Chapter -10

Practical Geometry

Introduction to lines and points Line:

A line is a straight figure which doesn’t have an endpoint and extends infinitely in opposite directions.

Properties of angles associated with parallel lines

Parallel lines are the lines that do not intersect or meet each other at any point in a plane.

hen a transversal intersects two parallel lines, then pairs of angles are formed, such as:

• Corresponding angles
• Alternate interior angles
• Alternate exterior angles
• Vertically opposite angles
• Linear pair

If two lines are intersecting each other at a point, in a plane, they are called intersecting lines. If they meet each other at 90 degrees, then they are called perpendicular lines.

Construction of a line parallel to the given line

Step 1. Take a line l and a point A outside the line l.

Step 2. Take any point B on l and join B to A.

Step 3. With B as centre and a convenient radius, draw an arc cutting line l at C and BA at D.

Step 4. Now with A as centre and the same radius as in Step 3, draw an arc EF cutting AB at G.

Step 5. Place the pointed tip of the compasses at C and adjust the opening so that the pencil tip is at D.

Step 6. With the same opening as in Step 5 and with G as centre, draw an arc cutting the arc EF at H.

Step 7. Now, join AH to draw a line m.

Since ABC and BAH are alternate interior angles

Properties of triangle

The triangles are classified based on sides or angles and the following important properties concerning triangles:

(i) The exterior angle of a triangle is equal in measure to the sum of interior opposite angles.

(ii) The total measure of the three angles of a triangle is 180°.

(iii) Sum of the lengths of any two sides of a triangle is greater than the length of the third side.

(iv) In any right-angled triangle, the square of the length of hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Two triangles are said to be congruent if their sides have same length and angles have same measure.

Construction of triangles

The construction of triangles is based on the rules of congruent triangles. A triangle can be drawn if-

• Three sides are given (SSS criterion).
• Two sides and an included angle are given (SAS criterion).
• Two angles and an included side are given. (ASA criterion).
• A hypotenuse and a side are given for right angle triangle (RHS criterion).

Construction of a triangle using SSS Criterion

Construction of a triangle with three given sides (SSS criterion)

Example

Draw a triangle ABC with the sides AB = 6 cm, BC = 5 cm and AC = 9 cm.

Solution

Step 1: First of all draw a rough sketch of a triangle, so that we can understand how to go ahead.

Step 2: Draw a line segment AB = 6 cm.

Step 3: From point A, C is 9 cm away so take A as a centre and draw an arc of 9 cm.

Step 4: From point B, C is 5 cm away so take B as centre and draw an arc of 5 cm in such a way that both the arcs intersect with each other.

Step 5: This point of intersection of arcs is the required point C. Now join AC and BC.

ABC is the required triangle.

SAS Criterion

Construction of a triangle if two sides and one included angle is given (SAS criterion)

Example

Construct a triangle LMN with LM = 8 am, LN = 5 cm and NLM = 60°.

Solution

Step 1: Draw a rough sketch of the triangle according to the given information.
Step 2: Draw a line segment LM = 8 cm.
Step 3: draw an angle of 60° at L and make a line LO.
Step 4: Take L as a centre and draw an arc of 5 cm on LO.
Step 5: Now join NM to make a required triangle LMN.

ASA Criterion

Construction of a triangle if two angles and one included side is given (ASA criterion)

Example

Draw a triangle ABC if BC = 8 cm, B = 60°and C = 70°.

Solution

Step 1: Draw a rough sketch of the triangle.

Step 2: Draw a line segment BC = 8 cm.

Step 3: Take B as a centre and make an angle of 60° with BC and join BP.

Step 4: Now take C as a centre and draw an angle of 70° using a protractor and join CQ. The point where BP and QC intersects is the required vertex A of the triangle ABC.

ABC is the required triangle ABC.

RHS Criterion

Construction of a right angle triangle if the length of the hypotenuse and one side is given (RHScriterion)

Example

Draw a triangle PQR which is right angled at P, with QR =7 cm and PQ = 4.5 cm.

Solution

Step 1: Draw a rough sketch of the triangle.

Step 2: Draw a line segment PQ = 4.5 cm.

Step 3: At P, draw PS PQ. This shows that R must be somewhere on this perpendicular.

Step 4: Take Q as a centre and draw an arc of 7 cm which intersects PS at R.

PQR is the required triangle.

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###### Medha Sharma

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