Practical Geometry Class 7 Notes: Chapter 10

Lines That Don’t Meet

Method of construction of a line parallel to a given line, using only a sheet of paper

  1. Take a piece of paper.
  2. Fold it in half and unfold the line l. Mark a point A on paper outside l.
  3. Fold the paper perpendicular to the line such that this perpendicular passes through A. Name the perpendicular AN.
  4. Make a fold perpendicular to AN through point A. Name the new perpendicular line as m.
  5. Now,  l || m.

Steps of construction of a line parallel to a given line

  1. Take a line l and a point A outside l.
  2. Take any point B on l and join it to A.
    Steps of construction of a line parallel to a given line
  3. With B as the centre and a convenient radius, cut an arc on l at C and BA at D.
    Steps of construction of a line parallel to a given line
  4. With A as the centre and same radius as in Step 3, cut an arc EF to cut AB at G.
    Steps of construction of a line parallel to a given line
  5. Measure the arc length CD by placing pointed tip of the compass at C and pencil tip opening at D.
  6. With this opening, keep G as centre and draw an arc to cut arc EF at H
    Steps of construction of a line parallel to a given line
  7. Join AH to draw a line m
    Steps of construction of a line parallel to a given line

∠ABC and ∠BAH are alternate interior angles. Therefore, m || l

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Let’s Build Triangles

Classification of triangles based on sides and angles

Triangles can be classified based on their:

  1. SIDES:
    • Equilateral triangle: All three sides are equal in measure.
    • Isosceles triangle: Two sides have equal measure.
    • Scalene triangle: All three sides have different measures.
  2. ANGLES:
    • Acute triangle: All angles measure less than 900.
    • Obtuse triangle: One angle is greater than 900.
    • Right triangle: One angle is 900.

For More Information On Classification of triangles based on sides and angles, Watch The Below Video.


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Important properties of triangles

  1. The exterior angle is equal to the sum of interior opposite angles.
  2. The sum of all interior angles is 180°
  3. Sum of the lengths of any two sides is greater than the length of the third side.
  4. Pythagoras theorem: In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Properties of triangles

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Triangles can be constructed if any of the following measurements are given

  1. Three sides.
  2. Two sides and an angle between them.
  3. Two angles and a side between them.
  4. The hypotenuse and a leg in case of a right-angled triangle.

Construction of triangles given a criterion are listed below:

  • Construction of a triangle with SSS criterion.

    • Construct a triangle ABC, given that AB = 4.5 cm, BC = 5 cm and AC = 6 cm.
      Steps:

      1. Make a rough sketch for your reference
      2. Draw a line segment BC = 5 cm
      3. With B as centre, draw an arc of radius 4.5 cm
      4. With C as centre, draw an arc of radius 6 cm and cut the previous arc
      5. Mark the point of intersection of arcs as A. Join AB and AC. ΔABC is now ready
    • Note: SSS congruency rule:-  If three sides of one triangle are equal to the corresponding three sides of another triangle, then the two triangles are congruent
  • Construction of a triangle with SAS criterion

    • Construct ΔPQR with QR = 7.5 cm, PQ = 5 cm and ∠Q = 600.
      Steps:

      1. Make a rough sketch for your reference
      2. Draw a line segment QR = 7.5 cm
      3. At Q, draw QX making 600 with QR
      4. With Q as centre, draw an arc of radius 5 cm. It cuts QX at P.
      5. Join AB. ΔPQR is now ready
  • Construction of a triangle with ASA criterion

    • Construct ΔXYZ with ∠X = 300, ∠Y = 1000 and XY = 5.8 cm.
      Steps:

      1. Make a rough sketch for your reference
      2. Draw XY = 5.8cm
      3. At X, draw a ray XP making an angle of 300 with AB.
      4. At Y, draw a ray YQ making an angle of 1000 with XY.
      5. The point of intersection of the two rays is Z.
      6. ΔXYZ is now completed
  • Construction of a triangle with RHS criterion

    • Construct ΔLMN, where ∠M = 900, MN = 8cm and LN = 10 cm.
      Steps:

      1. Make a rough sketch for your reference
      2. Draw MN = 8 cm
      3. At M, draw MX ⊥ MN.
      4. With N as centre, draw an arc of radius 10 cm to cut MX at L
      5. Join LN.
      6. ΔLMN is now completed

To know more about Construction of triangles, visit here.

Basics of Practical Geometry

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Introduction to Constructions of basic figures

Basic constructions:

– To draw a line segment of given length
– a line perpendicular to a given line segment
– an angle
– an angle bisector
– a circle

  • In geometry, you have already studied basic constructions.
  • To recap, some of them are:
    • Drawing a line segment of given length
    • Drawing a line perpendicular to a given line segment.
    • Angles
    • Angle bisectors
    • Circles
  • Tools used for simple constructions are ruler, protractor and a compass.

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