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.
3. With B as the centre and a convenient radius, cut an arc on l at C and BA at D.
4. With A as the centre and same radius as in Step 3, cut an arc EF to cut AB at G.
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
7. Join AH to draw a line m
∠ABC and ∠BAH are alternate interior angles. Therefore, m || l
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.
83,988
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
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
Basics of Practical Geometry
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.
