We will discuss about some of the properties of angles of a triangle.
1. The three angles of a triangle are together equal to two right angles.
ABC is a triangle.
Then ∠ZXY + ∠XYZ + ∠YZX = 180°
Using this property, let us solve some of the examples.
Solved examples:
(i) In ∆XYZ, ∠X = 55° and ∠Y = 75°. Find ∠Z.
Solution:
∠X + ∠Y + ∠Z = 180°
or, 55° + 75° + ∠Z = 180°
or, 130° + ∠Z = 180°
or, 130°  130° + ∠Z = 180°  130°
Therefore, ∠Z = 50°
(ii) In the ∆XYZ, ∠Y = 5∠Z and ∠X= 3∠Z. Find the angles of the triangle.
Solution:
∠X + ∠Y + ∠Z = 180°
or, 3∠Z + 5∠Z + ∠Z = 180°
or, 9∠Z = 180°
or, \(\frac{9∠Z}{9}\) = \(\frac{180°}{9}\)
Therefore, ∠Z = 20°
We know, ∠X= 3∠Z
Now, plugin the value of ∠Z
∠X= 3 × 20°
Therefore, ∠X= 60°
Again we know, ∠Y= 5∠Z
Now, plugin the value of ∠Z
∠Y= 5 × 20°
Therefore, ∠Y= 100°
Hence, the angles of the triangle are ∠X = 60°, ∠Y = 100° and ∠Z = 20°.
2. If one side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.
The side QR of the ∆PQR is produced to S.
Then ∠PRS = ∠RPQ + ∠PQR
Corollary 1: An exterior angle of a triangle is greater than either of the interior opposite angles.
In ∆PQR, QR is produced to S.
Therefore, ∠PRS > ∠RPQ and ∠PRS ∠PQR
Corollary 2: A triangle can have only one right angle.
Corollary 3: A triangle can have only one obtuse angle.
Corollary 4: A triangle must have at least two acute angles.
Corollary 5: In a rightangled triangle, the acute angles are complementary.
Now, using this property, let us solve some of the following examples.
Solved examples:
(i) Find ∠Q from the given figure.
Solution:
∠P + ∠Q = ∠PRS
Given, ∠P = 50° and ∠PRS = 120°
or, 50° + ∠Q = 120°
or, 50°  50° + ∠Q = 120°  50°
or, ∠Q = 120°  50°
Therefore, ∠Q = 70°
(ii) From the given figure find all the angles of ∆ABC, given that ∠B = ∠C.
Solution:
Given, ∠B = ∠C
We know, ∠DAC = 150°
∠DAC + ∠CAB = 180°, as they form a linear pair
or, 150° + ∠CAB = 180°
or, 150°  150° + ∠CAB = 180°  150°
or, ∠CAB = 30°
Let ∠B = ∠C = x°
Therefore, x° + x° = 150°, as the exterior angle of a triangle is equal to the sum of the interior opposite angles.
or, 2x° = 150°
or, \(\frac{2x°}{2}\) = \(\frac{150°}{2}\)
or, x° = 75°
Therefore, ∠B = ∠C = 75°.
From Properties of Angles of a Triangle to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.