CBSE Class 9 > Class 9 Notes > Mathematics (Maths) > Points to Remember: Triangles
Points to Remember Triangles - (Maths) Class 9 PDF Download
Properties of Triangles
Properties of Triangle
- Two congruent figures have exactly the same shape and size.
- If two triangles are congruent, then their corresponding parts are equal.
- If two sides and included angle of one triangle are equal to two sides and the included angle of the other triangle, then the two triangles are congruent. [SAS congruence rule]
- If two angles and the included side of one triangle are equal to two angles and included side of the other triangle, then the two triangles are congruent. [ASA congruence rule]
- If two angles and one side of a triangle are equal to two angles and the corresponding side of the other triangles, then the two triangles are congruent. [AAS congruence rule]
- If three sides of one triangle are equal to three sides of other triangle, then the two triangles are congruent. [SSS congruence rule]
- If in two right triangles, hypotenuse and one side of a triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent. [RHS congruence rule]
- Angles opposite to equal sides of a triangle are equal.
- Sides opposite to equal angles are equal.
- The measure of each angle of an equilateral triangle is 60º.
- In a triangle, the sum of any two sides is always greater than the third side.
- In a triangle, the angle opposite to the longer side is (greater) larger.
- In a triangle, the side opposite to the greater angle is longer.
Congruent Figures
If two geometrical figures have exactly the same shape and size then they are called congruent figures.
Note: (i) Two line segments are congruent only when their lengths are equal.
(ii) Two angles are congruent only when their degree measures are equal. ∠ABC ≌ ∠PQR
(iii) The symbol '≌' is used to represent congruence.
MULTIPLE CHOICE QUESTIONTry yourself: Which rule states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent?
Congruent Triangles
Two triangles are congruent if and only if one of them can be made to superimpose on the other, such as to cover it exactly.
In the figure, ΔABC ≌ ΔDEF.
Note: (i) When ΔABC ≌ ΔDEF, then sides of ΔDEF fall on corresponding equal sides of ΔABC, i.e.
DE covers AB or DE ↔ AB; EF covers BC or EF ↔ BC and FD covers CA or FD ↔ CA.
(ii) In case ΔABC ≌ ΔDEF,
∠D covers ∠A or ∠D ↔ ∠A.
∠E covers ∠B or ∠E ↔ ∠B.
∠F covers ∠C or ∠F ↔ ∠C.
(iii) In case ΔABC ≌ ΔDEF
D corresponds to A or D ↔ A.
E corresponds to B or E ↔ B.
F corresponds to C or F ↔ C.
(iv) In congruent triangles, corresponding parts are equal and we write in short 'c.p.c.t.' "for Corresponding Parts of Congruent Triangles".
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FAQs on Points to Remember: Triangles
| 1. What are the properties of congruent triangles? |  |
Ans.Congruent triangles are triangles that are identical in shape and size, meaning their corresponding sides and angles are equal. The main properties include: 1. Corresponding sides of congruent triangles are equal in length. 2. Corresponding angles of congruent triangles are equal in measure. 3. If two triangles are congruent, any transformation (like rotation, reflection, or translation) applied to one will produce the other.
| 2. How can we prove that two triangles are congruent? | |
Ans.Two triangles can be proven congruent using several criteria: 1. Side-Side-Side (SSS) Criterion: If all three sides of one triangle are equal to the three sides of another triangle, they are congruent. 2. Side-Angle-Side (SAS) Criterion: If two sides and the included angle of one triangle are equal to those of another triangle, the triangles are congruent. 3. Angle-Side-Angle (ASA) Criterion: If two angles and the side between them in one triangle are equal to those in another triangle, the triangles are congruent. 4. Angle-Angle-Side (AAS) Criterion: If two angles and a non-included side of one triangle are equal to those in another triangle, they are congruent.
| 3. What is the significance of the congruence criteria in solving triangle problems? | |
Ans.The congruence criteria are significant in solving triangle problems as they provide a systematic approach to establishing the equality of triangles. They help in: 1. Determining unknown side lengths and angles in geometric problems. 2. Proving that certain triangles within a diagram are congruent, which simplifies the analysis of the figure. 3. Facilitating the use of congruent triangles in applications such as construction, design, and real-world problem-solving.
| 4. Can two triangles be congruent if they have the same angles but different side lengths? | |
Ans.No, two triangles cannot be congruent if they have the same angles but different side lengths. This scenario describes similar triangles, which have the same shape but not necessarily the same size. For triangles to be congruent, all corresponding sides must be equal in length, as well as all corresponding angles.
| 5. What are some real-life applications of congruent triangles? | |
Ans.Congruent triangles have various real-life applications, including: 1. Architecture and engineering, where congruent triangles are used in the design of stable structures. 2. Art and design, where congruent triangles help in creating balance and symmetry in visual compositions. 3. Navigation, where congruent triangles assist in triangulating positions and distances using landmarks.
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