IB MYP 4-5 Maths- Similarity and congruence- Study Notes - New Syllabus
IB MYP 4-5 Maths- Similarity and congruence – Study Notes
Standard
- Similarity and congruence
IB MYP 4-5 Maths- Similarity and congruence – Study Notes – All topics
Congruent Triangles & Similar Triangles
Congruent Triangles
Two triangles are said to be congruent if they have the same size and shape. This means their corresponding sides and angles are equal.
Key Properties:
- Corresponding sides are equal in length.
- Corresponding angles are equal in measure.
- Congruent triangles can be placed on each other exactly by rigid motions (rotation, reflection, translation).
Conditions for Congruence:
- SSS (Side-Side-Side): If three sides of one triangle are equal to three sides of another.
- SAS (Side-Angle-Side): If two sides and the included angle are equal.
- ASA (Angle-Side-Angle): If two angles and the included side are equal.
- AAS (Angle-Angle-Side): Two angles and a non-included side are equal.
- RHS (Right angle-Hypotenuse-Side): For right triangles, if the hypotenuse and one side are equal.
Example :
Two triangles have sides AB = PQ, AC = PR, and BC = QR. Are they congruent?
▶️ Answer/Explanation
All three sides are equal. So the triangles are congruent by the SSS rule.
Example :
In ΔABC and ΔPQR, AB = PQ, ∠B = ∠Q, and BC = QR. Prove they are congruent.
▶️ Answer/Explanation
Two sides and the included angle are equal, so ΔABC ≅ ΔPQR by the SAS rule.
Similar Triangles
Two triangles are said to be similar if they have the same shape but not necessarily the same size. Their corresponding angles are equal, and their corresponding sides are in proportion.
Key Properties:
- All corresponding angles are equal.
- The ratio of corresponding sides is constant (scale factor).
Conditions for Similarity:
- AAA (Angle-Angle-Angle): If two angles of one triangle are equal to two angles of another.
- SAS (Side-Angle-Side): If two sides are in proportion and the included angle is equal.
- SSS (Side-Side-Side): If all corresponding sides are in the same ratio.
Example :
Two triangles have angles 40°, 60°, and 80°, and another triangle has angles 40°, 60°, and 80°. Are they similar?
▶️ Answer/Explanation
All angles are equal, so the triangles are similar by the AAA rule.
Example :
Two similar triangles have corresponding sides in ratio 2:3. If one side of the smaller triangle is 8 cm, find the corresponding side in the larger triangle.
▶️ Answer/Explanation
Scale factor = 2:3. So, corresponding side = \( \dfrac{8 \times 3}{2} = 12\ \text{cm} \).