Congruent and Similar Triangles
When triangles are identical or proportional
Congruent vs Similar
Congruent Triangles
Same shape AND same size
All corresponding sides are equal
All corresponding angles are equal
Symbol: \(\triangle ABC \cong \triangle DEF\)
Similar Triangles
Same shape, different sizes
Corresponding sides are proportional
All corresponding angles are equal
Symbol: \(\triangle ABC \sim \triangle DEF\)
Key Difference
Congruent = Exact copies (like photocopies)
Similar = Scaled copies (like enlargements or reductions)
Conditions for Congruence
Two triangles are congruent if any ONE of these conditions is met:
SSS
Side-Side-Side
All three sides are equal
SAS
Side-Angle-Side
Two sides and included angle are equal
ASA
Angle-Side-Angle
Two angles and included side are equal
RHS
Right-Hypotenuse-Side
Right angle, hypotenuse, and one side are equal
Conditions for Similarity
Two triangles are similar if any ONE of these conditions is met:
AA
Angle-Angle
Two pairs of angles are equal (the third must also be equal since angles sum to 180°)
SSS (ratio)
Sides in Proportion
All three pairs of sides are in the same ratio
SAS (ratio)
Two Sides Proportional + Included Angle
Two pairs of sides in same ratio with equal included angle
Working with Similar Triangles
Scale Factor
\[\text{Scale Factor} = \frac{\text{Side of larger triangle}}{\text{Corresponding side of smaller triangle}}\]
All corresponding sides share the same scale factor!
Example 1: Finding Unknown Sides
Triangles ABC and DEF are similar. AB = 6 cm, BC = 8 cm, AC = 10 cm, and DE = 9 cm. Find EF and DF.
\[\text{Scale Factor} = \frac{DE}{AB} = \frac{9}{6} = 1.5\]
\[EF = BC \times 1.5 = 8 \times 1.5 = 12 \text{ cm}\]
\[DF = AC \times 1.5 = 10 \times 1.5 = 15 \text{ cm}\]
Example 2: Setting Up Proportions
In the diagram, triangles PQR and PST are similar. PQ = 12 cm, PS = 8 cm, and QR = 15 cm. Find ST.
\[\frac{PQ}{PS} = \frac{QR}{ST}\]
\[\frac{12}{8} = \frac{15}{ST}\]
\[ST = \frac{15 \times 8}{12} = \frac{120}{12} = 10 \text{ cm}\]
Interactive Similar Triangles Calculator
Find Unknown Sides
Enter the sides of two similar triangles to find the missing side
Scale Factor: 1.5
Missing sides: b' = 12, c' = 15
Area of Similar Triangles
Area Ratio
\[\frac{\text{Area}_1}{\text{Area}_2} = \left(\frac{\text{Side}_1}{\text{Side}_2}\right)^2 = k^2\]
If the scale factor is k, the area ratio is k²
Example 3: Area of Similar Triangles
Two similar triangles have sides in the ratio 2:3. If the smaller triangle has area 24 cm², find the area of the larger triangle.
\[\text{Area ratio} = k^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4}\]
\[\text{Larger area} = 24 \times \frac{9}{4} = 54 \text{ cm}^2\]
Practice Problems
Question 1: State whether triangles with sides 3, 4, 5 and 6, 8, 10 are congruent, similar, or neither.
Show Solution
Check ratios: 6/3 = 2, 8/4 = 2, 10/5 = 2
All ratios are equal, so the triangles are Similar (scale factor 2).
They are NOT congruent because the sides are different sizes.
Question 2: Triangles ABC and XYZ are similar. AB = 5 cm, XY = 7.5 cm, BC = 8 cm. Find YZ.
Show Solution
Scale factor = 7.5 ÷ 5 = 1.5
YZ = BC × 1.5 = 8 × 1.5 = 12 cm
Question 3: Two similar triangles have areas 16 cm² and 49 cm². What is the ratio of their corresponding sides?
Show Solution
Area ratio = 16 : 49
Side ratio = √16 : √49 = 4 : 7
Question 4: Which congruence condition (SSS, SAS, ASA, or RHS) would you use if you know two angles and the side between them are equal?
Show Solution
ASA (Angle-Side-Angle) — two angles with the included side.
CSEC Exam Tips
- Always state which condition you're using (SSS, SAS, etc.)
- Match corresponding vertices correctly when writing similarity/congruence
- For similar triangles, set up proportions carefully
- Remember: Area ratio = (side ratio)²
- Draw diagrams to identify corresponding parts