A triangle is a three-sided polygon. It has three sides, threeÂ vertices and three angles. We know that a unique triangle can be constructed if

(i) all three sides are given

(ii) two sides and included angle are given

(iii) two angles and the included side is given

(iv) the measure of the hypotenuse and a side is given in the right triangle.

For constructing triangles from given data, we generally make use of the given congruency conditions and construct the required triangle. Let us discuss the technique of constructing different types of triangles when specific conditions are given:

## Construction of Triangles – Case 1

Given the base of a triangle, its baseÂ angle and sum of other two sides

For constructing âˆ†ABC such that base BC, base angle âˆ B and the sum of other two sides, i.e. AB + AC are given, the following steps of construction is followed:

**Steps of Construction of a Triangle**

1. Draw the base BC of âˆ†ABC as given and construct âˆ XBC of the required measure at B as shown.

2. Keeping the compass at point B cut an arc from the ray BX such that its lengthÂ equals to AB + AC at point P and join it to C as shown

3. Now measure âˆ BPC and from C, draw an angle equal to âˆ BPC as shown

âˆ†ABC is the required triangle. This can be proved as follows:

Sl.No |
Statement |
Reason |

Base BC and âˆ B are drawn as given. Now in âˆ†ACP, | ||

1 | âˆ ACP = âˆ APC | By Construction |

2 | AC = AP | âˆ†ACP is isosceles |

3 | AB = BP â€“ AP = BP â€“ AC | From Statement 2 |

4 | AB + AC = BP | Proved |

## Construction of Triangles – Case 2

Given the base of a triangle, its base angle and difference of the other two sides

For constructing âˆ†ABC such that base BC, base angleâˆ B and difference of the other two sides, i.e. AB – AC orÂ AC-AB is given, then for constructing triangles such as these two cases can arise:

- AB > AC
- AC > AB

The following steps of construction are followed for the two cases:

**Steps of Construction if AB > AC:**

1. Draw the base BC of âˆ†ABC as given and construct âˆ XBC of the required measure at B as shown.

2. From the ray, BX cut an arc equal to AB – AC at point P and join it to C as shown

3. Draw the perpendicular bisector of PC and let it intersect BX at point A as shown:4. Join AC, âˆ†ABC is the required triangle.

**Steps of Construction if AC > AB:**

1. Draw the base BC of âˆ†ABC as given and construct âˆ XBC of the required measure at B as shown.

2. From the ray BX cut an arc equal to AB – AC at point P and join it to C. In this case P will lie on the opposite side to the ray BX. Draw the perpendicular bisector of PC and let it intersect BX at point A as shown

3. Join the points A and C, and hence âˆ†ABC is the required triangle.