Which property helps in finding unknown angle in triangle?

The angle sum property: add known angles and subtract from 180°.

Define a scalene triangle.

A triangle with all sides of different lengths and all angles different.

Give an MCQ: What is the sum of angles of a triangle?

(A) 180°, (B) 90°, (C) 360°, (D) 270°. Answer: (A) 180°

How are triangles used in computer graphics?

Triangles are building blocks for rendering 3D shapes and textures.

Name a triangle property used in navigation.

Triangulation, which helps in finding exact positions using angles and distances.

What is a real-world scenario using triangle inequality?

To check if three rods can make a triangle before construction.

Which triangle has all angles equal?

Equilateral triangle $60^\circ, 60^\circ, 60^\circ $.

Why is the chapter important for board exams?

It forms a basis for proofs, constructions, and advanced mathematical concepts.

How is the triangle congruence symbol written?

The symbol for congruence is $\cong$.

How do transformations apply to triangles?

They help show congruence and symmetry, and are used in coordinate proofs.

Suggest an example question for triangles.

"Triangle ABC has sides 5 cm, 6 cm, 7 cm. Find the perimeter and area."

What exam questions are common from this chapter?

Proving congruence, calculating area/perimeter, applying angle/side properties, giving real-life examples.

How do triangles help in trigonometry?

Triangles form the basis of sine, cosine, and tangent calculations.

What principle supports triangle area calculation?

The base-height relationship for every triangle.

How do you spot right triangles in geometry problems?

Look for a 90° angle box or clues like “perpendicular.”

What is the first step to solve a triangles question?

Read the problem carefully, note all given values, and draw or label the triangle.

How can students practice triangle theorems?

By solving NCERT exercises, extra questions, and drawing diagrams.

What are LSI keywords for triangles?

Area of triangle, angles in triangles, triangle calculator, properties of triangle class 9.

If two triangles have equal areas, are they congruent?

No. Congruence requires matching sides and angles, not just area.

How is area calculated for a triangle with vertices (x1, y1), (x2, y2), (x3, y3)?

Use coordinate geometry: \[A = \frac{1}{2} \Bigl[ x_1(y_2 - y_3)\\ + x_2(y_3 - y_1)\\ + x_3(y_1 - y_2) \Bigr]\]

What tricks help identify congruence in exam questions?

Look for side and angle markings, right angles, and parallel lines in diagrams.

Why is the triangle the “strongest shape”?

Because its sides support each other, making structures stable and rigid.

What are common devices using triangle properties?

Engineering bridges, surveying equipment, architecture frames.

What is the transformation property in triangles?

Congruent triangles can be mapped onto each other using rigid motions (translation, rotation, reflection).

How to prove two triangles are congruent?

Use any congruence criteria (SSS, SAS, ASA, AAS, RHS) with the given measurements.

Explain a real-life application of triangles.

Triangles are used in construction for stability (trusses, roof supports) and navigation (triangulation).

State the angle sum property with a formula.

$\angle A + \angle B + \angle C = 180^\circ$

What is an isosceles triangle’s key property?

Angles opposite equal sides are also equal.

Give an example of applying triangle congruence.

Proving two triangles in a geometric figure are equal for construction or calculation.

How do you classify triangles based on sides and angles together?

Example: Acute scalene triangle, Right isosceles triangle, etc.

What is the incentre?

The point where the angle bisectors meet; it’s the center of the incircle.

What is the circumcentre?

The point where the perpendicular bisectors of the triangle’s sides meet; it's the center of the circumcircle.

What is the centroid of a triangle?

The point of intersection of the medians; also the triangle’s center of mass.

Define orthocentre of a triangle.

The point where all three altitudes of a triangle meet.

What is an altitude of a triangle?

A perpendicular segment from a vertex to the opposite side (or its extension).

What is a median of a triangle?

A line segment drawn from a vertex to the midpoint of the opposite side.

How do you calculate the perimeter of a triangle?

By adding the lengths of all three sides. Perimeter = a + b + c

Explain the RHS congruence rule.

For right triangles, if the hypotenuse and one side are equal, the triangles are congruent.

State the ASA congruence rule.

If two angles and the included side are equal, the triangles are congruent.

What is the SAS congruence rule?

If two sides and the included angle are equal, the triangles are congruent.

What is the SSS congruence rule?

Triangles with all three sides equal are congruent.

List the criteria for triangle congruence.

$\text{SSS (Side-Side-Side),}\\ \text{SAS (Side-Angle-Side),}\\ \text{ASA (Angle-Side-Angle),}\\ \text{AAS (Angle-Angle-Side),}\\\small\text{RHS (Right angle-Hypotenuse-Side).}$

Define congruent triangles.

Triangles with equal corresponding sides and angles are congruent and can be superimposed on each other.

What is the triangle inequality theorem?

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

State the exterior angle property of a triangle.

The exterior angle of a triangle equals the sum of the two opposite interior angles.

What is the sum of interior angles of a triangle?

The sum of all interior angles of a triangle is always 180 degrees.

Name the types of triangles by sides.

Equilateral (all sides equal), Isosceles (two sides equal), Scalene (all sides different).

A triangle is a polygon with three sides, three vertices, and three angles.

NCERT Class 9 Maths Chapter 7 Triangles: Properties, Theorems & Solutions

November 9, 2025 | By Academia Aeternum

TRIANGLES-Notes

Maths - Notes

Congruency

Congruency in geometry means two figures or objects are identical in shape and size or one is the mirror image of the other. They can be transformed into each other through rigid motions: translation, rotation, and reflection, without resizing. This means one figure can be repositioned and flipped to coincide exactly with the other. Congruent figures have equal corresponding sides, angles, diagonals, perimeters, and areas (e.g., congruent triangles have three pairs of equal sides and angles). Congruency is denoted by the symbol "≅". Unlike similarity, congruent figures are the same size as well as shape. This concept applies to various geometric shapes such as line segments, angles, triangles, and circles, where if the measurements match, the shapes are congruent

Congruence of Triangles

Two triangles are said to be congruent if they are exactly equal in both shape and size. This means that all the corresponding sides and angles of the two triangles are equal. Congruent triangles can be perfectly superimposed on each other without any gaps or overlaps.

Note that in congruent triangles corresponding parts are equal and we write in short ‘CPCT’ for corresponding parts of congruent triangles.

Criteria for Congruence of Triangles

The criteria for congruence of triangles are five essential rules used to prove that two triangles are exactly equal in shape and size. These criteria are:

SSS (Side-Side-Side) Criterion:
If all three sides of one triangle are equal to the corresponding three sides of another triangle, then the two triangles are congruent.
SAS (Side-Angle-Side) Criterion:
If two sides and the included angle (the angle between those two sides) of one triangle are equal to the corresponding two sides and included angle of another triangle, the triangles are congruent.
ASA (Angle-Side-Angle) Criterion:
If two angles and the included side (the side between those two angles) of one triangle are equal to the corresponding two angles and included side of another triangle, the triangles are congruent.
AAS (Angle-Angle-Side) Criterion:
If two angles and a non-included side of one triangle are equal to the corresponding two angles and side of another triangle, then the triangles are congruent.
RHS (Right angle-Hypotenuse-Side) Criterion:
In right-angled triangles, if the hypotenuse and one other side of one triangle are equal to the hypotenuse and one corresponding side of another triangle, the triangles are congruent.

Axiom 7.1 (SAS congruence rule)

In Fig. 7.8, OA = OB and OD = OC. Show that (i) $\Delta AOD \cong \Delta BOC$ and (ii) $AD \parallel BC$

Given $$\begin{aligned}AO&=BO\\ DO&=CO\\ \angle OB&=\angle AOD\end{aligned}$$ (i) To prove $$\triangle AOD\cong \triangle BOC$$ Proof

In $\triangle AOD$ and $\triangle BOC$

$$\begin{aligned}AO&=BO\\ DO&=CO\\ \angle COB&=\angle AOD\end{aligned}$$ $$\triangle AOD\cong \triangle BOC$$ (By SAS rule of Congruency)
Hence Proved. $$ AD\parallel BC$$ (ii) To prove $$\angle OCB=\angle ODA$$ (By CPCT) (Alternate Angles for the lines AD and CB) $$ \therefore AD\parallel BC$$ Hence Proved

Example

$AB$ is a line segment and line $l$ is its perpendicular bisector. If a point P lies on $l$, show that $P$ is equidistant from $A$ and $B$.

Given:

$AB$ is a line segment and $l$ is $\perp \;AB$

To Prove
\[AP=BP\]
Prrof

In $\triangle APC$ and $\triangle BPC$
($C$ is a midpoint as $l$ is perpedicular bisector of $AB$) : Given

$$\therefore AC=CB$$ $$\begin{aligned}\angle ACP&=\angle BCP\\ &=90^{\circ }\end{aligned}$$ PC = PC (common) $$\triangle APC\cong \triangle BPC \\\text{ (By SAS Rule of Congruency)}$$ \[AP = BP\text{ (By CPCT)}\]
Hence Proved

Example

Line-segment $AB$ is parallel to another line-segment $CD$. $O$ is the mid-point of $AD$ (see Fig. 7.15). Show that (i) $\triangle AOB \cong \triangle DOC$ (ii) $O$ is also the mid-point of $BC$.

Given:

$AB\parallel CD$ and $AO = OD$ ($O$ is midpoint of $AD)$

To prove $$\triangle AOB\cong \triangle DOC$$ Proof
$$AO = OD \;\text{ (Given)}$$ $$\angle DCO=\angle ABO\\\text{(Alternate Angle)}$$ $$\angle AOB=\angle DOC\\\text{(Vertically opposite angles)}$$ $$\Delta AOB\cong \triangle DOC\\\text{(By AAS rule of Congruency)}$$ Hence Proved. $$CO = OB \\\text{(by CPCT)}$$

$O$ is also mid-point of $BC$

Proved.

Isosceles Triangle

A triangle in which two sides are equal is called an isosceles triangle.

Theorem

Angles opposite to equal sides of an isosceles triangle are equal.

Proof:

Given $\triangle ABC$ is an isocles triangle $AB = AC$

To Prove
$$\angle ABC=\angle ACD$$ Construction:

Draw a angle bisector of $\angle A$, and let $D$ be the point of intersection of this angle bisector

In $\triangle BAD$ and $\triangle CAD$

$$AB=AC \text{ (Given)}$$ $$AD=AD\text{ (Common)}$$ $$\angle BAD=\angle CAD\text{ (By Construction)}$$ $$\triangle BAD\cong \triangle CAD\text{ (By SAS Rule)}$$ $$\angle ABC=\angle ACD$$ Hence Proved.

Theorem

The sides opposite to equal angles of a triangle are equal.

Example

In $\triangle ABC$, the bisector $AD$ of $\angle A$ is perpendicular to side $BC$ (see Fig. 7.27). Show that $AB = AC$ and $\triangle ABC$ is isosceles.

In $\triangle ABD$ and $\triangle ACD$

$$\angle BAD=\angle CAD\text{ (Given)}$$ \[AD = AD \text{ (common)}\] $$\angle ADB=\angle ADC=90^{\circ }\text{ (Given)}$$ $$\triangle ABD\cong \triangle ACD\\\text{(ASA Rule of Congruency)}$$ $$\therefore AB=AC\text{ (By CPCT)}$$

$\triangle ABC$ is an isosceles triangle

Exmple

E and F are respectively the mid-points of equal sides AB and AC of ∆ ABC (see Fig. 7.28). Show that BF = CE.

Solution:

In $\triangle ABF$ and $\triangle ACE$

\[\begin{align} AB &= AC\text { (Given)}\\ AF &= AE \text{ (Half of AB and AG)}\\ \angle A&=\angle A\text{ (Common)}\end{align}\] \[\triangle ABF\cong \triangle ACE\\\text{(SAS Rule of Congruency)}\] therefore \[BF = CE \text{ (By CPCT)}\]

Example

AB is a line-segment. P and Q are points on opposite sides of AB such that each of them is equidistant from the points A and B (see Fig. 7.37). Show that the line PQ is the perpendicular bisector of AB.

Solution:

$AP = BP$ (Given)
$AQ = BQ$ (Given)

To show that
$PQ$ is a ⟂ bisector of $ AB$

We need to prove that
$$AC = BC\\ \angle ACP=\angle BCP=90^{\circ }$$ Proof:

In $\triangle PAQ$ and $\triangle PBQ$

\[\begin{aligned}AP &= BP\text{ (Given)} \\AQ &= BQ \text{ (Given)} \\PQ &= PQ \text{ (Common)}\end{aligned}\] $$\therefore \Delta PAQ\cong \Delta PBQ\\\text{ (SSS Rule)}$$ $$\Rightarrow \angle APC=\angle BPC\text{ (By CPCT)}\tag{1}$$ \[\small\begin{aligned}\text{In } \triangle PAC \text{ and} \triangle PBC\\ AP &= BP \text{ (Given)}\\ PC &= PC \text{ (Common)}\\ \angle APC&=\angle BPC\text{ (From Eqn (I))}\end{aligned}\] $$\Delta PAC\cong \Delta PBC\text{ (SAS Rule)}$$ $$\therefore \angle PCA=\angle PCB\text{ (By CPCT)}$$ $$AC=BC\text{ (By CPCT)}$$ $$\begin{aligned}\angle ACP+\angle BCP&=180^{\circ}\text{ (Linear Pair)}\\ 2\angle ACP&=180^{\circ}\\ \angle ACP&=90^{\circ}\end{aligned}$$ Proved.

TRIANGLES-Notes

TRIGONOMETRIC FUNCTIONS-Exercise 3.2

TRIGONOMETRIC FUNCTIONS-Exercise 3.1