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CBSE · Class 7 · Mathematics

Chapter 7: A Tale of Three Intersecting Lines

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

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Constructing triangles from three sides

From daily life

A carpenter joins three sticks of 4, 5 and 6 feet; the corners are fixed and the frame cannot wobble.

Simple definition

A triangle has 3 sides, 3 vertices (corners) and 3 angles. Given three side lengths, draw one side as the base, then draw two arcs from its ends with the other two lengths; the arcs cross at the third vertex.

6 cm4 cm5 cmABC

In the figure: Base AB = 6 cm; arcs of 4 cm from A and 5 cm from B meet at C.

Rule 1: Steps with compass and ruler — Draw the base, set the compass to a side, draw an arc from one end, repeat from the other end.

  • Easy: Sides 4, 5, 6: base 6 cm.
  • Medium: Join C to A and B.
  • Exam-type: Check by measuring the sides again.

Rule 2: Equilateral triangle — All three sides are equal, so use the same compass setting for both arcs; each angle is 60°.

  • Easy: Side 5 cm: arcs of 5 cm from both ends.
  • Medium: Perimeter = 3 × 5 = 15 cm.
  • Exam-type: Perimeter 30 cm → side 10 cm.

Rule 3: Perimeter — Perimeter = sum of the three sides.

  • Easy: 4 + 5 + 6 = 15 cm.
  • Medium: Isosceles 7, 7, 6: 20 cm.
  • Exam-type: Perimeter 20, equal sides 7 each → base 6.
Remember

The two arcs cross at the third vertex; join it to both ends.

Common mistake

Changing the compass opening between the two arcs of an equilateral triangle.

Shortcut

Draw the longest side as base so that the arcs cross cleanly. Memory clue: "Base first, arcs next, join last."

Spot the difference

TriangleSidesCompass setting
Equilateral5, 5, 5same for both arcs
Scalene4, 5, 6different for each arc

Practice questions

Easy
  1. 1How many vertices does a triangle have?
  2. 2Find the perimeter of a triangle with sides 5, 6 and 7 cm.
  3. 3What is each angle of an equilateral triangle?
Medium
  1. 4An equilateral triangle has perimeter 27 cm. Find its side.
  2. 5Isosceles triangle: equal sides 9 cm, perimeter 25 cm. Find the base.
  3. 6Which tool draws the arcs?
Challenging
  1. 7Sides 6, 8, 10 cm: which side is the best base?
  2. 8Perimeter of a triangle with sides x, x and 4 is 18. Find x.
Show answers (try them yourself first)
  1. 13
  2. 218 cm
  3. 360°
  4. 49 cm
  5. 57 cm
  6. 6Compass
  7. 710 cm (longest)
  8. 87

The triangle inequality

From daily life

Two sticks of 2 and 3 feet cannot reach across a 6-foot gap; with sticks 3 and 4 they can reach a 5-foot gap.

Simple definition

Three lengths form a triangle only if the sum of any two is greater than the third. This is the triangle inequality. It is enough to check that the two smaller sides together exceed the largest.

3 + 4 > 5: triangle2 + 3 < 6: gap, no triangle

In the figure: Bars 3 and 4 reach beyond 5 (triangle); bars 2 and 3 fall short of 6 (no triangle).

Live figure

The triangle inequality

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Rule 1: Test three lengths — Add the two smaller lengths; compare with the largest.

  • Easy: 3, 4, 5: 3 + 4 = 7 > 5 → triangle.
  • Medium: 2, 3, 6: 2 + 3 = 5 < 6 → no triangle.
  • Exam-type: 5, 5, 10: 5 + 5 = 10, not greater → flat line, no triangle.

Rule 2: Range of the third side — The third side is more than the difference and less than the sum of the other two.

  • Easy: Sides 4 and 9: third between 5 and 13.
  • Medium: Whole-number choices: 6, 7, …, 12.
  • Exam-type: Sides 7 and 7: third between 0 and 14.

Rule 3: Why it is true — The straight path between two points is the shortest, so one side is shorter than the other two together.

  • Easy: Walking A to C directly is shorter than A to B to C.
  • Medium: So AC < AB + BC.
  • Exam-type: Equal means all three points lie on one line.
Remember

Only the two smaller sides need to beat the largest.

Common mistake

Saying 5, 5, 10 forms a triangle because 5 + 5 = 10. The sum must be greater, not equal.

Shortcut

Sort the lengths first, then add the two small ones. Memory clue: "Small two must beat the big one."

Spot the difference

LengthsSmall twoTriangle?
3, 4, 57 > 5Yes
2, 3, 65 < 6No
5, 5, 1010 = 10No

Practice questions

Easy
  1. 1Can 3, 4, 5 be the sides of a triangle?
  2. 2Can 2, 3, 6 form a triangle?
  3. 3Can 7, 8, 9 form a triangle?
Medium
  1. 4Sides 4 and 9: between which numbers is the third side?
  2. 5Can 5, 5, 10 form a triangle?
  3. 6How many whole-number lengths can the third side take if the other two are 6 and 8?
Challenging
  1. 7A triangle has sides 8, 8 and x. Give the range of x.
  2. 8For 10, 15, 30, how much must 10 grow so a triangle exists (smallest whole number)?
Show answers (try them yourself first)
  1. 1Yes
  2. 2No
  3. 3Yes
  4. 4between 5 and 13
  5. 5No
  6. 611 (3, 4, …, 13)
  7. 70 < x < 16
  8. 86 (16 + 15 = 31 > 30)

Constructing with sides and angles

From daily life

To fix a roof beam, a builder knows two rafter lengths and the angle between them, and that single triangle is then fixed.

Simple definition

A triangle is fixed by two sides and the angle between them (SAS), or by two angles and the side between them (ASA). In ASA the third angle is 180° minus the two given angles. Use a protractor to mark the angles.

6 cm5 cm50°SAS2 sides + angle between

In the figure: Sides 6 cm and 5 cm with 50° between them: draw base, mark 50°, cut off 5 cm, join.

Rule 1: SAS construction — Draw one side, make the given angle at one end, mark the second side on that arm, then join.

  • Easy: 6 cm, 5 cm and 50° between.
  • Medium: The third side is found by joining the ends.
  • Exam-type: Use a protractor at the right end.

Rule 2: ASA construction — Draw the side, make one angle at each end; the rays meet at the third vertex.

  • Easy: Base 7 cm, angles 40° and 60°.
  • Medium: Third angle = 180° – 40° – 60° = 80°.
  • Exam-type: Angles 70° and 50° → third angle 60°.

Rule 3: When angles are too big — Two angles must add to less than 180°, otherwise the rays never meet.

  • Easy: 100° + 70° = 170° < 180°: possible.
  • Medium: 110° + 80° = 190°: impossible.
  • Exam-type: 90° + 90° = 180°: rays are parallel, no triangle.
Remember

The angle in SAS lies between the two given sides.

Common mistake

Putting the angle at the wrong corner. It must be where the two given sides meet.

Shortcut

Mark the base-line zero of the protractor exactly at the end point. Memory clue: "Side-Angle-Side: the angle sits in the sandwich."

Spot the difference

CaseGivenThird part
SSS3 sides3 angles by measuring
SAS2 sides + angle betweenjoin ends
ASA2 angles + side between180° – sum

Practice questions

Easy
  1. 1In SAS, where does the angle lie?
  2. 2Angles 40° and 60° are given. Find the third angle.
  3. 3Which tool measures angles?
Medium
  1. 4Angles 70° and 50° are given. Find the third angle.
  2. 5Can a triangle have angles 110° and 80°?
  3. 6Two angles are 35° and 45° with side 8 cm between. Name the case.
Challenging
  1. 7Two angles of a triangle are equal and the third is 40°. Find them.
  2. 8Angles 90° and 90° with a side between: possible?
Show answers (try them yourself first)
  1. 1Between the two sides
  2. 280°
  3. 3Protractor
  4. 460°
  5. 5No: sum is 190°
  6. 6ASA
  7. 770° each
  8. 8No: rays are parallel

Altitudes of a triangle

From daily life

A plumb line hangs straight down from the ridge of a roof to the floor: it shows the height of the roof.

Simple definition

An altitude is the perpendicular from a vertex to the opposite side (or its extension). Its length is the height. Every triangle has 3 altitudes and they pass through one common point.

h

In the figure: Acute triangle: altitude inside. Obtuse triangle: the altitude falls outside, on the extended base.

Rule 1: Drawing an altitude — Place a set square on the base, slide until the right angle edge touches the vertex, then draw.

  • Easy: Pick a vertex and the opposite base.
  • Medium: Mark the right angle at the foot.
  • Exam-type: Repeat for the other two vertices.

Rule 2: Height and area — Area = half × base × height, where height is the altitude to that base.

  • Easy: Base 10, height 6: area 30.
  • Medium: Base 8, height 5: area 20.
  • Exam-type: Area 36, base 9 → height 8.

Rule 3: Where altitudes lie — Acute: all inside. Right: two are the legs. Obtuse: two lie outside.

  • Easy: Equilateral: all 3 inside.
  • Medium: Right triangle: the legs are altitudes.
  • Exam-type: Obtuse: extend the base to draw the altitude.
Remember

An altitude is perpendicular to the side; a median goes to the midpoint.

Common mistake

Using the slanting side as height. Height is always the perpendicular distance.

Shortcut

Mark the small square at the foot to remember it is 90°. Memory clue: "Altitude = Always at Angle 90°."

Spot the difference

LineGoes toAngle
Altitudeopposite side (perpendicular)90°
Medianmidpoint of opposite sideany

Practice questions

Easy
  1. 1What is the angle between an altitude and its base?
  2. 2How many altitudes does a triangle have?
  3. 3Base 10, height 6: find the area.
Medium
  1. 4Area 36, base 9: find the height.
  2. 5In a right triangle with legs 6 and 8, one altitude is 6. Which side is its base?
  3. 6For an obtuse triangle, where can an altitude fall?
Challenging
  1. 7Right triangle with legs 6 and 8: its area?
  2. 8Area 60 and height 10: find the base.
Show answers (try them yourself first)
  1. 190°
  2. 23
  3. 330
  4. 48
  5. 5The side of length 8
  6. 6Outside, on the extended base
  7. 724
  8. 812

Types of triangles and angle sum

From daily life

A tent end, a set square and a slice of pizza are all triangles, but each looks different in sides and angles.

Simple definition

By sides: equilateral (3 equal), isosceles (2 equal), scalene (none equal). By angles: acute (all less than 90°), right (one 90°), obtuse (one more than 90°). The three angles of any triangle add to 180°.

equilateralisoscelesscalene

In the figure: Equilateral, isosceles and scalene triangles.

Live figure

Types of triangles and angle sum

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Rule 1: Name by sides — Count how many sides are equal.

  • Easy: 5, 5, 5 → equilateral.
  • Medium: 6, 6, 4 → isosceles.
  • Exam-type: 3, 4, 5 → scalene.

Rule 2: Name by angles — Look at the largest angle.

  • Easy: 60°, 70°, 50° → acute.
  • Medium: 90°, 50°, 40° → right.
  • Exam-type: 110°, 40°, 30° → obtuse.

Rule 3: Angle sum 180° — Third angle = 180° minus the other two.

  • Easy: 50° + 60° → third 70°.
  • Medium: Right triangle with 35° → other acute angle 55°.
  • Exam-type: Isosceles with apex 40° → base angles 70° each.
Remember

Equilateral is also isosceles, but not the other way round.

Common mistake

Calling a triangle with a 100° angle "acute". One angle above 90° makes it obtuse.

Shortcut

A triangle can have only one right or obtuse angle. Memory clue: "Sides: 3-2-0 equal; angles: biggest decides."

Spot the difference

NameRuleExample
Equilateral3 equal sides5, 5, 5
Isosceles2 equal sides6, 6, 4
Scaleneno equal sides3, 4, 5

Practice questions

Easy
  1. 1Name the triangle with sides 5, 5, 5.
  2. 2What do the three angles of a triangle add to?
  3. 3Name the triangle with angles 90°, 50°, 40°.
Medium
  1. 4Two angles are 50° and 60°. Find the third.
  2. 5Isosceles triangle with apex angle 40°: find a base angle.
  3. 6Right triangle has one angle 35°. Find the other acute angle.
Challenging
  1. 7Angles are x, 2x and 3x. Find x.
  2. 8Can a triangle have two right angles? Why?
Show answers (try them yourself first)
  1. 1Equilateral
  2. 2180°
  3. 3Right
  4. 470°
  5. 570°
  6. 655°
  7. 730°
  8. 8No: 90° + 90° = 180° leaves nothing for the third angle