1
A quadrilateral has diagonals —
💡 Hint:
Opposite vertices.
Explanation:
Two.
Class 9 · Maths · Chapter 8 · बिहार बोर्ड (BSEB)CBSE · NCERT 2026-27
Quadrilaterals
Quadrilaterals
How to use this page:
1. Read — Parts · angle sum · types · Theorems 8.1 to 8.7 · Theorem 8.8 · Exercise 8.1 · mid-point · Exercise 8.2, diagram, worked example, board tip
2. Check — each lesson has its own questions; the number follows the lesson
3. Mastery ★ — all of that lesson correct. Redo the wrong ones
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- 1 Parts of a quadrilateral
- 2 The angle sum is 360°
- 3 Types of quadrilaterals
- 4 Properties of a parallelogram
- 5 One pair equal and parallel — Exercise 8.1
- 6 The mid-point theorem
- 7 Exercise 8.2 — the four mid-points
- 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
Parts of a quadrilateral
चतुर्भुज के भाग · Textbook 8.1 · side · angle · diagonal
Four sides, and the diagonals meet insideNotes
Four points make a quadrilateral only when no three lie on one line. There are four sides, four angles and four vertices. The two segments joining opposite vertices are the diagonals.
The book takes only the quadrilateral whose diagonals meet inside the figure. A crossed figure, whose diagonals fall outside, is not a quadrilateral of this chapter. Read the sides in order: AB, BC, CD, DA.
Worked exampleExample
Question: In ABCD the diagonal AC = 10 cm and O is its mid-point. Find AO.
Rule: the mid-point splits the segment into two equal parts.
Substitution: AO = AC / 2 = 10 cm / 2 = 5 cm.
10-second revision
- Four sides, four angles, two diagonals
- The diagonals meet inside
- The order is AB, BC, CD, DA
Board tip · BSEBBoard tip
A diagonal is not a side. AC is a side only when it is one of the four boundary sides.
Board tip · CBSEBoard tip
Do not apply the angle sum to a crossed figure by calling it a quadrilateral. The book keeps that figure apart.
Check your understandingall correct = mastery ★
2
A figure with three points on one line is a quadrilateral of this chapter.
Explanation:
False — no three points may be collinear, and only then is there a closed four-sided figure.
3
Half of a 10 cm diagonal is ______ cm.
💡 Hint:
10 / 2.
Explanation:
5 cm.
4
Write the names of the two diagonals and the four sides of quadrilateral ABCD.
Model answer (board format)
The diagonals are AC and BD. The sides are AB, BC, CD and DA.
Check your answer:
The angle sum is 360°
कोणों का योग 360° · Textbook 8.2 · diagonal · two triangles
Two triangles together give 360°Notes
A diagonal splits a quadrilateral into two triangles. Each triangle gives 180°, so the four angles add to 360°. The check is done by drawing a diagonal. It is not a numbered activity.
In a ratio, first add the parts, then share 360°. In 3 : 5 : 9 : 13 the parts are 3 + 5 + 9 + 13 = 30. One part = 360° / 30 = 12°.
Draw a diagonal and add the two trianglesActivity
Draw AC in ABCD. The angles of ΔADC and of ΔABC are each 180°. The two sums give 360°. Angle A splits into two pieces on AC, and so does angle C. The pieces join back into the four angles. This is the book’s check.
Worked exampleExample
Question: The angles are in the ratio 3 : 5 : 9 : 13. Find all four measures.
Formula: one part = 360° / (sum of the parts).
Substitution: the sum of the parts = 30. One part = 360° / 30 = 12°. The angles are 36°, 60°, 108° and 156°. The unit is the degree. Check: 36 + 60 + 108 + 156 = 360.
10-second revision
- The sum is 360°
- A diagonal makes two triangles
- 3 : 5 : 9 : 13 gives 36°, 60°, 108°, 156°
Board tip · BSEBBoard tip
Do not multiply each number of the ratio by 360°. First take 30, then 12°.
Board tip · CBSEBoard tip
Write the check 36 + 60 + 108 + 156 in the answer. If the sum is not 360°, the part is wrong.
Check your understandingall correct = mastery ★
1
The sum of the angles of a quadrilateral is —
💡 Hint:
Two triangles.
Explanation:
360°.
2
One diagonal splits a quadrilateral into two triangles.
Explanation:
True — so 180° + 180° = 360°.
3
In 3 : 5 : 9 : 13 one part is ______ degrees.
💡 Hint:
360 / 30.
Explanation:
12°.
4
The smallest angle in the ratio 3 : 5 : 9 : 13 is —
💡 Hint:
3 × 12.
Explanation:
36°.
5
Three angles are 80°, 90° and 110°. Find the fourth.
Model answer (board format)
80° + 90° + 110° = 280°. The fourth = 360° − 280° = 80°.
Check your answer:
6
The angles in the ratio 3 : 5 : 9 : 13 are 3°, 5°, 9° and 13°.
Explanation:
False — the sum is 360°. The angles are 36°, 60°, 108° and 156°.
Types of quadrilaterals
चतुर्भुज के प्रकार · Textbook 8.3 · trapezium · kite · square
The name follows from how many pairs are parallelNotes
If one pair of opposite sides is parallel, the figure is a trapezium. If both pairs are parallel, it is a parallelogram. A rectangle is a parallelogram with one angle 90°. A rhombus has all four sides equal. A square is both a rectangle and a rhombus.
A kite has two pairs of adjacent sides equal. A kite is not a parallelogram. A rectangle or a rhombus is not a square by itself. In a proof, take a trapezium to mean one pair of parallel sides. A parallelogram needs both pairs.
Worked exampleExample
Question: A parallelogram has all sides equal and one angle 90°. Name it.
Rule: four equal sides make a rhombus. One angle of 90° makes a rectangle. Both together make a square.
Conclusion: the figure is a square. A square is also a parallelogram.
10-second revision
- Trapezium: one pair parallel
- Square = rectangle and rhombus
- A kite is not a parallelogram
Board tip · BSEBBoard tip
Stopping at “rhombus” for a square is half an answer. Also write that one angle is 90°.
Board tip · CBSEBoard tip
The equal sides of a kite are adjacent. Writing them as opposite becomes a wrong reason for a parallelogram.
Check your understandingall correct = mastery ★
1
If both pairs of opposite sides are parallel, the figure is —
💡 Hint:
Two pairs.
Explanation:
A parallelogram.
2
A kite is a parallelogram.
Explanation:
False — a kite has equal adjacent sides. Both pairs of opposite sides are not taken as parallel.
3
Four equal sides and one angle of 90° together make a ______.
💡 Hint:
Both a rectangle and a rhombus.
Explanation:
A square.
4
A square is —
💡 Hint:
The book’s line.
Explanation:
Both a rectangle and a rhombus.
5
Write the difference in parallel sides among a trapezium, a parallelogram and a kite.
Model answer (board format)
A trapezium has one pair of opposite sides parallel. A parallelogram has both pairs parallel. A kite is not a parallelogram.
Check your answer:
Properties of a parallelogram
समांतर चतुर्भुज के गुण · Textbook 8.4 · Theorems 8.1 to 8.7
Opposite parts equal, diagonals halvedNotes
Theorem 8.1: a diagonal divides a parallelogram into two congruent triangles. Theorem 8.2: opposite sides are equal. Theorem 8.4: opposite angles are equal. Theorem 8.6: the diagonals bisect each other.
The converses also make a parallelogram. Theorem 8.3 uses both pairs of opposite sides equal. Theorem 8.5 uses the opposite angles. Theorem 8.7 uses the bisecting diagonals. The diagonals of a rectangle are equal. The diagonals of a rhombus cut at 90°. A square has both facts.
Worked exampleExample
Question: In a parallelogram ∠A = 70°. Find the other three angles.
Rule: opposite angles are equal, Theorem 8.4. Adjacent angles are supplementary, because the opposite sides are parallel and a side is a transversal.
Substitution: ∠C = 70°. ∠B = 180° − 70° = 110°. ∠D = 110°. The unit is the degree.
10-second revision
- Opposite sides and opposite angles are equal
- The diagonals bisect each other
- The converses are 8.3, 8.5 and 8.7
Board tip · BSEBBoard tip
The opposite of 70° is also 70°. The neighbour is 110°. Do not write all four as 70°.
Board tip · CBSEBoard tip
A rectangle has equal diagonals, and a rhombus has perpendicular diagonals. Write both properties on one figure only when it is a square.
Check your understandingall correct = mastery ★
1
If ∠A = 70° in a parallelogram, the opposite angle is —
💡 Hint:
Theorem 8.4.
Explanation:
70°.
2
The diagonals of a parallelogram bisect each other.
Explanation:
True — Theorem 8.6.
3
If ∠A = 70°, an adjacent angle is ______ degrees.
💡 Hint:
180 − 70.
Explanation:
110°.
4
Write Theorems 8.2, 8.4 and 8.6 in one line each.
Model answer (board format)
Opposite sides are equal. Opposite angles are equal. The diagonals bisect each other.
Check your answer:
One pair equal and parallel — Exercise 8.1
एक जोड़ा बराबर और समांतर — अभ्यास 8.1 · Textbook 8.5 · Theorem 8.8 · Exercise 8.1
Both equal and parallel are requiredNotes
Theorem 8.8: if one pair of opposite sides is both equal and parallel, the quadrilateral is a parallelogram. If the sides are only equal, or only parallel, this theorem does not apply. A trapezium has one parallel pair, and equality is not required.
Exercise 8.1 comes at this place. First the ratio of angles, then the tests for a parallelogram. When you choose a test, write the one among 8.3, 8.5, 8.7 and 8.8 that matches the given fact.
Exercise 8.1 — angles and the parallelogram testActivity
The first question is the angle ratio 3 : 5 : 9 : 13. Share 360° into 30 parts. In the later questions, see what is given: both pairs of sides, both pairs of angles, the mid-point of the diagonals, or one pair equal and parallel. Write one test and give the theorem number.
Worked exampleExample
Question: In ABCD, AB = CD and AB ∥ CD. Is ABCD a parallelogram?
Theorem: 8.8. One pair of opposite sides is equal and parallel.
Conclusion: yes, ABCD is a parallelogram. Writing only AB ∥ CD was not enough. Equality is given as well.
10-second revision
- Theorem 8.8: both equal and parallel
- Exercise 8.1 starts with the angle ratio
- One test is enough
Board tip · BSEBBoard tip
Do not apply 8.8 for parallel alone. The answer must contain both words, equal and parallel.
Board tip · CBSEBoard tip
In the first ratio question show 36°, 60°, 108° and 156°. Repeating 3 : 5 : 9 : 13 is not the measure.
Check your understandingall correct = mastery ★
1
If AB = CD and AB ∥ CD, then ABCD is —
💡 Hint:
Theorem 8.8.
Explanation:
A parallelogram.
2
If only one pair is parallel and the sides are not equal, Theorem 8.8 still applies.
Explanation:
False — 8.8 needs both equal and parallel.
3
The largest angle in the ratio 3 : 5 : 9 : 13 is ______ degrees.
💡 Hint:
13 × 12.
Explanation:
156°.
4
If the diagonals bisect each other, the theorem is —
💡 Hint:
The converse.
Explanation:
8.7.
5
Write the four tests for a parallelogram, with numbers.
Model answer (board format)
8.3 both pairs of opposite sides equal. 8.5 both pairs of opposite angles equal. 8.7 diagonals bisect. 8.8 one pair equal and parallel.
Check your answer:
6
Exercise 8.1 comes before the mid-point theorem.
Explanation:
True — in the book 8.1 comes after section 8.5, and the mid-point work is in 8.6.
The mid-point theorem
मध्य-बिंदु प्रमेय · Textbook 8.6 · Theorem 8.9 · Theorem 8.10
The middle segment is half and parallelNotes
Theorem 8.9: the segment joining the mid-points of two sides is parallel to the third side and half of it. The proof draws one more parallel and builds a congruent triangle and a parallelogram.
Theorem 8.10 is the converse. A line through the mid-point of one side, parallel to a second side, bisects the third side. Half means half the length, not half the angle.
The book’s construction — join the mid-pointsActivity
Draw a triangle and mark the mid-points E and F of two sides. Join EF. The book says EF will come out parallel to the third side and half of it. This is not a numbered activity. The measurement does not replace Theorem 8.9.
Worked exampleExample
Question: E and F are the mid-points of AB and AC. BC = 10 cm. Find EF.
Formula: EF = BC / 2, and EF ∥ BC. Theorem 8.9.
Substitution: EF = 10 cm / 2 = 5 cm. The unit is cm.
10-second revision
- Theorem 8.9: parallel and half
- Theorem 8.10 bisects the third side
- Half refers to the length
Board tip · BSEBBoard tip
Writing EF = BC is wrong. The mid-point segment is half, not the whole side.
Board tip · CBSEBoard tip
Show 10 cm / 2 = 5 cm. The word “half” without the measure can lose the mark.
Check your understandingall correct = mastery ★
1
The segment joining the mid-points is, of the third side, —
💡 Hint:
Theorem 8.9.
Explanation:
Half and parallel.
2
Theorem 8.10 is the converse of Theorem 8.9.
Explanation:
True — the parallel line bisects the third side.
3
If BC = 10 cm, the mid-segment is ______ cm.
💡 Hint:
Half.
Explanation:
5 cm.
4
E is the mid-point of AB and EF ∥ BC. Where is F?
💡 Hint:
Theorem 8.10.
Explanation:
The mid-point of AC.
5
Write Theorem 8.9 and find EF when BC = 18 cm.
Model answer (board format)
The segment joining the mid-points of two sides is parallel to the third and half of it. EF = 18 cm / 2 = 9 cm.
Check your answer:
Exercise 8.2 — the four mid-points
अभ्यास 8.2 — चारों मध्य-बिंदु · Textbook 8.6 · Exercise 8.2
Join the mid-points and a parallelogram appearsNotes
Exercise 8.2 asks you to join the mid-points of the four sides of a quadrilateral. Joined in order, the new quadrilateral is a parallelogram. The same mid-point language runs on a rectangle, a rhombus or a trapezium.
The reason is Theorem 8.9. In each triangle cut by a diagonal, the mid-segment is parallel and half. Two such half segments make one pair of parallel sides. Then Theorem 8.8 or 8.3 gives the parallelogram.
Exercise 8.2 — the quadrilateral of mid-pointsActivity
Take P, Q, R and S as the mid-points of the sides and join them in order. Show that PQRS is a parallelogram. If a rhombus or a rectangle is given, find the new angles or sides from the same theorem. In a trapezium with AB ∥ DC and E a mid-point, use 8.10 to catch the other mid-point.
Worked exampleExample
Question: PQRS is formed by the mid-points of ABCD. One diagonal AC = 12 cm. Find the mid-segment that is half of that diagonal.
Formula: mid-segment = third side / 2, when the third side is the diagonal of 12 cm.
Substitution: half = 12 cm / 2 = 6 cm. The unit is cm. PQRS is a parallelogram.
10-second revision
- Four mid-points form a parallelogram
- The reason is Theorem 8.9
- Exercise 8.2 comes after 8.1
Board tip · BSEBBoard tip
Do not write PQRS as a square unless a 90° angle or equal diagonals are given. The general answer is a parallelogram.
Board tip · CBSEBoard tip
With 12 cm / 2 = 6 cm write theorem number 8.9. Only 6 cm is half a reason.
Check your understandingall correct = mastery ★
1
Joining the mid-points of the sides of a quadrilateral in order gives —
💡 Hint:
The tenth line of the summary.
Explanation:
A parallelogram.
2
Exercise 8.2 is placed before Exercise 8.1.
Explanation:
False — 8.1 comes first, and 8.2 comes after the mid-point theorem.
3
The mid-segment of a 12 cm diagonal is ______ cm.
💡 Hint:
Half.
Explanation:
6 cm.
4
PQRS is formed from the mid-points. Write one reason for calling it a parallelogram.
Model answer (board format)
In each triangle cut by a diagonal, Theorem 8.9 makes the mid-segment parallel. Two parallel pairs appear, so PQRS is a parallelogram.
Check your answer:
❓ Full question bank — with answers and explanations — 66 questions
Pick a type. The 34 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
The sum of the angles of a quadrilateral is —
💡 Hint:
Two triangles.
Explanation:
360°.
2
The smallest angle of 3 : 5 : 9 : 13 is —
💡 Hint:
3 × 12.
Explanation:
36°.
3
A square is —
💡 Hint:
Both names.
Explanation:
Both a rectangle and a rhombus.
4
In a parallelogram opposite angles are —
💡 Hint:
Theorem 8.4.
Explanation:
Equal.
5
If ∠A = 70°, an adjacent angle is —
💡 Hint:
180 − 70.
Explanation:
110°.
6
If the diagonals bisect each other, the theorem is —
💡 Hint:
The converse.
Explanation:
8.7.
7
From AB = CD and AB ∥ CD it follows that —
💡 Hint:
Theorem 8.8.
Explanation:
A parallelogram.
8
The mid-point segment is, of the third side, —
💡 Hint:
Theorem 8.9.
Explanation:
Half.
9
If BC = 10 cm, EF is —
💡 Hint:
Half.
Explanation:
5 cm.
10
Joining four mid-points gives —
💡 Hint:
Exercise 8.2.
Explanation:
A parallelogram.
11
The diagonals of a rhombus cut at —
💡 Hint:
The summary line.
Explanation:
90°.
12
A kite is —
💡 Hint:
Adjacent sides.
Explanation:
Not a parallelogram.
13
The diagonals of a rectangle are —
💡 Hint:
The summary.
Explanation:
Equal.
True or false
0/81
The angles of a quadrilateral add to 180°.
Explanation:
False — the sum is 360°. 180° belongs to one triangle.
2
A square is also a rhombus.
Explanation:
True — and a rectangle as well.
3
Theorem 8.6 is about the diagonals bisecting each other.
Explanation:
True.
4
Theorem 8.8 asks only for a parallel side.
Explanation:
False — the side must also be equal.
5
In Theorem 8.9 the segment equals the third side.
Explanation:
False — it is parallel and half.
6
Exercise 8.2 is the one about mid-points.
Explanation:
True — 8.1, before it, asks for angles and the parallelogram tests.
7
The diagonals of a rectangle are always at right angles, as in a rhombus.
Explanation:
False — in a rectangle the diagonals are equal. Cutting at right angles is a property of a rhombus.
8
A crossed figure is a quadrilateral of this chapter.
Explanation:
False — the book takes the figure whose diagonals meet inside.
Fill in the blanks
0/81
The angle sum is ______ degrees.
💡 Hint:
Two triangles.
Explanation:
360.
2
In 3 : 5 : 9 : 13 one part is ______ degrees.
💡 Hint:
360 / 30.
Explanation:
12.
3
The adjacent angle of 70° is ______ degrees.
💡 Hint:
180 − 70.
Explanation:
110.
4
The theorem number for opposite sides is ______.
💡 Hint:
The converse is 8.3.
Explanation:
8.2.
5
If BC = 10 cm, the mid-segment is ______ cm.
💡 Hint:
Half.
Explanation:
5.
6
Half of a 12 cm diagonal is ______ cm.
💡 Hint:
12 / 2.
Explanation:
6.
7
The diagonals of a rhombus cut at ______ degrees.
💡 Hint:
A right angle.
Explanation:
90.
8
The theorem number for equal and parallel is ______.
💡 Hint:
One pair.
Explanation:
8.8.
Match
0/21
Match the theorem with the statement.
Column B: A. One pair equal and parallel · B. Opposite sides equal · C. Opposite angles equal · D. Diagonals bisect
1. 8.2
2. 8.4
3. 8.6
4. 8.8
Explanation:
8.2 is the sides, 8.4 the angles, 8.6 the diagonals, and 8.8 equal and parallel.
2
Match the figure with the diagonal fact.
Column B: A. Half of the third side · B. Bisect each other · C. Equal diagonals · D. Cut at 90°
1. Parallelogram
2. Rectangle
3. Rhombus
4. Mid-segment
Explanation:
A parallelogram bisects, a rectangle has equal diagonals, a rhombus cuts at 90°, and the mid-segment is half.
Assertion–reason
0/5Check both statements, then see whether the reason explains the assertion.
1
Assertion (A): The angles of a quadrilateral add to 360°.
Reason (R): One diagonal makes two triangles, and each triangle gives 180°.
Explanation:
Both are true and R is the reason.
2
Assertion (A): The diagonals of a parallelogram bisect each other.
Reason (R): The mid-point segment is half of the third side.
Explanation:
Both are true, but R does not explain the diagonal statement.
3
Assertion (A): A square is a rectangle.
Reason (R): Every rectangle is a square.
Explanation:
A is true. R is false — the sides of a rectangle need not be equal.
4
Assertion (A): In Theorem 8.9 the mid-segment equals the third side.
Reason (R): It is parallel to the third side and half of it.
Explanation:
A is false. R is true.
5
Assertion (A): AB = CD and AB ∥ CD make ABCD a parallelogram.
Reason (R): If one pair of opposite sides is equal and parallel, the quadrilateral is a parallelogram.
Explanation:
Both are true and R is Theorem 8.8.
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 quadrilateral 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 figure in the right group.
Square
Rectangle
Kite
Rhombus
Explanation:
A square, a rectangle and a rhombus are parallelograms. A kite is not.
2
Place each test on its theorem number.
Opposite sides equal
One pair equal and parallel
Mid-points, half segment
Both pairs of sides, not the diagonals
Explanation:
Opposite sides are 8.3. Equal and parallel is 8.8. The mid-point theorem is 8.9.
Very short answer
0/51
Write the sum of the angles of a quadrilateral.
Model answer (board format)
360°.
Check your answer:
2
Write Theorem 8.6.
Model answer (board format)
The diagonals of a parallelogram bisect each other.
Check your answer:
3
Write the four angles of the ratio 3 : 5 : 9 : 13.
Model answer (board format)
36°, 60°, 108° and 156°.
Check your answer:
4
In Theorem 8.9, what fraction of the third side is the segment?
Model answer (board format)
Half, and parallel.
Check your answer:
5
At what angle do the diagonals of a rhombus cut?
Model answer (board format)
At 90°.
Check your answer:
Short answer
0/41
In a parallelogram ∠A = 65°. Find the other angles.
Model answer (board format)
The opposite ∠C = 65°. An adjacent angle = 180° − 65° = 115°. So ∠B = ∠D = 115°.
Check your answer:
2
E and F are mid-points and BC = 14 cm. Write EF and its direction.
Model answer (board format)
EF = 14 cm / 2 = 7 cm, and EF ∥ BC. Theorem 8.9.
Check your answer:
3
AB = CD and AB ∥ CD. Which theorem calls ABCD a parallelogram? Why is parallel alone not enough?
Model answer (board format)
Theorem 8.8. Parallel alone can also make a trapezium. Equality is the second condition.
Check your answer:
4
Write one difference each for the diagonals of a rectangle and a rhombus. What happens in a square?
Model answer (board format)
The diagonals of a rectangle are equal. The diagonals of a rhombus cut at 90°. In a square the diagonals are equal and they also cut at 90°.
Check your answer:
Long answer
0/31
Write the reason for the angle sum. Then find the angles of 3 : 5 : 9 : 13 and show the check of the sum.
Model answer (board format)
A diagonal makes two triangles, 180° + 180° = 360°. The parts are 30, and one part is 12°. The angles are 36°, 60°, 108° and 156°. The sum is 360°.
Check your answer:
2
Write Theorems 8.2, 8.4, 8.6 and 8.8. Give the four angles of a parallelogram with ∠A = 80°.
Model answer (board format)
Opposite sides are equal. Opposite angles are equal. The diagonals bisect. One pair equal and parallel also gives a parallelogram. The angles are 80°, 100°, 80° and 100°.
Check your answer:
3
Write Theorems 8.9 and 8.10. Find EF when BC = 16 cm. Why does Exercise 8.2 produce a parallelogram?
Model answer (board format)
8.9: the mid-segment is parallel and half. 8.10: a parallel line bisects the third side. EF = 8 cm. At the four mid-points, Theorem 8.9 makes the parallel pairs.
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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
One angle of a parallelogram is 100°. An adjacent angle is —
💡 Hint:
180 − 100.
Explanation:
80°.
2
The segment in the mid-point theorem is —
💡 Hint:
Theorem 8.9.
Explanation:
Half.
3
After angles 70°, 80° and 120°, the fourth is ______ degrees.
💡 Hint:
360 − 270.
Explanation:
90.
4
Theorem 8.1 makes two congruent triangles from a diagonal.
Explanation:
True.
5
BC = 22 cm and E, F are mid-points. Write EF. Why is a measurement not the proof?
Model answer (board format)
EF = 11 cm and EF ∥ BC. A measurement makes the figure clear. The reason is Theorem 8.9.
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6
This is coefficient practice, not a result of the quadrilaterals 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 window is rectangular. The right fact about the diagonals is —
💡 Hint:
The rectangle summary.
Explanation:
Equal, and they bisect each other.
2
A farmer joins poles at the mid-points of the four sides of a field. The inner figure is —
💡 Hint:
Exercise 8.2.
Explanation:
The quadrilateral of the mid-points is a parallelogram.
3
Assertion (A): A square is a parallelogram.
Reason (R): Every parallelogram is a square.
Explanation:
A is true. R is false — the sides of a parallelogram need not be equal, nor the angles 90°.
4
The ratio 3 : 5 : 9 : 13 means the angles are 3° and 13° because a ratio is small.
Explanation:
False — the sum is 360°, so the angles are 36°, 60°, 108° and 156°.
5
The top of a table is a parallelogram. One corner measures 75°. Write the other corners. Why not treat the measure as the proof?
Model answer (board format)
The opposite corner is 75°, and the adjacent corners are 105° and 105°. The measure is a check. The reason is Theorem 8.4 and the adjacent sum 180°.
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6
A path between the mid-points of two sides of a triangular park is parallel to the third side of 24 m. Find the length of the path.
Model answer (board format)
By Theorem 8.9 the length = 24 m / 2 = 12 m. The path is parallel to the third side.
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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.
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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
- The four angles of a quadrilateral add to 360°, because a diagonal makes two triangles.
- In a parallelogram the opposite sides and the opposite angles are equal.
- The diagonals bisect each other, and the converse also makes a parallelogram.
- One pair of opposite sides equal and parallel also makes the figure a parallelogram.
- The diagonals of a rectangle are equal. The diagonals of a rhombus cut at right angles.
- The segment joining the mid-points is parallel to the third side and half of it.
| What | Keep this |
|---|---|
| Angle sum | 360° |
| Opposite sides | equal in a parallelogram |
| Diagonals | bisect each other |
| Rectangle | equal diagonals |
| Rhombus | diagonals at 90° |
| Mid-point | parallel and half |
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.