NCERT Notes for Class 9 Maths Chapter 5 Triangles
![NCERT Notes for Class 9 Maths Chapter 5 Triangles NCERT Notes for Class 9 Maths Chapter 5 Triangles](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiGcpAs6r3WqBgdoaPFyWasvAXVIS89yXcbapST5-61PN2XNgRSI2MGNKRTKRPfY7YTxBKJ7N4vuMed3yhYsgjVgs25iqXbc3Jg_xHRkWq5IV8LIvcx81ffWovEfFuG_ykmZkUsTumST_aBrfWTEUKtjNqBI52UeSU2dLBO5_e8JfJSrDCW78iFtncwNwo/s16000/triangles-revision-notes-class9-maths.jpg)
Class 9 Chapter 5 Triangles Notes
Chapter Name | Triangles Notes |
Class | CBSE Class 9 |
Textbook Name | NCERT Mathematics Class 9 |
Related Readings |
|
Triangle
A closed figure with three sides is called a Triangle. It has three vertex, sides and Angles.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAQ7PcbZ7vUCMB9uJk5WFmj5CS1nOfw9vG2QgWHW6H6TrCEIwceg8MYJiJxCagxSH__6Nf3GGMLSQoAJvO13q2Ow-BrSLSgliq-WlBq8upRIKZgjNCcyMzQ3juQXrHPzAXpHWHDGVGNW3_neyB8ZYotzvUV3VHZza06mNNYr4YtWM8L56YU1yAO5RVwZ4/w247-h200/Ncert%20Notes%20Class%209%20math%20Chapter%205%20Triangles%20img%201.jpg)
Types of Triangle
1. There are three types of triangles on the basis of the length of the sides.
Name of Triangle |
Property | Image |
Scalene |
Length of all sides are different | ![]() |
Isosceles |
Length of two sides are equal | ![]() |
Equilateral |
Length of all three sides are equal |
Name of Triangle |
Property |
Image |
Acute |
All the three angles are less than 90° | ![]() |
Obtuse |
One angle is greater than 90° |
|
Right |
One angle is equal to 90° |
Congruence
If the shape and size of two figures are same then these are called Congruent.
1. Two circles are congruent if their radii are same.![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFAN5IqNGnsaAmIifWMGB5dkBajhJ56PiuMxiOn6CEBFLndS3HPq3up6HLV8_8zT2By-4sw9J-Ap2S4IMEIyg1if4ghPBloDmku-kzU2kW6Fff8O7rl98KhXzVS7WMqbV8-YSG7exK1uN4xw-05ZAieH3CeAGPH265QIio6SOt9eEKA2UHtCdyS9h51As/w275-h129/Ncert%20Notes%20Class%209%20math%20Chapter%205%20Triangles%20img%208.jpg)
2. Two squares are congruent if their sides are equal.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1I-bsblAAqzXW6ATXSHgF1Btr5ZCfcjoici-vy_hpG1xnH-dpTmPgRw4XuVCvXRaMGAXjf8xmkCHVBoEQN9whqTjhcmChqfmeuH9NO4OEplH-dqfLcE2k6QkDRhUb-XCcqsVryjSahT1yZLsceEkP2M3IK5eZtJUMgebxDEEbYKcsI0ANgHKv8cZeSNw/w297-h117/Ncert%20Notes%20Class%209%20math%20Chapter%205%20Triangles%20img%209.jpg)
Congruence of Triangles
A triangle will be congruent if its corresponding sides and angles are equal.
The symbol of congruent is “≅”.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7RnfYu2FVVz71Se23EA0OqdZtOJrv27ZoQbWaCk26EXxAzXT4pervXn1EA6O8n_Aw1Ctphyz_FqAeu3MS8SOZ7kcJDh9GDevwr9yoaknywo0Gkykq5KCzkpEbTp0vHoilTmC4oi-3VHHjFCJj4DatJLhZ02Z7d2nScejc_taiGDVbOOVf6lzNv1IvhEY/w265-h171/Ncert%20Notes%20Class%209%20math%20Chapter%205%20Triangles%20img%2010.jpg)
AB = DE, BC = EF, AC = DF
m∠A = m∠D, m∠B = m∠E, m∠C = m∠F
Here, ∆ABC ≅ ∆DEF
Criteria for Congruence of Triangles
S.No. |
Rule |
Meaning |
Figure |
1. |
SAS (Side-Angle-Side) Congruence rule |
If the two sides and the including angle of one triangle is equal to another triangle then they are called congruent triangles. |
|
2. |
ASA (Angle-Side-Angle) Congruence rule |
If the two angles and the including side of one triangle is equal to another triangle then they are called congruent triangles. |
|
3. |
AAS (Angle-Angle-Side) Congruence rule |
If any two pairs of angles and a pair of the corresponding side is equal in two triangles then these are called congruent triangles. |
|
4. |
SSS (Side-Side-Side) Congruence rule |
If all the three sides of a triangle are equal with the three corresponding sides of another triangle then these are called congruent triangles. |
|
5. |
RHS (Right angle-Hypotenuse-Side) Congruence rule |
If there are two right-angled triangles then they will be congruent if their hypotenuse and any one side are equal. |
Remark
- SSA and ASS do not show the congruency of triangles.
- AAA is also not the right condition to prove that the triangles are congruent.
Example: Find the ∠P, ∠R, ∠N and ∠M if ∆LMN ≅ ∆PQR.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgpSn0EAVCbnK-duyjsFyZBPA1t1hc0nFX9TuVJYc9lWCqb4i6hwKH4Zwhz5wY4b3DNGM1cc3DplAlLCmjxNt-IvhxLuUmGwi4qTPKRf2fP6MP1Xc7y0keNL5Suj_kSqgARVkHqNgQmoLR-dPOs2xt6SV-Hm8PfSz1Z22cRMvRkuQ3K0tvjS7mO8P6pdhg/w209-h164/Ncert%20Notes%20Class%209%20math%20Chapter%205%20Triangles%20img%2016.jpg)
Solution
If ∆ LMN ≅ ∆PQR, then
∠L = ∠P
∠M = ∠Q
∠N = ∠R
So,
∠L = ∠P = 105°
∠M = ∠Q = 45°
∠M + ∠N + ∠L = 180° (Sum of three angles of a triangle is 180°)
45° + 105° + ∠N = 180°
∠N = 180°- 45° + 105°
∠N = 30°
∠N = ∠R = 30°
Some Properties of a Triangle
If a triangle has two equal sides then it is called an Isosceles Triangle.
1. Two angles opposite to the two equal sides of an isosceles triangle are also equal.![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVqJfd5dBb-lfW_iUoi0aU8vPWdN8K0oB90XdLJFqqcmuctdLU4bOXr1C4qcFqQEbFXZk-_eSUaa75o8klIGE-bguGeI6WZ6ddAwofC0-ogN6HP-U7O25YDX_3WUK1ILkO8YW-gWy5k8apyB8HABSX6QgWloBK7BjV86QDuyzfd06aOO6-5O3mVnemMPE/w150-h171/Ncert%20Notes%20Class%209%20math%20Chapter%205%20Triangles%20img%2017.jpg)
2. Two sides opposite to the equal angles of the isosceles triangle are also equal. This is the converse of the above theorem.
Inequalities in a Triangle
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbPPBP7ZP3gmnyNNrFKs0g_cDrqTyz7jGHCCjj597V2S43vj2sBiaoAb24qfPcsAcEYitLQt6zvgwGNxe0J4C0sVU4aMgu6-xGgQiu_sPt09wyjP2WXefrg2prNwjVULzMgqkgGNT8p8SXvTi2q3RlKy7LRbwsQqMe76EkU5p645EzzoV7tnZCPyDygKE/w170-h116/Ncert%20Notes%20Class%209%20math%20Chapter%205%20Triangles%20img%2018.jpg)
a > b, if and only if ∠A > ∠B
Longer sides correspond to larger angles.
Theorem 2: In the given triangle, the side opposite to the larger angle will always be longer. This is the converse of above theorem.
Theorem 3: The sum of any two sides of a triangle will always be greater than the third side.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrykI0dV5nu69pISmvIQV96e8lz1c9Zy7x7q8Yl1YkHrULdaKL-cbNmojc81OiGamwxgerUNGEswyhlDwTVwSKEAKrFuFUGgooeoVgd2-sbQ3IAqWlUPnORyFk-IMaFMqKo5A-AqH6CMoQWMEApS9VMV1ntQeBGknw-P7CT2M0-oPckMKJZNSZI9aNw3E/w317-h204/Ncert%20Notes%20Class%209%20math%20Chapter%205%20Triangles%20img%2019.jpg)
Example: Show whether the inequality theorem is applicable to this triangle or not?
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnXtpYSgdkpuWg_czAkh1z5IolpGZixIVnSLgnIywgiohy5UWbo7WpY91G9H94LnTKxGyFaLzNVrmNnxU2PNh66nkFGdBTN13ksdkQA3gSNnXqYn3qy11_V0Yi6WkVWPUhs4bJ5bafGpQ3EeIzrsYvIgXRH7_ND1G1mqaaTpchy3sryLlW49ELQ-vIivs/w164-h120/Ncert%20Notes%20Class%209%20math%20Chapter%205%20Triangles%20img%2020.jpg)
Solution
The three sides are given as 7, 8 and 9.
According to inequality theorem, the sum of any two sides of a triangle will always be greater than the third side.
Let’s check it
7 + 8 > 9
8 + 9 > 7
9 + 7 > 8
This shows that this theorem is applicable to all the triangles irrespective of the type of triangle.