Maharashtra Board Class 9 Maths Part II Chapter 3 Triangles PDF Download

Triangles

Theorem Of Remote Interior Angles Of A Triangle

Theorem: The measure of an exterior angle of a triangle is equal to the sum of its remote interior angles.

Given: PRS is an exterior angle of triangle PQR.

To Prove: PRS = PQR + QPR

Proof: The sum of all angles of a triangle is 180°.

Therefore, PQR + QPR + PRQ = 180° .......(I)

PRQ + PRS = 180° ..... angles in linear pair ......(II)

From (I) and (II)

PQR + QPR + PRQ = PRQ + PRS

Therefore, PQR + QPR = PRS ....... eliminating PRQ from both sides

Therefore, the measure of an exterior angle of a triangle is equal to the sum of its remote interior angles.

Teacher's Note

In your own home, if you have a triangle drawn on paper and extend one side, the outer angle equals the two angles inside that are far from it. This is the exterior angle theorem used in real buildings.

Exam Trick

Remember: Exterior angle = Sum of two remote interior angles. Just like how the outside is made from inside parts, the outside angle is made from inside angles.

Points To Remember

An exterior angle is formed when you extend one side of a triangle.
The two angles inside the triangle that are not touching this exterior angle are called remote interior angles.
The exterior angle is always bigger than any single remote interior angle.
This theorem helps us find unknown angles in triangles without using all three angles.

Activity

Draw a triangle of any measure on a thick paper. Take a point T on ray QR as shown in the figure. Cut two pieces of thick paper which will exactly fit the corners of angle P and angle Q. See that the same two pieces fit exactly at the corner of angle PRT as shown in the figure.

Property Of An Exterior Angle Of Triangle

The sum of two positive numbers a and b, that is (a + b) is greater than a and greater than b also. That is, a + b \(>\) a, a + b \(>\) b

Using this inequality we get one property related to exterior angle of a triangle.

If PRS is an exterior angle of triangle PQR then PRS \(>\) P, PRS \(>\) Q

Therefore, an exterior angle of a triangle is greater than its remote interior angle.

Solved Examples

Ex (1) The measures of angles of a triangle are in the ratio 5 : 6 : 7. Find the measures.

Solution: Let the measures of the angles of a triangle be 5x, 6x, 7x.

Therefore, 5x + 6x + 7x = 180°

18x = 180°

x = 10°

5x = 5 × 10 = 50° 6x = 6 × 10 = 60° 7x = 7 × 10 = 70°

Therefore, the measures of angles of the triangle are 50°, 60° and 70°.

Ex (2) Observe the figure and find the measures of angle PRS and angle RTS.

Solution: PRS is an exterior angle of triangle PQR.

So from the theorem of remote interior angles,

PRS = PQR + QPR

= 40° + 30°

= 70°

In triangle RTS

TRS + RTS + TSR = 180° ........ sum of all angles of a triangle

Therefore, 70° + RTS + 90° = 180°

Therefore, RTS + 90° = 180°

Therefore, RTS = 90°

Teacher's Note

When you see a triangle with one side extended, you can find the outside angle by adding the two far inside angles. This is useful when solving real geometry problems in construction.

Exam Trick

Always mark the exterior angle clearly. Write down the two remote interior angles first. Then add them to get the exterior angle quickly.

Points To Remember

The exterior angle theorem makes angle problems easier to solve.
You do not need to find all three angles if you know two of them.
The exterior angle is always on a straight line with the third angle of the triangle.
Practice drawing different triangles and extending sides to understand this better.
This concept is used in many geometry proofs.

This is a preview of the first 3 pages. To get the complete book, click below.