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.

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 ${900}^{}$.

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.

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

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.