We will discuss here about some of the properties of triangle.
Relation between the measures of three angles of a triangle.
The sum of three angles of every triangle is 180°.
In ∆ABC, ∠A + ∠B + ∠C = 180°,
Draw three triangles on your not book. Name them as ∆PQR, ∆ABC and ∆LMN. With the help of protector measure all the angles the angles and find them: In ∆ABC ∠ABC + ∠BCA + ∠CAB = 180°
In ∆PQR ∠PQR + ∠QRP + ∠RPQ = 180° In ∆LMN ∠LMN + ∠MNL + ∠NLM = 180° |
Here, we observe that in each case, the sum of the measures of three angles of a triangle is 180°.
Hence, the sum of the three angles of a triangle is equals to 180°.
Note: If two angles of a triangle are given, we can easily find out its third angle.
Solved Examples on Angle Sum Property of a Triangle:
1. In a right triangle, if one angle is 50°, find its third angle.
Solution:
∆ PQR is a right triangle, that is, one angle is right angle.
Given, ∠PQR = 90°
∠QPR = 50°
Therefore, ∠QRP = 180° - (∠Q + ∠ P)
= 180° - (90° + 50°)
= 180° - 140°
∠R = 40°
2. PQR is an equilateral triangle. Find the measure of its each angle.
Solution:
PQR is an equilateral triangle. ∠P = ∠Q = ∠R According to the angle sum property of a triangle, we get ∠P + ∠Q + ∠R = 180° ⟹ ∠P + ∠P + ∠P = 180°; [Since, ∠P = ∠Q = ∠R] ⟹ 3 ∠P = 180° ⟹ ∠P = \(\frac{180°}{3}\) ⟹ ∠P = 60° Thus, ∠P= ∠Q= ∠R = 60° |
Therefore, each angle of an equilateral triangle is 60°.
Triangle inequality property is the relation between lengths of the side of a triangle.
∆ABC has three sides namely AB, BC and CA.
For a shorter notation, the length of the side opposite to the vertex A is written as 'a'
i.e., a = BC
Similarly, b = CA and c = AB
If we measure the lengths of a, b and c, we find the following relation:
a + b > c
b + c > a
c + a > b
Now, we have the following:
The sum of any two sides in a triangle is greater than the third side.
Solved Examples on Triangle Inequality Property:
1. Draw a ∆ABC. Measure the length of its three sides.
Let the lengths of the three sides be AB = 5 cm, BC = 7 cm, AC = 8 cm.
Now add the lengths of any two sides compare this sum with the lengths of the third side.
(i) AB + BC = 5 cm + 7 cm = 12 cm
Since 12 cm > 8 cm
Therefore, (AB + BC) > AC
(ii) BC + CA = 7 cm + 8 cm = 15 cm
Since 15 cm > 5 cm
Therefore, (BC + CA) > AB
(iii) CA + AB = 8 cm + 5 cm = 13 cm
Since 13 cm > 7 cm
Therefore, (CA + AB) > BC
In the below figure we can see in each case, if we add up any two sides of the ∆, the sum is more than its third side.
Thus, we conclude that the sum of the length of any two sides of a triangle is greater than the length of the third side.
Solved Examples on Triangle Inequality Property:
1. Is it possible to have a triangle whose sides are 5 cm, 6 cm and 4 cm?
Solution:
The lengths of the sides are 5 cm, 6 cm, 4 cm,
(a) 5 cm + 6 cm > 4 cm.
(b) 6 cm + 4 cm > 5 cm.
(c) 5 cm + 4 cm > 6 cm.
Hence, a triangle with these sides is possible.
2. Which of the following can be the possible lengths (in cm) of a triangle?
(i) 3, 5, 3
(ii) 4, 3, 8
Solution:
(i) Since 3 + 5 (i.e., 8) > 3, 5 + 3 (i.e., 8) > 3 and 3 + 3 (i.e., 6) > 5, therefore 3, 5, 3 (in cm) can be the lengths of the sides of a triangle.
(ii) Since 4 + 3 (i.e., 7) < 8, therefore 4, 3, 8 (in cm) cannot be the lengths of the sides of a triangle.
Consider a triangle ABC. Produce its side BC to X.
∠ACX is called an exterior angle of ∆ABC at C.
Similarly, produce side CB to Y, then ∠ABY is an exterior angle of ∆ABC at B.
Now, ∠ACB i.e., ∠3 is called the interior adjacent angle for ∠ACX at C, whereas ∠CBA and ∠CAB are called interior opposite angles for ∠ACX at C.
Similarly, ∠ABC i.e., ∠2 is called the interior adjacent angle for ∠ABY and ∠ACB, BAC are the interior opposite angles for ∠ABY.
Let us find a relation between the exterior angle and its interior opposite angles of a ∆ABC shown in the above figure.
In ∆ABC, Also, ∠1 + ∠ 2+ ∠3 = 180 deg; [Angle Sum Property]
Also, ∠ACB + ∠ACX = 180°; [Linear Pair]
⟹ ∠3 + ∠ACX = 180°
⟹ ∠3 + ∠ACX = ∠1 + ∠2 + ∠3; (Since, 1 + ∠2 + ∠3 = 180°)
⟹ ∠ACX = ∠1 + ∠2
Thus, exterior ∠ACX = sum of its two interior opposite angles, where ∠1 (= angle A) and ∠2 (= angle B) are the two interior opposite angles of the exterior ∠ACX
Similarly, exterior ∠ABY = ∠1 + ∠3
i.e. exterior ∠ABY = sum of its two interior opposite angles
Now, we have the following:
1. In a triangle, an exterior angle is equal to the sum of its two interior opposite angles.
2. In a triangle, an exterior angle is greater than either of the two interior opposite angles.
To Construct a Triangle whose Three Sides are given.
To Construct a Triangle when Two of its Sides and the included Angles are given.
To Construct a Triangle when Two of its Angles and the included Side are given.
To Construct a Right Triangle when its Hypotenuse and One Side are given.
Worksheet on Construction of Triangles.
5th Grade Math Problems
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