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State Board · Class 6 · Mathematics

Lesson 8: Triangles and Squares

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

This lesson is also in: मराठी

Parts of a triangle

From daily life

A slice of pizza, a flag on a pole and a set square all have three corners and three straight edges. That is a triangle.

Simple definition

A triangle (त्रिकोण) is a closed shape made by joining three non-collinear points with line segments. It has 3 vertices, 3 sides and 3 angles.

Rule 1: Naming the parts — △XYZ has vertices X, Y, Z; sides XY, YZ, XZ; and angles ∠X, ∠Y, ∠Z. The middle letter of an angle name is its vertex, so ∠XYZ is the angle at Y.

Triangle PQR with its vertices, sides and angles.PQRside PQside PRside QR3 vertices3 sides3 angles

In the figure : In the figure triangle PQR with its vertices, sides and angles.

  • Easy : Easy: △ABC has 3 sides AB, BC, AC.
  • Medium : Medium: In ∠LMN the vertex is M.
  • Exam-type : Exam: ∠QRP and ∠PRQ both mean the angle at R.

Rule 2: Smallest shape and counting — Three points on one straight line cannot make a triangle. Three matchsticks are the fewest that make a closed shape. Counting tip: a line from A to 5 points on a base makes 10 triangles (choose 2 of 5 points: 4 + 3 + 2 + 1 = 10).

  • Easy : Easy: 3 matchsticks make 1 triangle.
  • Medium : Medium: Points A, B, C on one line do not form a triangle.
  • Exam-type : Exam: Base with 3 points and apex A: 3 + 2 + 1 = 6 triangles.
Remember

Triangle: 3 vertices, 3 sides, 3 angles. Angle name has its vertex in the middle.

Common mistake

Writing the angle name with the vertex at the end. The vertex is in the middle.

Shortcut

Memory clue: "कोनाचे नाव = मध्ये शिरोबिंदू".

Spot the difference

PartNumberIn △XYZ
Vertices3X, Y, Z
Sides3XY, YZ, XZ
Angles3∠X, ∠Y, ∠Z

Practice questions

Easy
  1. 1How many vertices does a triangle have?
  2. 2Name the sides of △XYZ.
  3. 3What is the vertex of ∠XYZ?
Medium
  1. 4Can three points on one line form a triangle?
  2. 5What is the fewest number of matchsticks to make a closed shape?
  3. 6Is ∠QRP the same as ∠PRQ?
Challenging
  1. 7Apex A is joined to 3 points on the base line. How many triangles?
  2. 8Apex A is joined to 5 points on the base line. How many triangles?
Show answers (try them yourself first)
  1. 13.
  2. 2XY, YZ and XZ.
  3. 3Y.
  4. 4No.
  5. 53.
  6. 6Yes, both are the angle at R.
  7. 73 + 2 + 1 = 6.
  8. 84 + 3 + 2 + 1 = 10.

Interior, exterior and region

From daily life

A child standing on the painted line of a triangle in the playground is on the triangle. One standing in the middle is inside it, and one standing far away is outside it.

Simple definition

Points inside a triangle form its interior (अंतर्भाग), points outside form its exterior (बाह्यभाग), and points on the sides and vertices are on the triangle.

Rule 1: Three positions — In the figure D lies on the triangle, E is in the interior and F is in the exterior. A point on a side is not inside.

E is in the interior, D is on the triangle and F is in the exterior.E insideD onF outsideTriangular region = triangle + interior

In the figure : In the figure D on the triangle, E inside, F outside.

  • Easy : Easy: A point on a vertex is on the triangle.
  • Medium : Medium: The centre of the triangle is in the interior.
  • Exam-type : Exam: A point on side AB is on the triangle, not in the interior.

Rule 2: Regions — A triangle together with its interior is the triangular region. In the same way a quadrilateral and its interior make the region of the quadrilateral. The area of a triangle means the area of its region.

  • Easy : Easy: A triangle plus its interior is a triangular region.
  • Medium : Medium: Colouring a triangle fully colours its region.
  • Exam-type : Exam: Area is measured for the region, not for the line.
Remember

On, inside, outside. Triangle + interior = region.

Common mistake

Saying a point on a side is inside the triangle. It is on the triangle.

Shortcut

Memory clue: "रेषेवर = त्रिकोणावर".

Spot the difference

PositionName
On the sidesOn the triangle
InsideInterior
OutsideExterior

Practice questions

Easy
  1. 1What is the interior of a triangle?
  2. 2What is the exterior of a triangle?
  3. 3A point lies on a vertex. Where is it?
Medium
  1. 4Is a point on a side in the interior?
  2. 5What is a triangular region?
  3. 6What is the region of a quadrilateral?
Challenging
  1. 7A child stands on the painted line of a triangle in the playground. Is the child inside it?
  2. 8Why do we measure area of the region and not of the line?
Show answers (try them yourself first)
  1. 1The points inside it.
  2. 2The points outside it.
  3. 3On the triangle.
  4. 4No, it is on the triangle.
  5. 5The triangle together with its interior.
  6. 6The quadrilateral together with its interior.
  7. 7No, the child is on the triangle.
  8. 8Area is the space covered, which is inside the boundary.

Triangles by sides

From daily life

A rope tied into a triangle can have all three sides equal, only two equal, or all different. The sides decide the name.

Simple definition

Equilateral (समभुज): all three sides equal. Isosceles (समद्विभुज): exactly two sides equal. Scalene (विषमभुज): all three sides different.

Rule 1: Naming by sides — Sides 8, 8, 8 cm give an equilateral triangle. Sides 5, 5, 3 cm give an isosceles triangle. Sides 4.5, 5.2, 6.1 cm give a scalene triangle.

Triangles by sides: all equal, two equal, none equal.EquilateralIsoscelesScaleneSame ticks = equal sides

In the figure : In the figure triangles by sides.

  • Easy : Easy: 7, 7, 7 is equilateral.
  • Medium : Medium: 6, 9, 6 is isosceles.
  • Exam-type : Exam: 3, 4, 5 is scalene.

Rule 2: Using the idea — A 15 m rope tied into an equilateral triangle has each side 15 ÷ 3 = 5 m. An isosceles triangle with sides 8, 8, 5 cm has perimeter 8 + 8 + 5 = 21 cm.

  • Easy : Easy: 12 m rope as an equilateral triangle: 4 m each side.
  • Medium : Medium: Equilateral side 6 cm gives perimeter 18 cm.
  • Exam-type : Exam: Isosceles 10, 10, 6 cm: perimeter 26 cm.
Remember

Equilateral: 3 equal. Isosceles: 2 equal. Scalene: none equal.

Common mistake

Calling an equilateral triangle isosceles in the exam. Use the "exactly two" rule.

Shortcut

Memory clue: "समभुज 3, समद्विभुज 2, विषमभुज 0".

Spot the difference

TriangleEqual sides
Equilateral3
Isosceles2
Scalene0

Practice questions

Easy
  1. 1Name a triangle with all sides equal.
  2. 2Name a triangle with exactly two equal sides.
  3. 3Name the triangle with sides 4.5, 5.2, 6.1 cm.
Medium
  1. 4Name the triangle with sides 6, 9, 6 cm.
  2. 5A 15 m rope is tied into an equilateral triangle. Find each side.
  3. 6Find the perimeter of an isosceles triangle with sides 8, 8, 5 cm.
Challenging
  1. 7The perimeter of an equilateral triangle is 27 cm. Find a side.
  2. 8An isosceles triangle has perimeter 22 cm and two equal sides of 8 cm. Find the third side.
Show answers (try them yourself first)
  1. 1Equilateral.
  2. 2Isosceles.
  3. 3Scalene.
  4. 4Isosceles.
  5. 515 ÷ 3 = 5 m.
  6. 621 cm.
  7. 727 ÷ 3 = 9 cm.
  8. 822 − 16 = 6 cm.

Triangles by angles

From daily life

A triangle can be sharp like a roof, square-cornered like a set square, or wide like a half-open book. Its angles decide the name.

Simple definition

Acute-angled (लघुकोनी): all angles less than 90°. Right-angled (काटकोनी): one angle exactly 90°. Obtuse-angled (विशालकोनी): one angle more than 90°.

Rule 1: Naming by angles — Angles 40°, 60°, 80° give an acute-angled triangle. Angles 90°, 50°, 40° give a right-angled triangle. Angles 100°, 45°, 35° give an obtuse-angled triangle. The three angles always add up to 180°.

Triangles by angles: acute, right and obtuse.Acute-angledall angles < 90°Right-angledone angle = 90°Obtuse-angledone angle > 90°

In the figure : In the figure triangles by angles.

Live figure

Triangles by angles

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

Open in the app
  • Easy : Easy: 50°, 60°, 70° is acute-angled (sum 180°).
  • Medium : Medium: 90°, 30°, 60° is right-angled.
  • Exam-type : Exam: 20°, 20°, 140° is obtuse-angled.

Rule 2: Limits and double names — A triangle cannot have two right angles (90° + 90° = 180°) or two obtuse angles. A right-angled triangle can be isosceles (90°, 45°, 45°). An equilateral triangle is always acute-angled (60° each). Angles 20°, 20°, 140° make it obtuse-angled and isosceles.

  • Easy : Easy: An equilateral triangle has 60° each, so it is acute-angled.
  • Medium : Medium: 90°, 45°, 45° is right-angled and isosceles.
  • Exam-type : Exam: Angles 70° and 50° leave 180° − 120° = 60° for the third.
Remember

Acute: all less than 90°. Right: one 90°. Obtuse: one more than 90°. Sum 180°.

Common mistake

Giving a triangle two obtuse angles. Their sum would pass 180°.

Shortcut

Memory clue: "मोठा कोन ठरवतो नाव".

Spot the difference

TriangleLargest angle
Acute-angledless than 90°
Right-angled90°
Obtuse-angledmore than 90°

Practice questions

Easy
  1. 1What is an acute-angled triangle?
  2. 2What is a right-angled triangle?
  3. 3What is an obtuse-angled triangle?
Medium
  1. 4Angles are 50°, 60°, 70°. Name the triangle.
  2. 5Two angles are 70° and 50°. Find the third.
  3. 6Can a triangle have two right angles?
Challenging
  1. 7Angles 20°, 20°, 140°: name the triangle in two ways.
  2. 8Why is an equilateral triangle always acute-angled?
Show answers (try them yourself first)
  1. 1All three angles are less than 90°.
  2. 2One angle is exactly 90°.
  3. 3One angle is more than 90°.
  4. 4Acute-angled.
  5. 560°.
  6. 6No, they would already add up to 180°.
  7. 7Obtuse-angled and isosceles.
  8. 8Each angle is 180° ÷ 3 = 60°, less than 90°.

Quadrilateral and its elements

From daily life

A school notice board, a window and a kite all have four corners. Such a shape is a quadrilateral.

Simple definition

A quadrilateral (चौकोन) has 4 vertices, 4 sides, 4 angles and 2 diagonals (कर्ण), lines joining opposite vertices.

Rule 1: Opposite and adjacent — In PQRS, sides with no common vertex are opposite (PQ and SR). Sides with a common vertex are adjacent (PQ and QR). There are 2 pairs of opposite sides, 4 pairs of adjacent sides, 4 pairs of adjacent angles and 2 pairs of opposite angles.

Quadrilateral ABCD with its diagonals AC and BD.ABCDABDCdiagonalsAC, BDAB, DC oppositeAB, BC adjacent

In the figure : In the figure quadrilateral ABCD with diagonals AC and BD.

  • Easy : Easy: PQ and SR are opposite sides.
  • Medium : Medium: PQ and QR are adjacent sides.
  • Exam-type : Exam: The diagonals of PQRS are PR and QS.

Rule 2: Naming a quadrilateral — Go round the vertices in order: 4 starting points × 2 directions = 8 names, such as PQRS, QRSP, RSPQ, SRQP. PRQS is wrong because it jumps across a diagonal.

  • Easy : Easy: PQRS is a correct name.
  • Medium : Medium: SRQP is also correct, going round the other way.
  • Exam-type : Exam: PRQS is wrong, because P and R are opposite vertices.
Remember

Quadrilateral: 4 vertices, 4 sides, 4 angles, 2 diagonals. Name it by going round.

Common mistake

Naming in a jumping order such as PRQS. Follow the edge.

Shortcut

Memory clue: "चौकोनाचे नाव कडेकडेने".

Spot the difference

ItemCount
Opposite sides pairs2
Adjacent sides pairs4
Diagonals2
Names of one quadrilateral8

Practice questions

Easy
  1. 1How many sides does a quadrilateral have?
  2. 2How many diagonals does it have?
  3. 3Name the diagonals of PQRS.
Medium
  1. 4Name a side opposite to PQ in PQRS.
  2. 5Name a side adjacent to PQ in PQRS.
  3. 6How many pairs of adjacent angles does a quadrilateral have?
Challenging
  1. 7How many different names can one quadrilateral have?
  2. 8Is PRQS a correct name for quadrilateral PQRS? Why?
Show answers (try them yourself first)
  1. 14.
  2. 22.
  3. 3PR and QS.
  4. 4SR.
  5. 5QR (or PS).
  6. 64.
  7. 74 starting points × 2 directions = 8.
  8. 8No, P and R are opposite vertices, so the order jumps across a diagonal.