NCERT MCQ Solutions for Class 9 Maths Chapter 7 Triangles

A triangle is defined as a closed figure formed by three intersecting line segments. It has three sides, three angles and three vertices. The word ‘tri’ means ‘three’ which clearly indicates the structure and elements of the triangle.

Congruent figures are identical in both shape and size. When two figures are congruent they can be superimposed perfectly over each other. The concept of congruence is used widely in geometry and practical situations like mold making.

If two equilateral triangles have equal side lengths then all corresponding sides and angles are equal. Therefore they are congruent. The chapter emphasizes that congruent triangles must have all corresponding parts equal which is satisfied here.

The congruence statement ∆ABC ≅ ∆DEF implies that all corresponding sides and angles of the two triangles are equal. This follows the rule that congruent triangles are identical in dimensions and orientation can be adjusted through rotation or flipping.

The SAS (Side-Angle-Side) rule applies when two sides and the included angle between them in one triangle are equal to the corresponding two sides and included angle in another triangle.

The line segment joining the midpoint of any side of a triangle to the opposite vertex is called a median. A median divides the triangle into two smaller triangles of equal area.

The length of the perpendicular drawn from the opposite vertex to any side of a triangle is called the altitude. An altitude is the shortest distance between a vertex and the line containing the opposite side.

Since two sides and the included angle are equal between ∆ABC and ∆DEF the triangles are congruent by SAS congruence rule. This rule is formally introduced and explained with diagrams for identifying triangle congruence.

In a triangle when two sides are equal, the angles opposite those sides are also equal.
Hence, if AB = AC, then the angle opposite AB (which is ∠C) equals the angle opposite AC (which is ∠B). 

The converse of the previous property states that if two angles of a triangle are equal, then the sides opposite to them are also equal. This using ASA congruence rule to show that AB = AC.

The ASA rule means two triangles are congruent if two angles and the included side in one triangle are respectively equal to the two angles and included side of another triangle. 

In an equilateral triangle all three sides are equal and each angle measures exactly 60°. This is a fundamental geometric property of equilateral triangles derived from angle sum and symmetry.

The point of intersection of all the altitudes of a triangle is called the orthocentre. The altitudes are the perpendiculars drawn from each vertex to the opposite side (or its extension).

The point of intersection of the perpendicular bisectors of all sides of a triangle is called the circumcentre. The circumcentre is equidistant from all three vertices of the triangle and is the center of the circumcircle.

Every triangle has a total of three interior angles and their sum is always 180°.
This is a basic geometric property of triangles that is often used for calculating unknown angles when two are given.

AAS rule implies if two angles and any one side (not between them) of one triangle are respectively equal to those of another triangle then the two triangles are congruent.

SSA rule is not valid for proving congruence because two sides and a non-included angle do not always ensure triangle uniqueness. It warns against using SSA as a congruence rule due to the possibility of multiple configurations.

The RHS (Right angle-Hypotenuse-Side) rule says that if the hypotenuse and one side of one right triangle are equal to the hypotenuse and one side of another then they are congruent. This rule applies specifically to right-angled triangles.

If AB = AC and AD is drawn perpendicular to BC, then triangle is isosceles and the perpendicular bisects both BC and ∠A. This is proved using ASA rule and congruence of two right triangles ABD and ACD.

The point of intersection of all the medians of a triangle is called the centroid.
The centroid divides each median in the ratio 2:1 and is the center of mass of the triangle.

In a triangle, the angle opposite to the longer side is always larger. This is based on the property that in any triangle, the larger angle is opposite the longer side.

An equilateral triangle has three equal sides and angles which means it has three lines of symmetry. Each line of symmetry passes through a vertex and bisects the opposite side. This characteristic can be discussed when proving triangle properties.

The SSS (Side-Side-Side) rule states that if three sides of one triangle are equal to three sides of another triangle then the triangles are congruent. This is a definitive congruence rule included in the chapter with examples and proofs.

Congruent triangles are identical in shape and size. That means their corresponding angles and sides are equal and they can be placed over one another perfectly.

In a triangle, the angles opposite to equal sides are equal. This is one of the important results derived and proved using triangle congruence and symmetry principles.

An isosceles triangle has two equal sides and hence two equal angles opposite those sides. The term ‘isosceles’ is used to describe such triangles and various congruence properties are demonstrated with these.

In a triangle, the side opposite to the larger angle is always longer. This is based on the property that in any triangle, the larger angle is opposite the longer side.

In a triangle, the sum of its two sides is always greater than the third side. This is known as the triangle inequality theorem.

The given condition matches the SAS rule. Two sides and the included angle in one triangle are equal to two sides and the included angle in another triangle. Therefore, the triangles are congruent as per the SAS rule.

Since ∠P = ∠Q, the sides opposite to these angles must be equal. That means PR = PQ. This is a direct consequence of the triangle property that sides opposite to equal angles are equal.

Knowing two sides and the included angle (SAS) uniquely determines a triangle. This combination ensures one specific triangle can be constructed and is discussed as a congruence rule. It is sufficient for proving triangle congruence in geometric problems.

The RHS rule is used only for right-angled triangles. If the hypotenuse and one side of two right triangles are equal then the triangles are congruent.

It covers five triangle congruence rules in detail—SAS, ASA, AAS, SSS and RHS. Each rule is introduced with diagrams examples and reasoning for their validity. SSA is not included as it does not always produce congruent triangles.

In a right-angled triangle, the hypotenuse is the longest side. It is the side opposite the right angle.

In a triangle, the sides opposite to two equal angles are always congruent. This is because equal angles correspond to equal sides.