Triangle is the simplest polygon of all the closed figures formed in a plane by three line segments. It is a closed figure formed by three line segments having six elements - three angles (i) ∠ABC or ∠B (ii) ∠ACB or ∠C (iii) ∠CAB or ∠A and three sides - (iv) AB (v) BC (vi) CA.
Triangle: A plane figure bounded by three line segments.
Types of Triangles
Based on Sides
- Scalene Triangle: A triangle in which all the sides are of different lengths.
- Isosceles Triangle: A triangle having two sides equal.
- Equilateral Triangle: A triangle having all sides equal.
Based on Angles
- Right-angled Triangle: A triangle in which one of the angles is right angle.
- Obtuse angled triangle: A triangle in which one of the angles is obtuse angle.
- Acute angled triangle: A triangle in which all the three angles are acute.
Angle Sum Property of a Triangle
The sum of the three interior angles of a triangle is 180º.
∠A + ∠B + ∠C = 180º
Exterior Angles of a Triangle
The angle formed by a produced side of the triangle and another side of the triangle is called an exterior angle of the triangle.
An exterior angle of a triangle is equal to the sum of the two interior opposite angles.
Congruence of Triangles
Two figures, which have the same shape and same size are called congruent figures and this property is called congruence.
- Two line segments are congruent when they are of equal length.
- Two squares are congruent if their sides are equal.
- Two triangles are congruent, if all the sides and all the angles of one are equal to the corresponding sides and angles of other.
For example, in triangles PQR and XYZ
Q = XY, PR = XZ, QR = YZ
∠P = ∠X, ∠Q = ∠Y, ∠R = ∠Z
Thus, ∆PQR is congruent to ∆XYZ and we write ∆PQR ≅ ∆XYZ where ≅ is symbol of congruence.
Criteria of Congruence
In order to prove, whether two triangles are congruent or not, you need to know that all the six parts of one triangle are equal to the corresponding parts of the other triangle. It is possible to prove the congruence of two triangles, even if you are able to know the equality of three of their corresponding parts.
- SAS Criterion of Congruence: If the two sides and the included angle of one triangle are equal to the corresponding sides and included angle of the other triangle, the two triangles are congruent.
- ASA or AAS Criterion of Congruence: If any two angles and one side of a triangle are equal to corresponding angles and the side of the another triangle, then the two triangles are congruent.
- SSS Criterion of Congruence: If the three sides of one triangle are equal to the corresponding sides of another triangle, then the two triangles are congruent.
- RHS Criterion of Congruence: If the hypotenuse and a side of one right triangle are respectively equal to the hypotenuse and a side of another right triangle, then the two triangles are congruent.
Properties or Theorems of Triangles
- The angles opposite to equal sides of a triangle are equal.
- The sides opposite to equal angles of a triangle are equal.
- Perpendiculars or altitudes drawn on equal sides, from opposite vertices of an isosceles triangle are equal.
- If two sides of a triangle are unequal, then the longer side has the greater angle opposite to it.
- In a triangle, the greater angle has longer side opposite to it.
- Sum of any two sides of a triangle is greater than the third side.
Similarity of Triangles
Objects which have the same shape but different sizes are called similar objects. Any two polygons, with corresponding angles equal and corresponding sides proportional are similar.
Two triangles are similar if
- their corresponding angles are equal
- their corresponding sides are proportional
∆ABC ∼ ∆DEF if ∠A = ∠D, ∠B = ∠E, ∠C = ∠F and AB/DE = BC/EF = CA/FD.
The symbol ∼ stands for is similar to.
Criteria for Similarity
- AAA Criterion for Similarity: If in two triangles the corresponding angles are equal, the triangles are similar.
- SSS Criterion for Similarity: If the corresponding sides of two triangles are proportional, the triangles are similar.
- SAS Criterion for Similarity: If one angle of a triangle is equal to one angle of the other triangle and the sides containing these angles are proportional, the triangles are similar.
Basic Proportionality Theorem
If a line drawn parallel to one side of a triangle intersects the other two sides at distinct points, the other two sides of the triangles are divided proportionally. If a line divides any two sides of a triangle in the same ratio, the line is parallel to third side of the triangle.
Bisector of Angle of Triangle
The internal bisector of any angle of a triangle divides the opposite side in the ratio of sides containing the angle.
Important Results
- If a perpendicular is drawn from the vertex containing right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to each other and to the original triangle.
- The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.
- Pythagoras Theorem: In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
- In a triangle, if the square on one side is equal to sum of the squares on the other two sides, the angle opposite to first side is a right angle.