theorems on triangles

In this lesson, we are going to cover some important theorems on triangles. 

• If two sides of a triangle are of unequal lengths, then the greater side has the greater angle opposite to it. Conversely, if two angles of a triangle are of unequal measures, then the greater angle has the greater side opposite to it. 
  

AB > AC ⇒ Therefore, ∠ACB > ∠ABC [Angles opposite to side AB and AC]

Conversely, as  ∠BAC > ∠ABC ⇒ Therefore, BC > AC [Sides opposite to angle A and B]

Corollaries

1. The sum of the lengths of any two sides of a triangle is greater than the length of the third side. 
2. The difference between the lengths of any two sides of a triangle is less than the length of the third side. 

• Pythagoras Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the remaining two sides. 

△ABC is a right-angled triangle where ∠C = 90°, AB is the hypotenuse then
AB² = AC² + BC² 

• The line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to one-half of it. 

D is the midpoint of AB and E is the midpoint of AC, then DE || BC and DE = ½ BC

• The measure of the exterior angle of a triangle is equal to the sum of the corresponding interior angles.

Exterior angle ∠ABD = ∠BAC + ∠ACB

• Angular Bisector and Incenter: The incenter of the triangle always lies in the interior of a triangle. A line that bisects any interior angle of a triangle is called its angular bisector and the point at which the three angular bisectors meet is called the incenter of the triangle.

AD, BE, and CF are the three angular bisectors of triangle ABC. The three angular bisectors are concurrent at point I which is called the incenter of the triangle. The point I will always lie in the interior of a triangle. 

• Median and Centroid: The centroid of a triangle divides each median internally in the ratio of 2:1.

AD, BC, and CF are the three medians of a triangle ABC. The line segment joining the vertex to the mid-point of the opposite side of a triangle is called a median of a triangle. The three medians are concurrent at G which is called the centroid of the triangle, then ​​​​​\(\frac{BG}{GE} = \frac{AG}{GD} = \frac{CG}{GF} = \frac{2}{1}\)

• Altitude and Orthocenter: The orthocenter of an acute-angled triangle lies in the interior of the triangle. The orthocenter of the obtuse-angled triangle lies in the exterior of the triangle. The orthocenter of the right-angled triangle lies at the vertex, where the triangle is a right angle. A line segment drawn from a vertex, perpendicular to the opposite side of a triangle is called an altitude of the triangle. The three altitudes are concurrent at a point called the orthocenter.

AD, BE and CF are the three altitudes of triangle ABC.  H is the orthocenter of the triangle. Here as  △ ABC is an acute-angled triangle hence the orthocenter lies inside the triangle. 

The altitudes of an equilateral triangle are equal. The altitudes to the equal sides of an isosceles triangle are equal. 

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Example 1: State whether the following triangle is right-angled or not. 

Check if 13² = 5² + 12²

As it satisfies the Pythagoras theorem, therefore, the given triangle is a right-angled triangle. 

Example 2: Find ∠x in the given figure. 

∠x = 40 + 60 = 100° (The measure of the exterior angle of a triangle is equal to the sum of the corresponding interior angles.)