PROPERTIES OF TRIANGLE.

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ANGLE SUM PROPERTIES OF TRIANGLE.

The sum of the angles of a triangle is equal to 180°.

Let us verify the above result by means of an experiment :

1. Let us draw any triangle ABC. Cut the angles along three lines as shown in the figure and place them one after the other to form adjacent angles as shown in the second figure.

On doing this, it will be seen that the outer arms of the first and the last angle are in a straight line.
Which shows that the sum of the angles is 180°.
Hence, the sum of the angles of a triangle is equal to 180°.

2. Let us prove our contention with the help of set-squares. There are two set-squares.

First set-square has angles 30°, 60°, 90° [fig. (i)]
Their sum = 30° + 60° + 90° = 180°
Second set-square has angles 45°, 45°, 90° [fig. (ii)
Their sum = 45° + 45° + 90° = 180°These set-squares represent the triangles.
In each case, sum of angles of the triangular figures is 180°.

3. The above result can be proved theoretically as well.
In

are the interior angles of the triangle.
We are to prove that

Produce BC to D and through C, draw CE || AB
Now, CE || AB and AC cuts them.

Illustrative Examples

Example 1. Find the value of x  in the following diagrams :

Solution.

Example 2. Find the value of x and y in the following diagrams :

Solution.

Example 3. Two angles of a triangle are 30° and 80°. Find the third angle.

Solution.

Example 4. One of the angles of a triangle is 80° and the other two angles are equal. Find the measure of each of the equal angles.

Solution.

Example 5. The three angles of a triangle are in the ratio 1 :2 :1 . Find all the angles of the triangle. Classify the triangle in two different ways.

Solution. Let the angles of the triangle be x, 2x and x.

Hence, the angles of the triangle are 45°, 90° and 45°.
Thus, the triangle is a right-angled triangle and it can also be named as an isosceles right triangle.

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