Triangles & Angles
Triangle: A triangle is a three-sided polygon that consists of three edges and three vertices. The most important property of a triangle is that the sum of the internal angles of a triangle is equal to 180 degrees. This property is called the angle sum property of a triangle.
Properties of Triangles
Let us discuss some of the properties of triangles.
- A triangle has three sides and three angles.
- The sum of the angles of a triangle is always 180 degrees.
- The exterior angles of a triangle always add up to 360 degrees.
- The sum of consecutive interior and exterior angle is supplementary.
- The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Similarly, the difference between the lengths of any two sides of a triangle is less than the length of the third side.
- The shortest side is always opposite the smallest interior angle. Similarly, the longest side is always opposite the largest interior angle.
Types of Triangles
On the basis of the length of the sides, triangles are classified into three categories:
- Scalene Triangle
- Isosceles Triangle
- Equilateral Triangle
On the basis of measurement of the angles, triangles are classified into three categories:
- Acute Angle Triangle
- Right Angle Triangle
- Obtuse Angle Triangle
Angle: The angle is formed when two rays originate from the same endpoint. The rays making an angle are called the arms of the angle and the endpoint is called the vertex of the angle. Different types of angles, such as
1.acute angle- An acute angle measures between 0° and 90°
2. right angle- A right angle is exactly equal to 90°.
3.obtuse angle- An angle greater than 90° but less than 180° is called an obtuse angle.
4.straight angle- A straight angle is equal to 180°.
5. reflex angle- An angle which is greater than 180° but less than 360° is called a reflex angle.
Two angles whose sum is 90° are called complementary angles, and two angles whose sum is 180° are called supplementary angles.
Practice Time
Question 1: The measures of two angles of a triangle are 72° and 58°. Find the measure of the third angle.
Solution:
The measures of two angles of a triangle are 72° and 58°.
Let the third angle be x.
Now, the sum of the measures of all the angles of a triangle is 180o.
∴∴ x + 72o + 58o = 180o
⇒ x + 130o = 180o
⇒ x = 180o −- 130o
⇒ x = 50o
The measure of the third angle of the triangle is 50o.
Question 2: The angles of a triangle are in the ratio 1 : 3: 5. Find the measure of each of the angles.
Solution:
The angles of a triangle are in the ratio 1:3:5.
Let the measures of the angles of the triangle be (1x), (3x) and (5x)
Sum of the measures of the angles of the triangle = 180o
∴ 1x + 3x + 5x = 180o
⇒ 9x = 180o
⇒ x = 20o
1x = 20o
3x = 60o
5x = 100o
The measures of the angles are 20o, 60o and 100o.
Question 3: One of the acute angles of a right triangle is 50°. Find the other acute angle.
Solution:
In a right angle triangle, one of the angles is 90o.
It is given that one of the acute-angled of the right-angled triangles is 50o.
We know that the sum of the measures of all the angles of a triangle is 180o.
Now, let the third angle be x.
Therefore, we have:
90o + 50o + x = 180o
⇒ 140o + x = 180o
⇒ x = 180o −- 140o
⇒ x = 40o
The third acute angle is 40o.
Question 4:
One of the angles of a triangle is 110° and the other two angles are equal. What is the measure of each of these equal angles?
ANSWER: ∠A = 110o and ∠B = ∠C
Now, the sum of the measures of all the angles of a triangle is 180o.
∠A + ∠B + ∠C = 180o
⇒ 110o + ∠B + ∠B = 180o
⇒ 110o + 2∠B = 180o
⇒ 2∠B = 180o −- 110o
⇒ 2∠B = 70o
⇒ ∠B = 70o / 2
⇒ ∠B = 35o
∴ ∠C = 35o
The measures of the three angles:
∠A = 110o, ∠B = 35o, ∠C = 35o
Question 5: The angles of a triangle are in the ratio 2 : 3: 4. The largest angle is
(a) 60°
(b) 80°
(c) 76°
(d) 84°
ANSWER:
(b) 80o
Let the measures of the given angles be (2x)o, (3x)o and (4x)o.
∴ (2x)o + (3x)o + (4x)o = 180o
⇒ (9x)o = 180o
⇒ x = 180 / 9
⇒ x = 20o
∴ 2x = 40o, 3x = 60o, 4x = 80o
Hence, the measures of the angles of the triangle are 40o, 60o, 80o. Thus, the largest angle is 80o.
MP TET Previous Year Question Paper (2012)
Question 1. If two angles of a triangle are 70 and 45, then the third angle is:
A. 40
B. 55
C. 45
D. 65
Ans. D.
Third angle = 180 – (70 + 45) = 65°
Question 2. Which of the following is not an acute angle?
A. 70
B. 75
C. 80
D. 95
Ans. D.
The acute angle is an angle smaller than 90ᵒ. Therefore, 95ᵒ is an acute angle
Thanks!
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