Congruency & Similarity - An Introduction
This chapter introduces the core concepts of congruency and similarity in triangles, helping students understand how and why triangles can be the same shape or the same shape in proportion. These fundamental ideas form the foundation for proving triangle congruency and similarity in the EMaths For GCE O Level Course.
Related Lessons:
Congruency & Its Properties
Congruency means both objects have the same size and same shape. When 2 triangles are congruent:
- Corresponding sides have the same length
- Corresponding angles are equal
- We use the “three bar” symbol “≡” to denote congruency
When we say ∆ABC≡ ∆XYZ , the angles and sides must correspond to each other. And we have 6 conclusions:
- ∠ABC=∠XYZ
- ∠ACB=∠XZY
- ∠BAC=∠YXZ
- AB=XY
- AC=XZ
- BC=YZ
Example On Applying Congruency
In the diagram above, triangle ABC and BDE are congruent. Given that DE=6cm, and ∠BAC=35°, find
- the length of CD
- ∠BED
Answer:
a) CD=2cm
b) \(\angle BED=55°\)
Similarity & Its Properties
Similarity means both objects have the different size and same shape. When 2 triangles are similar:
- Corresponding angles are equal
- Ratio of sides are equal
- There is no mathematical symbol to denote similarity
When we say ∆ABC & ∆XYZ are similar, we have 6 conclusions:
- ∠ABC=∠XYZ
- ∠ACB=∠XZY
- ∠BAC=∠YXZ
- \(\frac{𝐴𝐵}{𝑋𝑌}=\frac{𝐴𝐶}{𝑋𝑍}\)
- \(\frac{𝐴𝐵}{𝑋𝑌}=\frac{B𝐶}{Y𝑍}\)
- \(\frac{𝐴C}{𝑋Z}=\frac{B𝐶}{Y𝑍}\)
Example On Applying Similarity
In the diagram above, triangle ABC and XYZ are similar. Find
- BC
- ∠ACB
Answers:
a) \(\angle ACB=74°\)
b) BC=2cm
Calculating Scale Factor
To calculate the scale factor, we use the following formula:
\(\mathrm{scale\ factor\ }=\ \frac{\mathrm{dimension\ of\ new\ shape}}{\mathrm{dimension\ of\ old\ shape}}\)
Now,
- When the new shape is larger than the original shape, the scale factor is greater than 1. This transformation is called an enlargement.
- When the new shape is smaller than the original shape, the scale factor is less than 1. This transformation is called a reduction.
Watch: Full Concept Video
This chapter introduces the concepts of congruency and similarity. It is important for students to develop a strong understanding of these concepts, as they form the foundation for learning how to prove congruency and similarity in the later stages of the Elementary Mathematics course.