Properties of 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 Sum Property of a Triangle
The sum of
the angles of a triangle is 180°.
In triangle
ABC, ∠a + ∠b + ∠c = 180°
We can use
simple paper cutting to show the Angle Sum Property of a Triangle.
1. Draw a
triangle ABC on a sheet of paper.
2. Cut out
the marked angles of the triangle as shown.
3. Put the
pieces with angles a and b next to the piece with angle c. The 3 angles lie on
a 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°
Exterior Angle Property of a Triangle
When a side
of a triangle is extended outwards from a vertex of a triangle, an angle
outside the triangle is formed. This angle formed is called an exterior
angle of the triangle. In the given figure, ∠ACD is
an exterior angle as the side BC of triangle ABC is extended outwards to D.
“The measure
of an exterior angle of a triangle is equal to the sum of the measures of the
two opposite interior angles of the triangle.”
In the given
figure, ∠x = ∠a + ∠b (ext.
∠ of triangle)
where ∠a and ∠b are the two opposite interior
angles of the exterior angle x.
We can also
establish the fact that the measure of an exterior angle of a triangle is equal
to the sum of the measures of the two opposite interior angles by the following
proof.
In the above
figure, 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
Example
2: In the given
figure, PQR is a straight line. Find ∠PQS.
Solution: ∠PQS = ∠QSR + ∠QRS (ext. ∠ of triangle)
= 48° +
90°
= 138°
The Triangle Inequality Property
In a
triangle ABC, the sides opposite the angles A, B and C are denoted by a, b and
c respectively.
The sides of a triangle satisfy an important property.
Consider the
three triangles below:
In each
triangle above, c + a > b. Also, the length of b is gradually increased in
each triangle.
What would
happen if c + a = b?
Obviously,
the side b would merge into the straight line formed by c and a.
Further, we
can also say that, c + a < b is not possible in a triangle. Thus, the sum of
any two sides of a triangle is greater than the third side.
Thus, the triangle inequality property states that,
If we are
given three lengths, then a triangle with these three lengths is possible only
if 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 the
largest of the three lengths. If the sum of the other two lengths is greater
than 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 following
sets of numbers could be the lengths of the sides of a triangle.
a. 5 cm, 6
cm, 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 largest
of these lengths is 8 cm.
The sum of
the 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 three
given lengths are 3.5 cm, 2.5 cm, 7 cm.
The largest
of these lengths is 7 cm.
The sum of
the 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.