1
In adjacent angles the non-common arms lie —
💡 Hint:
On one side they are not adjacent.
Explanation:
On opposite sides of the common arm.
Class 9 · Maths · Chapter 6 · बिहार बोर्ड (BSEB)CBSE · NCERT 2026-27
Lines and Angles
Lines and Angles
How to use this page:
1. Read — Terms · linear pair · vertically opposite · Exercise 6.1 · transversal · corresponding · alternate · co-interior · Exercise 6.2 · angle sum · Exercise 6.3, 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
4. The memory figure shows two parallel lines cut by a transversal, with one pair of alternate angles marked equal.
In the 15-chapter Bihar book this is chapter 6. Ganita Manjari 2026-27 uses a different order. Progress stays in this browser.
🏅 Your achievements — 0/7 badges
🔥 0day streak · longest 0
⭐ 0total XP · today 0
🎓 1level
Next level: 150 XP to goright answer +10 · after a hint +5 · review +6 · badge +25
Last 7 days:
Everything is saved in this phone/browser — no login.
- 1 Lines, angles, and adjacent angles
- 2 The linear pair — Axioms 6.1 and 6.2
- 3 Vertically opposite angles — Theorem 6.1
- 4 A transversal and corresponding angles
- 5 Alternate and co-interior angles
- 6 Lines parallel to the same line
- 7 The angle sum of a triangle and the exterior angle
- 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
Lines, angles, and adjacent angles
रेखा, कोण और आसन्न कोण · Textbook 6.2 · adjacent · complementary
The name is fixed by the contextNotes
A piece with two ends is a segment. A ray has one end. A line runs both ways. The book may write the segment, the ray, the length and the line all as AB. Take the meaning from the sentence. Three or more points on one line are collinear, otherwise non-collinear.
Two rays from one endpoint make an angle. Acute is between 0° and 90°, a right angle is exactly 90°, obtuse is between 90° and 180°, a straight angle is 180°, and reflex is between 180° and 360°. If the sum is 90° they are complementary. If the sum is 180° they are supplementary. Adjacent angles share the vertex and one arm, and the other arms lie on opposite sides of the common arm.
Worked exampleExample
Question: One angle is 35°. Find its complement and its supplement.
Relations: complement = 90° − the given angle. Supplement = 180° − the given angle.
Substitution: complement = 90° − 35° = 55°. Supplement = 180° − 35° = 145°.
10-second revision
- Adjacent: common vertex, common arm, others opposite
- Complementary 90°, supplementary 180°
- Take the meaning of AB from the sentence
Board tip · BSEBBoard tip
Swapping 90° and 180° for complement and supplement is the common slip. Write both words in the answer.
Board tip · CBSEBoard tip
Do not write a sum of 180° just because the angles are adjacent. That condition belongs to a linear pair.
Check your understandingall correct = mastery ★
2
Two angles whose sum is 90° are called supplementary.
Explanation:
False — a sum of 90° is complementary. Supplementary angles add to 180°.
3
The measure of a straight angle is ______ degrees.
💡 Hint:
A right angle is 90°.
Explanation:
180°.
4
The complement of 70° is —
💡 Hint:
Subtract from 90.
Explanation:
90° − 70° = 20°.
5
Write the three conditions for adjacent angles. Is every adjacent pair a linear pair?
Model answer (board format)
A common vertex, a common arm, and the other arms on opposite sides of the common arm. No. A linear pair exists only when the other arms form a line.
Check your answer:
The linear pair — Axioms 6.1 and 6.2
रैखिक युग्म — अभिगृहीत 6.1 और 6.2 · Textbook 6.4 · linear pair axiom
If a ray stands on a line, the sum is 180°Notes
Axiom 6.1: if a ray stands on a line, the two adjacent angles formed add to 180°. Such angles are a linear pair. The converse is Axiom 6.2: if two adjacent angles add to 180°, their non-common arms form a line. Together they are called the linear pair axiom.
A converse swaps the given fact and the conclusion. In 6.1, “the ray stands on a line” is given and the sum is the conclusion. In 6.2 the sum is given and the line is the conclusion. Both are axioms here, not theorems.
Worked exampleExample
Question: One angle of a linear pair is 65°. Find the other.
Axiom: a linear pair adds to 180°.
Substitution: the other = 180° − 65° = 115°.
10-second revision
- Axiom 6.1: a ray on a line gives 180°
- Axiom 6.2 is the converse
- Together they are the linear pair axiom
Board tip · BSEBBoard tip
Both 6.1 and 6.2 are axioms. Writing the converse as a theorem is wrong in this chapter.
Board tip · CBSEBoard tip
In the 115° answer show 180° − 65°. Only 115° can lose half the marks.
Check your understandingall correct = mastery ★
1
The sum of a linear pair is —
💡 Hint:
A straight angle.
Explanation:
180°.
2
If two adjacent angles add to 180°, the non-common arms form a line.
Explanation:
True — this is Axiom 6.2.
3
If one angle of a linear pair is 40°, the other is ______ degrees.
💡 Hint:
Subtract from 180.
Explanation:
140°.
4
Write Axiom 6.1 and its converse in your own words.
Model answer (board format)
If a ray stands on a line, the adjacent angles add to 180°. The converse: if adjacent angles add to 180°, the other arms form a line. Both are the linear pair axiom.
Check your answer:
Vertically opposite angles — Theorem 6.1
शीर्षाभिमुख कोण — प्रमेय 6.1 · Textbook 6.4 · Theorem 6.1 · Exercise 6.1
The facing angles are equalNotes
Two lines crossing at O make two pairs of vertically opposite angles. Theorem 6.1: they are equal. The proof uses a linear pair. One angle makes 180° with each neighbour. Subtract the same angle and the facing angles are left equal.
In a ratio question, take the 180° of the linear pair first, then share the parts. A vertically opposite angle repeats that measure across the crossing. It does not create a new measure.
Worked exampleExample
Question: Two lines cross at O. ∠POR : ∠ROQ = 5 : 7. Find all four angles.
Linear pair: ∠POR + ∠ROQ = 180°. The parts are 5 + 7 = 12.
Substitution: ∠POR = (5/12) × 180° = 75°. ∠ROQ = (7/12) × 180° = 105°.
Vertically opposite: the facing angles are also 105° and 75°.
10-second revision
- Theorem 6.1: vertically opposite angles are equal
- The proof uses a linear pair
- 5 : 7 and 180° give 75° and 105°
Board tip · BSEBBoard tip
Do not multiply each part of the ratio by 180°. First take the sum of the parts, 12, then 5/12 and 7/12.
Board tip · CBSEBoard tip
If four angles are asked, write the vertically opposite repeats. Only two angles are half an answer.
Check your understandingall correct = mastery ★
1
When two lines cross, vertically opposite angles are —
💡 Hint:
Theorem 6.1.
Explanation:
Equal.
2
The proof of vertically opposite angles rests on the linear pair axiom.
Explanation:
True — both neighbouring pairs are 180°.
3
In the linear pair in the ratio 5 : 7, the smaller angle is ______ degrees.
💡 Hint:
5/12 × 180.
Explanation:
75°. The larger is 105°.
4
One vertically opposite angle is 105°. The facing angle is —
💡 Hint:
The facing measure is the same.
Explanation:
105°. The neighbour is 75°.
5
Two lines cross and one angle is 40°. Find the other three.
Model answer (board format)
The vertically opposite angle is 40°. Each neighbour, from a linear pair, is 180° − 40° = 140°. The four angles are 40°, 140°, 40° and 140°.
Check your answer:
6
In the pair 5 : 7 the angles are 5° and 7°.
Explanation:
False — the sum is 180°. The angles are 75° and 105°.
A transversal and corresponding angles
तिर्यक और संगत कोण · Textbook 6.5 · Axiom 6.3
One line cuts two lines at different pointsNotes
A line that cuts two or more lines at distinct points is a transversal. Four angles appear at each cut. Those outside the strip are exterior and those between are interior. Angles in the same position are corresponding.
The perpendicular distance between parallel lines is the same everywhere. Axiom 6.3: if a transversal cuts parallel lines, each corresponding pair is equal. The converse also holds: if one corresponding pair is equal, the lines are parallel.
The book’s measurement — the parallel lines of the notebookActivity
Draw parallels along two ruled lines of the notebook and cut them with a transversal. Measure a corresponding pair. They come out equal. This is not a numbered activity. The book says to repeat the checks of earlier classes here. A measurement does not replace the axiom, but it makes the figure clear.
Worked exampleExample
Question: m ∥ n and one corresponding angle on the transversal is 72°. Find its corresponding partner and the co-interior partner.
Axiom 6.3: the corresponding partner = 72°.
Co-interior: 180° − 72° = 108°.
10-second revision
- A transversal cuts at distinct points
- Axiom 6.3: if parallel, corresponding angles are equal
- One equal corresponding pair makes the lines parallel
Board tip · BSEBBoard tip
Corresponding angles are not alternate angles. Look at the same position, not the crossed place inside the strip.
Board tip · CBSEBoard tip
When the converse is used to prove parallel, one pair is enough. Measuring all four pairs is not required.
Check your understandingall correct = mastery ★
1
A transversal is a line that —
💡 Hint:
The cuts are separate.
Explanation:
It cuts two or more lines at distinct points.
2
Corresponding angles on parallel lines are equal.
Explanation:
True — Axiom 6.3.
3
If a corresponding angle is 72° and the lines are parallel, the co-interior partner is ______ degrees.
💡 Hint:
180 − 72.
Explanation:
108°.
4
If one corresponding pair is equal, then —
💡 Hint:
The converse of 6.3.
Explanation:
The lines are parallel.
5
In the check along the ruled lines, what will you measure, and which axiom is linked to it?
Model answer (board format)
I will draw two parallel ruled lines and a transversal and measure one corresponding pair. Equal measures are linked to Axiom 6.3. The measurement itself is not the proof.
Check your answer:
Alternate and co-interior angles
एकांतर और सह-आंतरिक कोण · Textbook 6.5 · Theorems 6.2 to 6.5
Two kinds of pairs in the inner stripNotes
Inside the strip, angles on opposite sides of the transversal are alternate interior angles. The book sometimes calls them only alternate angles. Interior angles on the same side are co-interior. They are also called consecutive interior angles or allied angles.
If the lines are parallel, alternate angles are equal and co-interior angles add to 180°. Theorems 6.2 and 6.4 say this. The converses 6.3 and 6.5 prove the lines parallel. One pair is enough.
Worked exampleExample
Question: m ∥ n. One alternate interior angle is 65°. Find the other alternate and the co-interior angle.
Theorem 6.2: the other alternate = 65°.
Theorem 6.4: co-interior = 180° − 65° = 115°.
10-second revision
- Alternate angles are equal when the lines are parallel
- Co-interior angles add to 180°
- The converse proves the lines parallel
Board tip · BSEBBoard tip
Do not write co-interior angles as equal. Their sum is 180°. Alternate and corresponding angles are the equal ones.
Board tip · CBSEBoard tip
On the figure an alternate pair is inside the strip on opposite sides of the transversal. A corresponding pair sits in the same position.
Check your understandingall correct = mastery ★
1
On parallel lines, alternate interior angles are —
💡 Hint:
Theorem 6.2.
Explanation:
Equal.
2
Co-interior angles add to 180° when the lines are parallel.
Explanation:
True — Theorem 6.4.
3
If an alternate angle is 65°, the co-interior angle is ______ degrees.
💡 Hint:
180 − 65.
Explanation:
115°.
4
If one co-interior pair adds to 180°, then —
💡 Hint:
Theorem 6.5.
Explanation:
The lines are parallel.
5
m ∥ n. One interior angle is 110°. Write its co-interior and alternate partners. Give the theorem numbers.
Model answer (board format)
The co-interior angle = 180° − 110° = 70°, Theorem 6.4. The alternate partner = 110°, Theorem 6.2.
Check your answer:
6
Alternate angles are another name for co-interior angles.
Explanation:
False — alternate angles are on opposite sides and are equal. Co-interior angles are on the same side and are supplementary.
Lines parallel to the same line
एक ही रेखा के समांतर रेखाएँ · Textbook 6.6 · Theorem 6.6 · Exercise 6.2
Parallelism carries forwardNotes
Theorem 6.6: lines which are parallel to the same line are parallel to each other. The same fact runs for more than two lines. A helper parallel is drawn and corresponding or alternate angles are matched.
An exercise often gives one angle 135° and another 40° and asks for the angle between. The helper line makes a co-interior or alternate link with both parallels. 180° − 135° = 45°, then 45° + 40° = 85°. The unit is the degree.
Worked exampleExample
Question: PQ ∥ RS. One co-interior angle is 135° and a further alternate angle is 40°. Find the angle between.
Step: 180° − 135° = 45°. This is the piece on the helper line.
Sum: 45° + 40° = 85°.
10-second revision
- Lines parallel to one line are parallel to each other
- A helper parallel splits the angle into pieces
- In the 135° and 40° form the answer is 85°
Board tip · BSEBBoard tip
Do not add 135° straight to 40°. First find 180° − 135° = 45°, then add 40°.
Board tip · CBSEBoard tip
Do not write “both are parallel” for Theorem 6.6 without the reason. Write that both are parallel to a third line.
Check your understandingall correct = mastery ★
1
If m ∥ l and n ∥ l, then —
💡 Hint:
Theorem 6.6.
Explanation:
m ∥ n.
2
Theorem 6.6 is limited to exactly two lines.
Explanation:
False — the book says the fact also runs for more than two lines.
3
After the co-interior piece of 135°, add 40°. The angle between is ______ degrees.
💡 Hint:
45 + 40.
Explanation:
85°.
4
AB ∥ CD and CD ∥ EF. What will you say about AB and EF? Name the reason.
Model answer (board format)
AB ∥ EF, because lines parallel to the same line are parallel to each other. This is Theorem 6.6.
Check your answer:
The angle sum of a triangle and the exterior angle
त्रिभुज का कोण योग और बाह्य कोण · Textbook 6.7 · Theorem 6.7 · Theorem 6.8 · Exercise 6.3
A parallel brings the three angles onto one lineNotes
Theorem 6.7: the three angles of a triangle add to 180°. In the proof a line is drawn through a vertex parallel to the opposite side. Alternate angles match the other two angles, and the three make a straight angle.
Theorem 6.8: if a side is extended, the exterior angle equals the sum of the two interior angles far from it. So it is greater than each of those two. In a right triangle, 90° + one angle + the other = 180°.
Worked exampleExample
Question: Two angles of a triangle are 50° and 60°. Find the third and the exterior angle far from them.
Theorem 6.7: 50° + 60° + x = 180°, so x = 70°.
Theorem 6.8: exterior angle = 50° + 60° = 110°.
10-second revision
- The three angles add to 180°
- Exterior angle = sum of the two far interior angles
- The exterior angle is greater than each far interior angle
Board tip · BSEBBoard tip
Do not write the exterior angle equal to the interior angle beside it. It makes 180° with that one as a linear pair. The two far angles are the ones that add.
Board tip · CBSEBoard tip
70° and 110° are two parts of one answer. Writing only the third angle leaves out the exterior-angle part.
Check your understandingall correct = mastery ★
1
The sum of the angles of a triangle is —
💡 Hint:
Theorem 6.7.
Explanation:
180°.
2
The exterior angle equals the interior angle next to it.
Explanation:
False — with the next angle it is a linear pair. The exterior angle equals the sum of the two far interior angles.
3
If the angles are 50° and 60°, the third angle is ______ degrees.
💡 Hint:
180 − 110.
Explanation:
70°.
4
The exterior angle far from 50° and 60° in a triangle is —
💡 Hint:
The sum of both.
Explanation:
50° + 60° = 110°.
5
In a right triangle one acute angle is 35°. Find the other acute angle. Also write one line for the exterior angle if the side of the right angle is extended and the far angles are 35° and 55°.
Model answer (board format)
90° + 35° + y = 180°, so y = 55°. The exterior angle = 35° + 55° = 90°, by Theorem 6.8.
Check your answer:
❓ Full question bank — with answers and explanations — 65 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, LA110 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 non-common arms of adjacent angles lie —
💡 Hint:
Opposite sides of the common arm.
Explanation:
On opposite sides.
2
The supplement of 40° is —
💡 Hint:
From 180.
Explanation:
140°.
3
The sum of a linear pair is —
💡 Hint:
Axiom 6.1.
Explanation:
180°.
4
Vertically opposite angles are —
💡 Hint:
Theorem 6.1.
Explanation:
Equal.
5
The larger angle of the 5 : 7 pair is —
💡 Hint:
7/12 × 180.
Explanation:
105°.
6
Corresponding angles on parallel lines are —
💡 Hint:
Axiom 6.3.
Explanation:
Equal.
7
Alternate interior angles on parallels are —
💡 Hint:
Theorem 6.2.
Explanation:
Equal.
8
The sum of co-interior angles is —
💡 Hint:
Theorem 6.4.
Explanation:
180°.
9
From m ∥ l and n ∥ l it follows that —
💡 Hint:
Theorem 6.6.
Explanation:
m ∥ n.
10
The sum of the angles of a triangle is —
💡 Hint:
Theorem 6.7.
Explanation:
180°.
11
An exterior angle equals —
💡 Hint:
Theorem 6.8.
Explanation:
The sum of the two far interior angles.
12
If one corresponding pair is equal, the lines are —
💡 Hint:
The converse.
Explanation:
Parallel.
13
The in-between angle in the 135° and 40° case is —
💡 Hint:
First 45, then add 40.
Explanation:
85°.
True or false
0/81
Every adjacent pair adds to 180°.
Explanation:
False — 180° is the condition for a linear pair.
2
Axiom 6.2 is the converse of Axiom 6.1.
Explanation:
True — together they are called the linear pair axiom.
3
Vertically opposite angles are supplementary, not equal.
Explanation:
False — Theorem 6.1 says they are equal. Their neighbours are supplementary.
4
A transversal cuts both lines at the same point.
Explanation:
False — the cuts are at distinct points.
5
Co-interior angles are equal.
Explanation:
False — their sum is 180°.
6
Lines parallel to the same line are parallel to each other.
Explanation:
True — Theorem 6.6.
7
An exterior angle of a triangle equals only one far angle.
Explanation:
False — it equals the sum of both far interior angles.
8
A right angle is 90° and a straight angle is 180°.
Explanation:
True.
Fill in the blanks
0/81
Complementary angles add to ______ degrees.
💡 Hint:
90.
Explanation:
Supplementary is 180.
2
In a linear pair the partner of 65° is ______ degrees.
💡 Hint:
115.
Explanation:
180 − 65.
3
In 5 : 7 the smaller angle is ______ degrees.
💡 Hint:
75.
Explanation:
5/12 × 180.
4
The axiom about corresponding angles is number ______.
💡 Hint:
6.3.
Explanation:
The linear pair is 6.1 and 6.2.
5
If the alternate angle is 65°, the co-interior angle is ______ degrees.
💡 Hint:
115.
Explanation:
180 − 65.
6
After 50° and 60° the third angle is ______ degrees.
💡 Hint:
70.
Explanation:
180 − 110.
7
In that situation the exterior angle is ______ degrees.
💡 Hint:
110.
Explanation:
50 + 60.
8
After the piece from 135°, the in-between angle is ______ degrees when the other piece is 40°.
💡 Hint:
85.
Explanation:
45 + 40.
Match
0/21
Match the pair with its rule. Assume parallel lines.
Column B: A. Sum 180° · B. Equal, same position · C. Equal, facing at a crossing · D. Equal, opposite sides in the strip
1. Corresponding
2. Alternate
3. Co-interior
4. Vertically opposite
Explanation:
Corresponding and alternate are equal, co-interior are supplementary, and vertically opposite are equal at a crossing.
2
Match the number with the statement.
Column B: A. The first linear-pair statement · B. Corresponding angles · C. Parallel to the same line · D. Angle sum of a triangle
1. 6.1
2. 6.3
3. 6.6
4. 6.7
Explanation:
6.1 is the linear pair, 6.3 corresponding, 6.6 parallelism, and 6.7 the angle sum.
Assertion–reason
0/5Check both statements, then see whether the reason explains the assertion.
1
Assertion (A): A linear pair adds to 180°.
Reason (R): If a ray stands on a line, the adjacent angles make a straight angle.
Explanation:
Both are true and R is the correct reason.
2
Assertion (A): Vertically opposite angles are equal.
Reason (R): The angles of a triangle add to 180°.
Explanation:
Both are true, but R does not explain vertically opposite angles.
3
Assertion (A): Alternate angles on parallel lines are equal.
Reason (R): Co-interior angles are equal too.
Explanation:
A is true. R is false — co-interior angles add to 180°.
4
Assertion (A): The exterior angle equals the interior angle next to it.
Reason (R): The exterior angle equals the sum of the two far interior angles.
Explanation:
A is false. R is true, Theorem 6.8.
5
Assertion (A): m ∥ l and n ∥ l imply m ∥ n.
Reason (R): Lines parallel to the same line are parallel to each other.
Explanation:
Both are true and R explains A as Theorem 6.6.
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 an angle 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 pair under its rule. Assume the lines are parallel.
Corresponding
Alternate
Co-interior
Linear pair
Explanation:
Corresponding and alternate are equal. Co-interior and a linear pair give 180°.
2
Place each statement on the right theorem or axiom.
A ray on a line gives 180°
Parallel to the same line
Angles of a triangle are 180°
The converse also makes a line
Explanation:
The linear pair is 6.1. Parallelism is 6.6. The angle sum is 6.7.
Very short answer
0/51
Define a linear pair in one line.
Model answer (board format)
Adjacent angles whose other arms form a line are a linear pair, and their sum is 180°.
Check your answer:
2
Write Theorem 6.1.
Model answer (board format)
If two lines intersect, the vertically opposite angles are equal.
Check your answer:
3
Write the complement and the supplement of 70°.
Model answer (board format)
The complement is 20°. The supplement is 110°.
Check your answer:
4
What is a transversal?
Model answer (board format)
A line that cuts two or more lines at distinct points is a transversal.
Check your answer:
5
Write Theorem 6.8 in one line.
Model answer (board format)
An exterior angle of a triangle equals the sum of the two far interior angles.
Check your answer:
Short answer
0/31
∠POR : ∠ROQ = 5 : 7 and they are a linear pair. Find all four angles.
Model answer (board format)
(5/12) × 180° = 75° and (7/12) × 180° = 105°. The vertically opposite angles are 75° and 105° as well.
Check your answer:
2
m ∥ n and one corresponding angle is 58°. Write the corresponding, alternate and co-interior measures.
Model answer (board format)
Corresponding 58°, alternate 58°, co-interior 180° − 58° = 122°.
Check your answer:
3
The angles of a triangle are 40° and 75°. Find the third angle and the exterior angle far from these two.
Model answer (board format)
The third = 180° − 115° = 65°. The exterior = 40° + 75° = 115°.
Check your answer:
Long answer
0/31
Write the linear pair axiom and Theorem 6.1. Then find the four angles of a linear pair in the ratio 2 : 3.
Model answer (board format)
If a ray stands on a line the sum is 180°, and the converse is also an axiom. Vertically opposite angles are equal. (2/5) × 180° = 72° and (3/5) × 180° = 108°. The four angles are 72°, 108°, 72° and 108°.
Check your answer:
2
Write the rules for corresponding, alternate and co-interior angles on parallel lines. If one angle is 70°, give the three measures. How does the converse give parallelism?
Model answer (board format)
Corresponding equal, alternate equal, co-interior sum 180°. The measures are 70°, 70° and 110°. If one corresponding or alternate pair is equal, or one co-interior pair adds to 180°, the lines are parallel.
Check your answer:
3
Write Theorems 6.6, 6.7 and 6.8 in one line each. Then show the calculation of the in-between angle for 135° and 40°.
Model answer (board format)
Lines parallel to the same line are parallel to each other. The angles of a triangle add to 180°. An exterior angle equals the sum of the two far interior angles. 180° − 135° = 45°, and 45° + 40° = 85°.
Check your answer:
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
A vertically opposite angle is 130°. An adjacent angle is —
💡 Hint:
A linear pair.
Explanation:
180° − 130° = 50°.
2
Co-interior angles on parallel lines are —
💡 Hint:
The sum is 180°.
Explanation:
Supplementary.
3
In a triangle, after 80° and 60°, the third angle is ______ degrees.
💡 Hint:
40.
Explanation:
180 − 140.
4
Axiom 6.3 is about corresponding angles.
Explanation:
True.
5
m ∥ n. An alternate angle is 48°. Find the co-interior angle and name the theorem.
Model answer (board format)
180° − 48° = 132°. Alternate angles are equal by Theorem 6.2 and co-interior angles are supplementary by Theorem 6.4.
Check your answer:
6
This is coefficient practice, not a result of lines and angles. 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 writes co-interior angles as equal. The correction is —
💡 Hint:
The sum is 180°.
Explanation:
On parallel lines, co-interior angles are supplementary.
2
On a map two roads are each parallel to a third. It is right to say —
💡 Hint:
Theorem 6.6.
Explanation:
Lines parallel to the same line are parallel to each other.
3
Assertion (A): The three angles of a triangle add to 180°.
Reason (R): A linear pair also adds to 180°.
Explanation:
Both are true, but R is not the reason for the angle sum. The reason uses a parallel and alternate angles.
4
The ratio 5 : 7 means the angles are 5° and 7° because a ratio gives small angles.
Explanation:
False — the linear pair shares 180°, so the angles are 75° and 105°.
5
Two horizontal bars of a window are parallel. A slanted rod makes one angle 68°. Write the corresponding and co-interior angles. Why is this measurement not the proof?
Model answer (board format)
The corresponding angle is 68° and the co-interior angle is 112°. The measurement makes the figure clear. The proof comes from Axiom 6.3 and Theorem 6.4.
Check your answer:
6
Two angles of a triangular park are 48° and 67°. Find the third angle and one exterior angle. Write one line on why the exterior angle is larger.
Model answer (board format)
The third = 180° − 115° = 65°. The exterior = 48° + 67° = 115°. 115° is larger than both 48° and 67°, because it is their sum.
Check your answer:
🏛️ 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 →
The verified label will be used only when a source page for the question is available.
CBSE-style · competency-based
These are case and assertion-reason practice items. Do not treat them as past CBSE questions.
🔁 Spaced review — today’s questions
Wrong questions return soon; correct ones return after a few days.
🧠 What you learned + equation sheet
What you learned
- Adjacent angles share one arm, and the other arms lie on opposite sides of it.
- A linear pair adds to 180°, and the converse is also an axiom.
- When two lines cross, the vertically opposite angles are equal.
- A transversal on parallel lines makes corresponding and alternate angles equal.
- Co-interior angles add to 180°. The converse makes the lines parallel.
- The angles of a triangle add to 180°. An exterior angle equals the sum of the two far interior angles.
| What | Keep this |
|---|---|
| Linear pair | 180° |
| Vertically opposite | equal |
| Corresponding | equal if parallel |
| Alternate | equal if parallel |
| Co-interior | sum 180° |
| Triangle | angle sum 180° |
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.