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**10. Practical Geometry**

**Construction of Triangles**

The sum of the measures of the three angles .

1) The sum of the measures of the three angles of a triangle is

2) **The sum of the lengths** of any **two sides of a triangle **is** always greater than **the** length of the third side.**

**AB + BC > CA; BC + CA > AB and CA + AB > BC**

3) **The difference between the lengths **of any** two sides of a triangle** is always less than the length of the **third side.**

**AB - BC < CA; BC - CA < AB and CA - AB < BC**

**Page 39**

4) The **measure of an exterior angle is** equal to the sum of the two remote or **opposite interior angles.**

**Practical Geometry - Construction of Triangles**

The properties of triangles are as follows:

**To construct a triangle, we should know any one of the following:**

**•** Length of the three sides**•** Two sides and the included angle**•** Two angles and the included side**•** Length of the hypotenuse and one side in case of a right-angled triangle.

**Construction of Parallel Lines**

Two lines in a plane that never meet each other at any point are said to be** parallel** to each other.

**Page 40**

Two lines in a plane that never meet each other at any point are said to be parallel to each other. Any line intersecting a pair of parallel lines is called a** transversal.**

**Properties of angles formed by parallel lines and transversal:**

1. All pairs of **alternate interior angles** formed by parallel lines and a transversal are equal.

2. All pairs of** corresponding angles** formed by parallel lines and a transversal are equal.

3. All pairs of **alternate exterior angles** formed by parallel lines and a transversal are equal.

4.** The interior angles** formed on the same side of the transversal are** supplementary** (the sum of their measures is 1800).

**Steps to construct parallel lines using the alternate interior angle property:**

1. Draw a line, l.

2. Mark point A outside line l.

3. Mark point B on line l.

4. Draw line joining points A and B.

5. Draw an arc with B as the centre, such that it intersects lines l and n at points D and E, respectively.

6. Draw another arc with the same radius and A as the centre, such that it intersects line n at F.

7. Ensure that the arc drawn from A cuts line n between A and B.

8. Measure distance DE with the help of the compass.

9. Draw another arc with F as the centre and radius equal to DE.

10. Mark the point of intersection of this arc and the previous arc as G.

11. Draw line m passing through A and G.

12. Lines l and m are parallel.

**Page 41**

**Verification of the construction:**

If the pair of alternate interior angles ∠ABC and ∠BAG are equal in measure, then line l //line m. Hence, the construction is verified.

**Steps to construct parallel lines using the corresponding angle property:**

1. Draw line l and point P outside it.

2. Mark point Q on line l.

3. Draw line joining point P and point Q.

4. Draw an arc with Q as the centre, such that it intersects line l at R and line n at S.

5. Draw another arc with the same radius and P as the centre, such that it intersects line n at X.

6. Ensure that the arc drawn from P cuts line n outside QP.

7. Draw another arc withX as the centre and distance RS as the radius, such that it intersects the previous arc at Y.

8. Draw line m passing through points P and Y.

9. Lines l and m are parallel.

**Verification of the construction:**

If the pair of corresponding angles ∠PQR and ∠XPY are equal in measure, then line l II line m.

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