# NCERT Solutions for Class 9 Maths Chapter 7 – Triangles PDF

### Free PDF of NCERT Solutions for Class 9 Maths Chapter 7 – Triangles

Includes all the questions provided in NCERT Books prepared by Mathematics expert teachers as per CBSE NCERT guidelines from wallindia.com .
To download our free pdf of Chapter 7 Triangles Maths NCERT Solutions for Class 9 to help you to score more marks in your board exams and as well as competitive exams.

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### Properties of Triangle

All geometrical shapes have different properties related to sides and angles that help us to identify them. The important properties of a triangle are listed below.

- A triangle has three sides, three vertices, and three interior angles.
- The angle sum property of a triangle states that the sum of the three interior angles of a triangle is always 180°. Observe the triangle PQR given above in which angle P + angle Q + angle R = 180°.
- The Triangle inequality theorem states that the sum of the length of the two sides of a triangle is greater than the third side.
- As per the Pythagoras theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides i.e., (Hypotenuse² = Base² + Altitude²)
- The side opposite the greater angle is the longest side.
- The Exterior angle theorem of a triangle states that the exterior angle of a triangle is always equal to the sum of the interior opposite angles.

### Triangle Formulas

In geometry, for every two-dimensional shape (2D shape), there are always two basic measurements that we need to find out, i.e., the area and perimeter of that shape. Therefore, the triangle has two basic formulas which help us to determine its area and perimeter. Let us discuss the formulas in detail.

### Perimeter of Triangle

The perimeter of a triangle is the sum of all three sides of the triangle.

### Basic facts

Elementary facts about triangles were presented by Euclid in books 1-4 of his Elements around 300 B.C.E. A triangle is a polygon and a 2-simplex (see polytope). All triangles are two-dimensional.

The angles of a triangle add up to 180 degrees. An exterior angle of a triangle (an angle that is adjacent and supplementary to an internal angle) is always equal to the two angles of a triangle that it is not adjacent/supplementary to. Like all convex polygons, the exterior angles of a triangle add up to 360 degrees.

The sum of the lengths of any two sides of a triangle always exceeds the length of the third side. That is the triangle inequality.

Two triangles are said to be similar if and only if the angles of one are equal to the corresponding angles of the other. In this case, the lengths of their corresponding sides are proportional. This occurs for example when two triangles share an angle and the sides opposite to that angle are parallel.

A few basic postulates and theorems about similar triangles:

Two triangles are similar if at least 2 corresponding angles are congruent.

If two corresponding sides of two triangles are in proportion, and their included angles are congruent, the triangles are similar.

If three sides of two triangles are in proportion, the triangles are similar.

For two triangles to be congruent, each of their corresponding angles and sides must be congruent (6 total). A few basic postulates and theorems about congruent triangles:

**SAS Postulate**: If two sides and the included angles of two triangles are correspondingly congruent, the two triangles are congruent.

SSS Postulate: If every side of two triangles are correspondingly congruent, the triangles are congruent.

**ASA Postulate**: If two angles and the included sides of two triangles are correspondingly congruent, the two triangles are congruent.

AAS Theorem: If two angles and any side of two triangles are correspondingly congruent, the two triangles are congruent.

Hypotenuse-Leg Theorem: If the hypotenuses and one pair of legs of two right triangles are correspondingly congruent, the triangles are congruent.

Using right triangles and the concept of similarity, the trigonometric functions sine and cosine can be defined. These are functions of an angle which are investigated in trigonometry.

In Euclidean geometry, the sum of the internal angles of a triangle is equal to 180°. This allows determination of the third angle of any triangle as soon as two angles are known.

A central theorem is the Pythagorean theorem, which states in any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two other sides. If the hypotenuse has length c, and the legs have lengths a and b, then the theorem states that

A²+B² = C²

**The converse is true:**if the lengths of the sides of a triangle satisfy the above equation, then the triangle is a right triangle.

### Some other facts about right triangles:

The acute angles of a right triangle are complementary.

If the legs of a right triangle are congruent, then the angles opposite the legs are congruent, acute and complementary, and thus are both 45 degrees. By the Pythagorean theorem, the length of the hypotenuse is the square root of two times the length of a leg.

In a 30-60 right triangle, in which the acute angles measure 30 and 60 degrees, the hypotenuse is twice the length of the shorter side.

For all triangles, angles and sides are related by the law of cosines and law of sines.

### Points, lines and circles associated with a triangle

There are hundreds of different constructions that find a special point inside a triangle, satisfying some unique property: see the references section for a catalogue of them. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent. Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are collinear: here Menelaus' theorem gives a useful general criterion. In this section just a few of the most commonly-encountered constructions are explained.

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