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TL;DR

Similarity means two shapes have the same shape but can be different sizes, while congruency means they're exactly the same shape and size. Think of similar shapes as scaled versions of each other, and congruent shapes as identical copies. You'll mostly deal with triangles for these concepts.

1. The Mental Model

Imagine you have a photograph. If you print it larger or smaller, all the objects in the photo are still the same shape, just different sizes – that's similarity. If you print two exact copies of the photo, they're identical – that's congruency.

2. The Core Material

What is Congruency?

Photo by Ann H on Pexels

Two shapes are congruent if they have the exact same size and shape. If you can place one shape perfectly on top of the other, they're congruent. This means all corresponding sides and all corresponding angles are equal.

For triangles, there are four main rules (the "congruency tests") to prove congruency without having to check every single side and angle:

• SSS (Side-Side-Side): If all three sides of one triangle are equal to the three corresponding sides of another triangle, then the triangles are congruent.
• SAS (Side-Angle-Side): If two sides and the included angle (the angle between those two sides) of one triangle are equal to the two corresponding sides and included angle of another triangle, then the triangles are congruent.
• ASA (Angle-Side-Angle): If two angles and the included side (the side between those two angles) of one triangle are equal to the two corresponding angles and included side of another triangle, then the triangles are congruent.
• RHS (Right-angle, Hypotenuse, Side): This applies only to right-angled triangles. If the hypotenuse and one other side of a right-angled triangle are equal to the hypotenuse and corresponding side of another right-angled triangle, then the triangles are congruent.

What is Similarity?

Photo by DS stories on Pexels

Two shapes are similar if they have the same shape but not necessarily the same size. This means:

1. All corresponding angles are equal.
2. The ratio of corresponding sides is constant (meaning one shape is an enlargement or reduction of the other). This constant ratio is called the scale factor.

For triangles, there are three main rules (the "similarity tests"):

• AA (Angle-Angle): If two angles of one triangle are equal to two corresponding angles of another triangle, then the triangles are similar. (The third angle must also be equal because angles in a triangle add up to 180°.)
• SSS (Side-Side-Side): If the ratios of all three pairs of corresponding sides are equal, then the triangles are similar.
• SAS (Side-Angle-Side): If the ratios of two pairs of corresponding sides are equal, and the included angles are equal, then the triangles are similar.

Here's a diagram to help visualize the relationship:

graph TD
    A["Shapes"] --> B["Congruent?"]
    B -- "Yes (Same shape & size)" --> C["All sides & angles equal"]
    B -- "No" --> D["Similar?"]
    D -- "Yes (Same shape, diff size)" --> E["All angles equal AND \n Sides in proportion (scale factor)"]
    D -- "No" --> F["Neither (Different shapes)"]
    C --> G["Tests: SSS, SAS, ASA, RHS"]
    E --> H["Tests: AA, SSS (ratio), SAS (ratio)"]

3. Worked Example

Let's look at two triangles, ABC and DEF.

Triangle ABC has angles A=50°, B=70°, C=60°. Side AB = 5cm, BC = 7cm, AC = 6cm.
Triangle DEF has angles D=50°, E=70°, F=60°. Side DE = 10cm, EF = 14cm, DF = 12cm.

Question: Are triangle ABC and triangle DEF congruent or similar?

Solution:
1. Check Angles:
* Angle A = Angle D = 50°
* Angle B = Angle E = 70°
* Angle C = Angle F = 60°
Since all corresponding angles are equal, the triangles could be similar. They are definitely not congruent if their sides are different.

2. Check Sides (for similarity - ratios):
   • Ratio of AB to DE: 5cm / 10cm = 1/2
   • Ratio of BC to EF: 7cm / 14cm = 1/2
   • Ratio of AC to DF: 6cm / 12cm = 1/2
     Since all corresponding sides are in the same proportion (ratio 1/2), and all corresponding angles are equal, the triangles are similar.
     They are not congruent because their sides are different lengths (DE is not equal to AB, etc.).

4. Key Takeaways

• Congruent shapes are identical in both shape and size.
• Similar shapes are identical in shape but can differ in size.
• For congruent shapes, all corresponding angles are equal, and all corresponding sides are equal.
• For similar shapes, all corresponding angles are equal, and corresponding sides are proportional (they have a constant scale factor).
• Congruency tests for triangles: SSS, SAS, ASA, RHS.
• Similarity tests for triangles: AA, SSS (ratios), SAS (ratios).
• If two shapes are congruent, they are also similar (with a scale factor of 1).
• If two shapes are similar, they are not necessarily congruent.

Common Mistakes to Avoid:
- Confusing similarity with congruency; remember size matters for congruency!
- Incorrectly identifying corresponding sides or angles when setting up ratios or comparisons.
- Using SAS or ASA for congruency without ensuring the angle is included between the two sides.
- Applying RHS to non-right-angled triangles.

5. Now Try It

Draw two triangles, ABC and PQR.
For triangle ABC, let AB = 4cm, angle B = 90°, and BC = 3cm.
For triangle PQR, let PQ = 8cm, angle Q = 90°, and QR = 6cm.

Determine if triangles ABC and PQR are congruent, similar, or neither. Explain your reasoning using one of the tests you've learned.

Success looks like: You correctly identify the relationship and state which congruency or similarity test you used, explaining why it applies (or doesn't).