Propertiesof Triangles
We know that a triangle is a 3-sided polygon. It has 3 sides, 3 angles and 3 vertices. There are some relations between the sides and the angles of a triangle. These relations are called the properties of triangles. In this article, I will discuss all the properties related to sides and angles of a triangle. Let us discuss these properties one by one.
Angle SumProperty of a Triangle
The sum ofthe angles of a triangle is 180°.
In triangleABC, ∠a + ∠b + ∠c = 180°
We can usesimple paper cutting to show the Angle Sum Property of a Triangle.
1. Draw atriangle ABC on a sheet of paper.
2. Cut outthe marked angles of the triangle as shown.
3. Put thepieces with angles a and b next to the piece with angle c. The 3 angles lie ona straight line, that is, the sum of these 3 angles is 180°.
Example 1: In the figure, AB = AC and AD // BC.Find the measures of angles x, y, and z.
Solution: ∠z = ∠CAD(alt. ∠, AD // BC) = 63°
∠y = ∠z (base∠s of isos. triangle) = 63°
∠x + ∠y + ∠z = 180° (∠ sum of triangle)
∠x + 63° + 63° = 180°
∴ ∠x = 180° – 63° – 63°
= 54°
ExteriorAngle Property of a Triangle
When a sideof a triangle is extended outwards from a vertex of a triangle, an angleoutside the triangle is formed. This angle formed is called an exteriorangle of the triangle. In the given figure, ∠ACD isan exterior angle as the side BC of triangle ABC is extended outwards to D.
“The measureof an exterior angle of a triangle is equal to the sum of the measures of thetwo opposite interior angles of the triangle.”
In the givenfigure, ∠x = ∠a + ∠b (ext.∠ of triangle)
where ∠a and ∠b are the two opposite interiorangles of the exterior angle x.
We can alsoestablish the fact that the measure of an exterior angle of a triangle is equalto the sum of the measures of the two opposite interior angles by the followingproof.
In the abovefigure, let ∠ACB be ∠c.
We have ∠a + ∠b + ∠c =180° (∠ sum of triangle) ... (1)
And ∠c + ∠x =180° (adj. ∠s on a st. line) ... (2)
Substituting(1) into (2), we have
∠c + ∠x = ∠a + ∠b + ∠c
∴ ∠x = ∠a + ∠b
Example2: In the givenfigure, PQR is a straight line. Find ∠PQS.
Solution: ∠PQS = ∠QSR + ∠QRS (ext. ∠ of triangle)
= 48° +90°
= 138°
TheTriangle Inequality Property
In atriangle ABC, the sides opposite the angles A, B and C are denoted by a, b andc respectively.
The sides of a triangle satisfy an important property.
Consider thethree triangles below:
In eachtriangle above, c + a > b. Also, the length of b is gradually increased ineach triangle.
What wouldhappen if c + a = b?
Obviously,the side b would merge into the straight line formed by c and a.
Further, wecan also say that, c + a < b is not possible in a triangle. Thus, the sum ofany two sides of a triangle is greater than the third side.
Thus, the triangle inequality property states that,
If we aregiven three lengths, then a triangle with these three lengths is possible onlyif the sum of every pair of lengths is greater than the third length.
In practice,to check whether a triangle can be formed out of three lengths, we pick thelargest of the three lengths. If the sum of the other two lengths is greaterthan this length, it is possible to form a triangle using these three lengths.If not, a triangle cannot be formed.
Example 3: Check to see whether the followingsets of numbers could be the lengths of the sides of a triangle.
a. 5 cm, 6cm, 8 cm b. 3.5 cm,2.5 cm, 7 cm
Solution: a. The three given lengths are 5 cm,6 cm, 8 cm.
The largestof these lengths is 8 cm.
The sum ofthe other two lengths is 5 cm + 6 cm = 11 cm
Since 5 + 6> 8, it is possible to form a triangle with sides of 5 cm, 6 cm, 8 cm.
b. The threegiven lengths are 3.5 cm, 2.5 cm, 7 cm.
The largestof these lengths is 7 cm.
The sum ofthe other two lengths is 3.5 cm + 2.5 cm = 6 cm.
Since 3.5 +2.5 < 7, it is not possible to form a triangle with sides of 3.5 cm, 2.5 cm,7 cm.
