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State Board · Class 9 · Mathematics Part 2

Lesson 2: Parallel Lines

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी

Angles formed by a transversal

From daily life

Railway tracks run side by side and never meet. A road that cuts across both tracks makes a set of angles at the two crossing points. These angles come in pairs with special names.

Simple definition

Parallel lines (समांतर रेषा) are coplanar lines that never meet. A line cutting two lines at different points is a transversal (छेदिका); it forms 8 angles.

Rule 1: Eight angles and four kinds of pairs — The 8 angles form pairs of corresponding angles (संगत कोन; same relative position), alternate interior angles (व्युत्क्रम अंतर्कोन; between the lines, opposite sides of the transversal), alternate exterior angles (व्युत्क्रम बाह्यकोन) and interior angles on the same side (छेदिकेच्या एकाच बाजूचे अंतर्कोन).

A transversal crossing two lines with the eight angles numbered 1 to 8A transversal t cuts the lines l and m: 8 angleslmt∠2∠1∠3∠4∠6∠5∠7∠8

In the figure : In the figure a transversal and the eight angles it forms with two lines.

The four kinds of angle pairs formed by a transversal, each shown in colourCorresponding anglesAlternate interior anglesAlternate exterior anglesInterior angles, same side

In the figure : In the figure the four kinds of angle pairs: corresponding, alternate interior, alternate exterior and interior angles on the same side.

  • Easy : Easy: A transversal forms 8 angles.
  • Medium : Medium: Angles between the two lines are interior angles; there are 4 of them.
  • Exam-type : Exam: Name the pair: the angles above the top line and above the bottom line on the left of the transversal. They are corresponding angles.

Rule 2: Angle facts we already know — When two lines intersect, vertically opposite angles are congruent and a linear pair is supplementary. If one pair of corresponding angles is congruent, all remaining pairs are congruent; the same holds for alternate angles. If one pair of interior angles on the same side is supplementary, the other pair is also supplementary.

  • Easy : Easy: A linear pair with 110° has the other angle 180° − 110° = 70°.
  • Medium : Medium: Vertically opposite to 70° is 70°.
  • Exam-type : Exam: If one of two intersecting lines forms an angle 2x with its vertically opposite angle 100°, then 2x = 100° and x = 50°.
Remember

Know the four pair types. Linear pair = 180°. Vertically opposite = equal.

Common mistake

Mixing up alternate and corresponding. Alternate angles are on opposite sides of the transversal.

Shortcut

Memory clue: "संगत = एकाच जागी; व्युत्क्रम = विरुद्ध बाजू".

Spot the difference

PairPosition
CorrespondingSame relative position
Alternate interiorBetween lines, opposite sides
Same-side interiorBetween lines, same side

Practice questions

Easy
  1. 1What are parallel lines?
  2. 2What is a transversal?
  3. 3How many angles does a transversal form?
Medium
  1. 4A linear pair has one angle 125°. The other?
  2. 5Name the four types of angle pairs.
  3. 6Where do alternate interior angles lie?
Challenging
  1. 7If one corresponding pair is congruent, what about the others?
  2. 8Two lines intersect, one angle is (3x + 10)° and its vertically opposite is 70°. Find x.
Show answers (try them yourself first)
  1. 1Coplanar lines that never meet.
  2. 2A line cutting two lines at different points.
  3. 38.
  4. 455°.
  5. 5Corresponding, alternate interior, alternate exterior, interior on the same side.
  6. 6Between the lines, on opposite sides of the transversal.
  7. 7All are congruent.
  8. 83x + 10 = 70, x = 20.

Interior angle theorem

From daily life

Imagine two roads that lean towards each other slightly. If you keep walking, they will meet. Euclid's fifth postulate says exactly this about angles on one side of a cutting line.

Simple definition

Interior angle theorem (अंतर्कोन प्रमेय): if a transversal cuts two parallel lines, the interior angles on the same side are supplementary (sum 180°).

Rule 1: Euclid's fifth postulate and the theorem — Fifth postulate: if the sum of the interior angles on one side of a transversal is less than two right angles, the lines meet on that side. Interior angle theorem: if two lines are parallel and a transversal cuts them, then ∠a + ∠b = 180° and ∠c + ∠d = 180°.

  • Easy : Easy: Interior angles 110° and 70° add to 180°.
  • Medium : Medium: If ∠a = 100°, the other interior angle on the same side is 80°.
  • Exam-type : Exam: Interior angles are 3x and 2x. Then 5x = 180°, x = 36°, and the angles are 108° and 72°.

Rule 2: Why it is true (indirect proof) — For ∠a + ∠b there are three possibilities: less than, more than, or equal to 180°. Less than 180° is impossible, since the lines would meet but they are parallel. More than 180° is impossible, since ∠a + ∠d = 180° and ∠c + ∠b = 180° give ∠c + ∠d = 360° − (∠a + ∠b), which would then be less than 180°. So ∠a + ∠b = 180°.

  • Easy : Easy: There are three possibilities: <, > or = 180°.
  • Medium : Medium: Less than 180° would make the lines meet, which is impossible.
  • Exam-type : Exam: It is called an indirect proof since the other two possibilities are denied by contradictions.
Remember

Parallel lines: same-side interior angles add up to 180°.

Common mistake

Using this rule for lines that are not parallel. It holds only for parallel lines.

Shortcut

Memory clue: "समांतर + एकाच बाजूचे अंतर्कोन = 180°".

Spot the difference

CaseResult
Sum < 180°Lines would meet, impossible
Sum > 180°Other side < 180°, impossible
Sum = 180°True

Practice questions

Easy
  1. 1State the interior angle theorem.
  2. 2What does Euclid's fifth postulate say?
  3. 3∠a = 70°. Find the interior angle on the same side.
Medium
  1. 4∠b = 85°. Find the other.
  2. 5Why is the proof called indirect?
  3. 6Interior angles are 4x and 5x. Find x and the angles.
Challenging
  1. 7Interior angles are (x + 30)° and (2x + 60)°. Find x.
  2. 8Why can ∠a + ∠b not be more than 180°?
Show answers (try them yourself first)
  1. 1For parallel lines, interior angles on the same side are supplementary.
  2. 2If interior angles on one side sum to less than 180°, the lines meet on that side.
  3. 3110°.
  4. 495°.
  5. 5Other possibilities are shown false by contradiction.
  6. 69x = 180, x = 20; 80° and 100°.
  7. 73x + 90 = 180, x = 30.
  8. 8Then ∠c + ∠d would be less than 180°, forcing the lines to meet.

Corresponding angles and alternate angles theorems

From daily life

When a ladder rests against two parallel poles, the angle it makes with the top of each pole is the same. Parallel lines pass an angle along unchanged.

Simple definition

For parallel lines, corresponding angles are equal and alternate angles are equal.

Rule 1: The two theorems — Corresponding angles: ∠a + ∠c = 180° (linear pair) and ∠b + ∠c = 180° (interior angles), so ∠a = ∠b. Alternate angles: ∠d + ∠c = 180° and ∠c + ∠b = 180°, so ∠d = ∠b.

Two parallel lines cut by a transversal: all blue angles are 40 degrees and all orange angles are 140 degreesIf l ∥ m and one angle is 40°lmt40°140°140°40°40°140°140°40°

In the figure : In the figure for parallel lines one known angle gives all the others: equal angles share its measure and the rest are 180° minus it.

Live figure

Corresponding angles and alternate angles theorems

In the app you can move this figure with your finger.

Open in the app
  • Easy : Easy: If one angle is 125°, its corresponding angle is 125°.
  • Medium : Medium: Its alternate interior angle is 125° and its same-side interior angle is 180° − 125° = 55°.
  • Exam-type : Exam: ∠a = (2x + 10)° and its corresponding angle is (3x − 20)°. Then 2x + 10 = 3x − 20, x = 30, and each angle is 70°.
Remember

Parallel lines: corresponding = equal. Alternate = equal. Same side interior = 180°.

Common mistake

Calling corresponding angles supplementary. They are equal.

Shortcut

Memory clue: "संगत, व्युत्क्रम = समान; एकाच बाजूचे = 180°".

Spot the difference

PairParallel lines
CorrespondingEqual
AlternateEqual
Interior, same sideSum 180°

Practice questions

Easy
  1. 1State the corresponding angles theorem.
  2. 2State the alternate angles theorem.
  3. 3One angle is 65°. Its corresponding angle?
Medium
  1. 4One angle is 65°. Its same-side interior angle?
  2. 5Alternate interior angle of 48°?
  3. 6∠a = (x + 20)° and its alternate angle is 65°. Find x.
Challenging
  1. 7Corresponding angles are (4x − 15)° and (3x + 10)°. Find x.
  2. 8Prove the corresponding angles are equal for parallel lines.
Show answers (try them yourself first)
  1. 1Corresponding angles are equal for parallel lines.
  2. 2Alternate angles are equal for parallel lines.
  3. 365°.
  4. 4115°.
  5. 548°.
  6. 6x = 45.
  7. 7x = 25; each angle is 85°.
  8. 8∠a + ∠c = 180° (linear pair) and ∠b + ∠c = 180° (interior angles), so ∠a = ∠b.

Angle sum property of a triangle

From daily life

Cut the three corners off a paper triangle and place them side by side: they form a straight line, a half turn of 180°. This is the angle sum property.

Simple definition

Theorem: the sum of the angles of a triangle is 180°.

Rule 1: Proof — Through A draw a line parallel to seg BC and take P and Q on it with P − A − Q. Seg AB and seg AC are transversals, so ∠ABC = ∠PAB and ∠ACB = ∠QAC (alternate angles). Adding ∠BAC to both sides: ∠ABC + ∠ACB + ∠BAC = ∠PAB + ∠BAC + ∠QAC = ∠PAC + ∠QAC = 180° (linear pair).

A triangle with a line through A parallel to BC; equal alternate angles x and z show that the three angles add up to 180 degreesABCPQxyzxzx + y + z = 180° (straight line), so ∠B + ∠A + ∠C = 180°Line PQ through A is parallel to BC: alternate angles are equal

In the figure : In the figure the line through A parallel to BC turns the angles x, y, z into a straight angle of 180°.

  • Easy : Easy: In ΔABC, ∠A = 62° and ∠B = 58°, so ∠C = 180° − 120° = 60°.
  • Medium : Medium: Angles in the ratio 1 : 2 : 3: 6x = 180°, x = 30°, so 30°, 60°, 90°.
  • Exam-type : Exam: In ΔPQR, ∠R = 60° and ∠P = 2∠Q. Then 3∠Q = 120°, so ∠Q = 40° and ∠P = 80°.
Remember

Triangle angles add to 180°. Find the third = 180° − the other two.

Common mistake

Subtracting from 360°. A triangle's angles add to 180°.

Shortcut

Memory clue: "त्रिकोण = 180°; तिसरा = 180° − दोन".

Spot the difference

StepReason
Draw line through A parallel to BCConstruction
∠ABC = ∠PABAlternate angles
∠ACB = ∠QACAlternate angles
Sum = 180°Linear pair

Practice questions

Easy
  1. 1What is the angle sum of a triangle?
  2. 2Two angles are 50° and 70°. Find the third.
  3. 3A right-angled triangle has an angle 35°. Find the third angle.
Medium
  1. 4Why is a line drawn through A parallel to BC?
  2. 5Angles of a triangle are x, x + 10, x + 20. Find x.
  3. 6Can a triangle have two right angles?
Challenging
  1. 7In an isosceles triangle, the apex angle is 40°. Each base angle?
  2. 8Which two theorems are used in the proof?
Show answers (try them yourself first)
  1. 1180°.
  2. 260°.
  3. 355°.
  4. 4To get alternate angles equal to ∠B and ∠C.
  5. 53x + 30 = 180, x = 50.
  6. 6No, 90° + 90° = 180° leaves 0° for the third.
  7. 7(180° − 40°) ÷ 2 = 70°.
  8. 8Alternate angles theorem and the linear pair.

Tests for parallel lines

From daily life

If a ladder meets the ground and the wall at the same angle at both rails, we can tell the two rails are parallel without measuring the distance. Angle tests let us decide parallel lines from angles alone.

Simple definition

Tests for parallel lines (समांतरतेच्या कसोट्या): supplementary same-side interior angles, equal alternate angles, or equal corresponding angles make the lines parallel.

Rule 1: The three tests — Interior angles test: if the interior angles on one side of a transversal of two lines are supplementary, the lines are parallel. Proof by contradiction: if they met at T, then in ΔPQT we would get ∠PTQ = 0°, impossible. Alternate angles test and corresponding angles test: if a pair of alternate (or corresponding) angles is congruent, the lines are parallel; both are proved using the interior angles test.

Three tests, each of which proves two lines parallelAny one of these gives parallel linesSupplementaryinterior angles on thesame side add to 180°Alternatealternate anglesare equalCorrespondingcorresponding anglesare equalResultthe two linesare parallel (l ∥ m)

In the figure : In the figure tests for parallel lines: supplementary interior angles, or equal alternate or corresponding angles.

  • Easy : Easy: Interior angles 100° and 80° add to 180°, so the lines are parallel.
  • Medium : Medium: Alternate angles 70° and 70° are equal, so the lines are parallel.
  • Exam-type : Exam: Corresponding angles 55° and 125° are not equal, so the lines are not parallel.

Rule 2: Using the tests and corollaries — For what x are the lines parallel if corresponding angles are (x + 20)° and (2x − 10)°? Put x + 20 = 2x − 10, so x = 30 and both angles are 50°. Corollary I: if a line is perpendicular to two lines in a plane, the two lines are parallel. Corollary II: two lines in a plane parallel to a third line in the plane are parallel to each other.

  • Easy : Easy: Two lines both perpendicular to a third are parallel.
  • Medium : Medium: If l ∥ n and m ∥ n, then l ∥ m.
  • Exam-type : Exam: Alternate angles (3x)° and (x + 40)° equal gives 2x = 40, x = 20, each 60°.
Remember

Supplementary same-side, or equal alternate, or equal corresponding angles mean parallel.

Common mistake

Using the tests when the angles are not in the right position. First name the pair type.

Shortcut

Memory clue: "कोन जुळले = रेषा समांतर".

Spot the difference

TestCondition
Interior anglesSum 180°
Alternate anglesEqual
Corresponding anglesEqual

Practice questions

Easy
  1. 1State the interior angles test.
  2. 2State the alternate angles test.
  3. 3Corresponding angles 80° and 80°. Parallel?
Medium
  1. 4Interior angles 95° and 90°. Parallel?
  2. 5State Corollary I.
  3. 6State Corollary II.
Challenging
  1. 7Find x if alternate angles (2x + 15)° and (4x − 25)° are equal.
  2. 8In the interior angles test proof, what is the contradiction?
Show answers (try them yourself first)
  1. 1If same-side interior angles are supplementary, the lines are parallel.
  2. 2If alternate angles are equal, the lines are parallel.
  3. 3Yes.
  4. 4No, sum is 185°.
  5. 5Two lines perpendicular to the same line are parallel.
  6. 6Lines parallel to the same line are parallel to each other.
  7. 72x = 40, x = 20; each 55°.
  8. 8The lines would meet at T and make ∠PTQ = 0°.

Quick revision

  • For parallel lines, corresponding angles and alternate angles are equal, and interior angles on the same side add up to 180°.
  • The sum of the three angles of a triangle is 180°.
  • Supplementary interior angles, or a congruent pair of alternate or corresponding angles, make the lines parallel.