triangle

A triangle is a simple closed curve made of three line segments. It has three vertices, three sides, and three angles. 

• \(\overline {AB}, \overline {BC}, \overline {AC}\) are the line segments
• \(\angle ABC, \angle BAC, \angle BCA\) are the three angles
• A, B, and C are three vertices. 

The side opposite to vertex A is \(\overline{BC}\), similarly, the side opposite to vertex B is \(\overline {AC}\) and the side opposite to vertex C is \( \overline {BA}\).

CLASSIFICATION

Classification of triangles based on sides

Scalene triangle - If none of the three sides of a triangle are equal to each other, it is called a Scalene Triangle.

Isosceles triangle - If in a triangle, any two sides are equal, then it is called an Isosceles Triangle.

Equilateral triangle - If all three sides of a triangle are equal, it is called an Equilateral Triangle.

Classification of triangles based on angles

Acute-angled triangle - If in a triangle each angle is less than 90°, then the triangle is called an acute-angled triangle.

Obtuse-angled triangle - If one of the angles is greater than 90°, then the triangle is called an obtuse-angled triangle.

Right-angled triangle - If one of the angles is a right angle then the triangle is called a right-angled triangle.

 

MEDIANS OF A TRIANGLE 

A median connects a vertex of a triangle to the mid-point of the opposite side. 

 

\(\overline {AD} \) is the median and \(\overline {BD} = \overline {DC}\)

How many medians a triangle can have? Solution: 3 ( medians from three vertices)

 

ALTITUDES OF A TRIANGLE

An altitude of a triangle is the perpendicular segment from a vertex of a triangle to the opposite side.

\(\overline {AO} \) is the altitude and  \(\angle AOC = 90^\circ, \angle AOB = 90^\circ\)

Will an altitude always lie in the interior of a triangle? Solution: No  An altitude is drawn from the vertex in an obtuse angle triangle, it lies in the exterior of the triangle.

 

EXTERIOR ANGLE OF A TRIANGLE

Draw a triangle ABC and produce one of its sides, say BC as shown in the below figure.

Observe the angle ACD formed at point C. This angle lies in the exterior of ∆ABC. We call it an exterior angle of the ∆ABC formed at vertex C. Clearly ∠BCA is an adjacent angle to ∠ACD. The remaining two angles of the triangle, namely ∠A and ∠B are called the two interior opposite angles.

Properties

An exterior angle of a triangle is equal to the sum of its interior opposite angles	∠ACD = ∠A + ∠B
The sum of the exterior angle and its adjacent interior angle is 180°	∠ACD + ∠ACB = 180°

 

Are the exterior angles formed at each vertex of a triangle equal? Solution: No (the exterior angle is equal to the sum of two interior opposite angles)

ANGLE SUM PROPERTY OF A TRIANGLE

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

Draw two triangles on plane paper and measure their angles using a protector. What do you observe? In ∆ABC, ∠A + ∠B + ∠C = ? In ∆DEF ∠D + ∠E + ∠F = ? (you can draw any kind of triangle, the sum of all three angles will be 180°)

Example 1: In ΔABC, BC is 10 cm long. AD is a median. Find the length of DC.

Solution:  AD is a median, therefore it cuts the side BC into two equal halves. DC = 10/2 = 5 cm 

Example 2: Find the value of the exterior angle:

   

Solution: An exterior angle is the sum of two interior opposite angles, therefore it is equal to 60° + 40° = 100°

Example 3: Find the value of \(\angle x\) :

Solution: As the sum of three angles is equal to 180°, therefore, \(\angle x = 180 - 70- 45 = 65\)