VARIOUS TYPES OF TRIANGLES AND WRITE THEIR
1. EXTERIOR ANGLE PROPERTY
2. ANGLE SUM PROPERTY
3. ISOCELES TRIANGLE PROPERTIES
4. PYTHAGORAS PROPERTY OF A RIGHT ANGLED TRIANGLE
Answers
Step-by-step explanation:
Exterior Angle of a Triangle and Its Property
Angle sum property of a Triangle
Two Special Triangles: Equilateral and Isosceles
Sum of the Lengths of Two Sides of a Triangle
Right-Angled Triangles and Pythagoras Property
Properties Of Triangles
Exterior Angle of a Triangle and Its Property
Draw a triangle ABC and produce one of its sides, say BC as shown in . Observe the angle ACD formed at the point C. This angle lies in the exterior of ∆ABC. We call it an exterior angle of the ∆ABC formed at vertex C. Clearly ∠BCA is an adjacent angle to ∠ACD. The remaining two angles of the triangle namely ∠A and ∠B are called the two interior opposite angles or the two remote interior angles of ∠ACD. Now cut out (or make trace copies of) ∠A and ∠B and place them adjacent to each other.
An exterior angle of a triangle is equal to the sum of its interior opposite angles.
Consider ∆ABC. ∠ACD is an exterior angle.
To Show: m∠ACD = m∠A + m∠B
Through C draw CE, parallel to BA
Justification
Steps Reasons
a) ∠1 = ∠x BA || CE and AC is a transversal. Therefore, alternate angles should be equal.
b) ∠2 = ∠y BA || CE and BD is a transversal.
c) ∠1 + ∠2 = ∠x + ∠y
d) Now, ∠x + ∠y = m ∠ACD.
Hence, ∠1 + ∠2 = ∠ACD From the figure above.
The above relation between an exterior angle and its two interior opposite angles is referred to as the Exterior Angle Property of a triangle.
Angle sum property of a Triangle
There is a remarkable property connecting the three angles of a triangle. You are going to see this through the following four activities.
Thus, the sum of the measures of the three angles of a triangle is 180°.
The same fact you can observe in a different way also. Take three copies of any triangle, say ΔABC.
Arrange them. What do you observe about ∠1 + ∠2 + ∠3? You will also see the ‘exterior angle property’.
Take a piece of paper and cut out a triangle, say, ΔABC .
Statement: The total measure of the three angles of a triangle is 180°.
To justify this let us use the exterior angle property of a triangle.
Given : ∠1, ∠2, ∠3 are angles of ΔABC. ∠4 is the exterior angle when BC is extended to D.
Justification : ∠1 + ∠2 = ∠4 (by exterior angle property) ∠1 + ∠2 + ∠3 = ∠4 + ∠3 (adding ∠3 to both the sides). But ∠4 and ∠3 form a linear pair so it is 180°. Therefore, ∠1 + ∠2 + ∠3 = 180°. Let us see how we can use this property in a number of ways.
Two Special Triangles: Equilateral and Isosceles
A triangle in which all the three sides are of equal lengths is called an equilateral triangle.
We conclude that in an equilateral triangle:
all sides have same length.
each angle has measure 60°.
A triangle in which two sides are of equal lengths is called an isosceles triangle.
From a piece of paper cut out an isosceles triangle XYZ, with XY=XZ. Fold it such that Z lies on Y. The line XM through X is now the axis of symmetry. You find that ∠Y and ∠Z fit on each other exactly. XY and XZ are called equal sides; YZ is called the base; ∠Y and ∠Z are called base angles and these are also equal.
Thus, in an isosceles triangle.
AB + BC > AC-------(i)
Similarly, if one were to start from B and go to A, he or she will not take the route BC and CA but will prefer BA . This is because
BC + CA > AB-----------(ii)
By a similar argument, you find that
CA + AB > BC-----------(iii)
These observations suggest that the sum of the lengths of any two sides of a triangle is greater than the third side.
Collect fifteen small sticks (or strips) of different lengths, say, 6 cm, 7 cm, 8 cm, 9 cm, ..., 20 cm.Take any three of these sticks and try to form a triangle. Repeat this by choosingdifferent combinations of three sticks. Suppose you first choose two sticks of length 6 cm and 12 cm. Your third stick has to be of length more than 12 – 6 = 6 cm and less than 12 + 6 = 18 cm.
Right-Angled Triangles and Pythagoras Property
Pythagoras, a Greek philosopher of sixth century B.C. is said to have found a very important and useful property of right-angled triangles given in this section. The property is hence named after him. In fact, this property was known to people of many other countries too. The Indian mathematician Baudhayan has also given an equivalent form of this property. We now try to explain the Pythagoras property. In a right angled triangle, the sides have some special names. The side opposite to the right angle is called the hypotenuse; the other two sides are known as the legs of the right-angled triangle.
The squares are identical; the eight triangles inserted are also identical. Hence the uncovered area of square A = Uncovered area of square B. i.e., Area of inner square of square A = The total area of two uncovered squares in square B.
a2= b2 + c2