is false. 180° is the sum of the angles, not of the sides.
4
If the three angles 40°, 60° and 80° match, the sides are always equal too.
Explanation:
False — matching angles give the shape. One side is needed for the size.
5
A triangular window frame is isosceles. The vertex angle is 96°. Find the base angles, and say why a measurement is not the proof.
Model answer (board format)
(180° − 96°) / 2 = 42°. Both base angles are 42°. A measurement makes the figure clear. The reason is Theorem 7.2 and the angle sum 180°.
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6
Two ladder frames make a right angle. The hypotenuse 15 m and one side 9 m are equal in both. Which rule says the frames are congruent?
Model answer (board format)
RHS, Theorem 7.5. The hypotenuse 15 m is opposite the right angle and the side 9 m is equal. Do not write SAS.
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🏛️ Board exam corner — Bihar Board (BSEB)— CBSE
Switch board with BSEB | CBSE above. The lessons follow the same NCERT chapter.
BSEB model · practiceBoard tip
This page has no verified annual-exam question, because no source page has been added. The model set below is practice in the board pattern.
🏛️ Model questions on one page →
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CBSE-style · competency-based
These are case and assertion-reason practice items. Do not treat them as past CBSE questions.
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🧠 What you learned + equation sheet
What you learned
- Congruent triangles match in both size and shape, and corresponding parts are equal.
- SAS is an axiom. ASA is a theorem. AAS uses the third angle. Three angles are not enough.
- Equal sides face equal angles, and the converse is also true.
- SSS uses all three sides. RHS uses the hypotenuse and one side in right triangles.
- The larger side faces the larger angle, and the larger angle faces the larger side.
- The sum of two sides is greater than the third. Exercises 7.1 to 7.5 stay in that order.
| What | Keep this |
|---|---|
| SAS | Axiom 7.1, included angle |
| ASA | Theorem 7.1 |
| SSS | Theorem 7.4 |
| RHS | Theorem 7.5 |
| Isosceles | Theorems 7.2 and 7.3 |
| Inequality | side sum greater |
The notes are original writing. The textbook was used only for activity order and numbers. “Verified” will be used only on a question that has a source page.