We will discuss here about some of the properties of triangle.
I. Angle Sum Property of a 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 andfind 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 onAngle 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°.
II.Triangle Inequality Property:
Triangle inequality property is therelation 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 onTriangle Inequality Property:
1. Draw a ∆ABC. Measure the length of its three sides.
Let thelengths of the three sides be AB = 5 cm, BC = 7 cm, AC = 8 cm.
Now add thelengths 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 onTriangle 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.
III. Properties of Exterior Angles and Interior Opposite Angles 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.
You might like these
Types of Quadrilaterals | Properties of Quadrilateral | Parallelogram
Types of quadrilaterals are discussed here: Parallelogram: A quadrilateral whose opposite sides are parallel and equal is called a parallelogram. Its opposite angles are equal.
Interior and Exterior of an Angle | Interior Angle | Exterior Angle
Interior and exterior of an angle is explained here. The shaded portion between the arms BA and BC of the angle ABC can be extended indefinitely.
Worksheet on Circle |Homework on Circle |Questions on Circle |Problems
In worksheet on circle we will solve different types of question in circle. 1. The following figure shows a circle with centre O and some line segments drawn in it. Classify the line segments as radius, chord and diameter:
Worksheet on Triangle | Homework on Triangle | Different types|Answers
In the worksheet on triangle we will solve different types of questions on triangle. 1. Take three non - collinear points L, M, N. Join LM, MN and NL. What figure do you get? Name: (a)The side opposite to ∠L. (b) The angle opposite to side LN. (c) The vertex opposite to
Worksheet on Angles | Questions on Angles | Homework on Angles
In worksheet on angles you will solve different types of questions on angles. Classify the following angles into acute, obtuse, right and reflex angle: (i) 35° (ii) 185° (iii) 90°
Perpendicular Lines | What are Perpendicular Lines in Geometry?|Symbol
In perpendicular lines when two intersecting lines a and b are said to be perpendicular to each other if one of the angles formed by them is a right angle. In other words, Set Square Set Square If two lines meet or intersect at a point to form a right angle, they are
What are Parallel Lines in Geometry? | Two Parallel Lines | Examples
In parallel lines when two lines do not intersect each other at any point even if they are extended to infinity. What are parallel lines in geometry? Two lines which do not intersect each other
Classification of Triangle | Types of Triangles |Isosceles|Equilateral
Triangles are classified in two ways: (i) On the basis of sides and, (ii) On the basis of angles.In classification of triangle there are six elements in a triangle, that is, three sides and three angles. So, classification of triangle is done on the base of these elements.
Pairs of Angles | Complementary Angles | Supplementary Angles|Adjacent
Pairs of angles are discussed here in this lesson. 1. Complementary Angles: Two angles whose sum is 90° are called complementary angles and one is called the complement of the other.
Circle | Interior and Exterior of a Circle | Radius|Problems on Circle
A circle is the set of all those point in a plane whose distance from a fixed point remains constant. The fixed point is called the centre of the circle and the constant distance is known
Relation between Diameter Radius and Circumference |Problems |Examples
Relation between diameter radius and circumference are discussed here. Relation between Diameter and Radius: What is the relation between diameter and radius? Solution: Diameter of a circle is twice
Medians and Altitudes of a Triangle |Three Altitudes and Three Medians
Here we will discuss about Medians and Altitudes of a Triangle. Median: The straight line joining a vertex of a triangle to the midpoint of the opposite side is called a median. A triangle has three medians. Here XL, YM and ZN are medians. A geometrical property of medians
5th Grade Geometry Worksheet | Angles | Triangles | Classification
In 5th Grade Geometry Worksheet we will classify the given angles as acute, right or obtuse angle; using a protractor, find the measure of the given angle, classify the given triangle and circle the numbers with right angles. 1. Write 3 examples of right angles. 2. Name the
Intersecting Lines | What Are Intersecting Lines? | Definition
Two lines that cross each other at a particular point are called intersecting lines. The point where two lines cross is called the point of intersection. In the given figure AB and CD intersect each other at point O.
Pairs of Lines | Parallel and Perpendicular Pairs of Lines | Example
Here we will learn pairs of lines. When pairs of lines are given in a plane, they maybe (i) parallel to each other. (ii) intersecting each other. Two lines in a same plane not intersecting each other are called parallel lines. The lines AB and CD never meet each other
● Triangle.
Classification of Triangle.
Properties of Triangle.
Worksheet on Triangle.
To Construct a Triangle whose Three Sides are given.
To Construct a Triangle when Two of its Sides and theincluded Angles are given.
To Construct a Triangle when Two of its Angles and the includedSide are given.
To Construct a Right Triangle when its Hypotenuse and One Sideare given.
Worksheet on Construction of Triangles.
5th Grade Geometry Page
5th Grade Math Problems
From Properties of Triangle to HOME PAGE
Didn't find what you were looking for? Or want to know more informationabout 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.
Share this page:What’s this? |