Congruence and Similarity

What makes two shapes "the same" or "alike"? Congruence and similarity help us compare geometric figures! These concepts connect transformations to real-world applications in construction, design, and nature.

---

Congruence

Congruent figures have the same size AND the same shape.

Think of it as:

• Exact copies
• Perfect matches
• One could fit exactly on top of the other

Symbol: ≅ (is congruent to)

Example: Triangle ABC ≅ Triangle DEF

What this means:

• All corresponding sides are equal
• All corresponding angles are equal
• One is a perfect copy of the other

---

Properties of Congruent Figures

For congruent figures:

Corresponding sides are equal:

• If triangle ABC ≅ triangle DEF, then:
• AB = DE
• BC = EF
• AC = DF

Corresponding angles are equal:

• ∠A = ∠D
• ∠B = ∠E
• ∠C = ∠F

Same perimeter and area:

• Perimeters are equal
• Areas are equal

---

Rigid Transformations and Congruence

Rigid transformations preserve size and shape:

• Translation (slide)
• Reflection (flip)
• Rotation (turn)

Key fact: If you can transform one figure into another using ONLY rigid transformations, the figures are CONGRUENT!

Example: Triangle A can be reflected and rotated to match Triangle B → They are congruent!

---

Testing for Congruence

For triangles, you don't need to check EVERYTHING!

Triangle Congruence Shortcuts:

SSS (Side-Side-Side):

• All three sides equal
• Triangles are congruent

SAS (Side-Angle-Side):

• Two sides and the included angle equal
• Triangles are congruent

ASA (Angle-Side-Angle):

• Two angles and the included side equal
• Triangles are congruent

AAS (Angle-Angle-Side):

• Two angles and a non-included side equal
• Triangles are congruent

HL (Hypotenuse-Leg) - Right Triangles Only:

• Hypotenuse and one leg equal
• Right triangles are congruent

NOT a shortcut:

• AAA - Same angles, but could be different sizes!
• SSA - Ambiguous (except HL for right triangles)

---

Examples of Congruence

Example 1: Are these triangles congruent? Triangle 1: sides 3, 4, 5 Triangle 2: sides 3, 4, 5

Solution: All three sides match → SSS Answer: Yes, congruent by SSS

Example 2: Are these triangles congruent? Triangle 1: sides 6, 8; included angle 50° Triangle 2: sides 6, 8; included angle 50°

Solution: Two sides and included angle match → SAS Answer: Yes, congruent by SAS

Example 3: Are these triangles congruent? Triangle 1: angles 40°, 60°, 80°; side 10 Triangle 2: angles 40°, 60°, 80°; side 5

Solution: Same angles but different side lengths! Answer: No, NOT congruent (they're similar though!)

---

Similarity

Similar figures have the same shape but NOT necessarily the same size.

Think of it as:

• One is a resized version of the other
• Same proportions
• Photographs of different sizes

Symbol: ~ (is similar to)

Example: Triangle ABC ~ Triangle DEF

What this means:

• Corresponding angles are equal
• Corresponding sides are proportional (same ratio)
• One is an enlarged or reduced copy

---

Properties of Similar Figures

For similar figures:

Corresponding angles are equal:

• ∠A = ∠D
• ∠B = ∠E
• ∠C = ∠F

Corresponding sides are proportional:

• AB/DE = BC/EF = AC/DF = k (scale factor)

Same shape, different size:

• Angles match
• Sides are in same ratio

---

Scale Factor

The scale factor (k) tells you how much bigger or smaller.

Formula: k = (length in new figure)/(length in original figure)

Example: Triangle 1 has side 6 cm, Triangle 2 has corresponding side 9 cm k = 9/6 = 3/2 = 1.5

Triangle 2 is 1.5 times larger!

If k > 1: Enlargement (bigger) If k = 1: Same size (congruent!) If 0 < k < 1: Reduction (smaller)

---

Testing for Similarity

For triangles:

AA (Angle-Angle):

• Two angles equal
• Triangles are similar
• (Third angle must be equal too, since angles sum to 180°)

SSS (Side-Side-Side Proportional):

• All three sides proportional
• Triangles are similar

SAS (Side-Angle-Side Proportional):

• Two sides proportional and included angle equal
• Triangles are similar

Note: AA is most common and easiest to use!

---

Examples of Similarity

Example 1: Are these triangles similar? Triangle 1: angles 50°, 60°, 70° Triangle 2: angles 50°, 60°, 70°

Solution: Two angles match (actually all three!) → AA Answer: Yes, similar by AA

Example 2: Are these triangles similar? Triangle 1: sides 3, 4, 5 Triangle 2: sides 6, 8, 10

Solution: Check ratios: 6/3 = 2, 8/4 = 2, 10/5 = 2 All ratios equal! → SSS Answer: Yes, similar by SSS with scale factor k = 2

Example 3: Triangle 1 has sides 4 and 6 with included angle 40° Triangle 2 has sides 8 and 12 with included angle 40°

Solution: Ratios: 8/4 = 2, 12/6 = 2 (proportional!) Included angle equal → SAS Answer: Yes, similar by SAS with scale factor k = 2

---

Dilations and Similarity

Key connection: Dilations create similar figures!

If you dilate a figure:

• The image is similar to the original
• Scale factor of dilation = scale factor of similarity

Example: Dilate triangle by k = 3

• Original and image are similar
• Corresponding sides ratio is 3:1
• Angles stay the same

---

Finding Missing Measures

Using similarity to find unknown lengths:

Example: Triangles ABC ~ DEF with scale factor k = 2 If AB = 5, what is DE?

Solution: k = DE/AB 2 = DE/5 DE = 10

Answer: DE = 10

Example: Triangles similar with sides 4, 6, x and corresponding sides 10, 15, 20 Find x.

Solution: Set up proportion: 4/10 = 6/15 = x/20

Using first ratio: 4/10 = x/20 Cross multiply: 4(20) = 10x 80 = 10x x = 8

Answer: x = 8

---

Perimeter and Area of Similar Figures

Perimeter: If scale factor is k, perimeter ratio is also k.

Example: Similar rectangles with k = 3 Original perimeter = 20 cm New perimeter = 20 × 3 = 60 cm

Area: If scale factor is k, area ratio is k².

Example: Similar triangles with k = 2 Original area = 10 cm² New area = 10 × 2² = 10 × 4 = 40 cm²

Remember: Area uses k squared!

---

Comparing Congruence and Similarity

| Property | Congruent | Similar | |----------|-----------|---------| | Same shape | ✓ Yes | ✓ Yes | | Same size | ✓ Yes | ✗ No | | Equal angles | ✓ Yes | ✓ Yes | | Equal sides | ✓ Yes | ✗ No (proportional) | | Scale factor | k = 1 | Any k > 0 | | Transformation | Rigid | Dilation (+ rigid) | | Symbol | ≅ | ~ |

Key insight: All congruent figures are similar (with k = 1), but not all similar figures are congruent!

---

Real-World Applications

Congruence:

Manufacturing: Identical parts

• All iPhone screens are congruent
• Mass-produced items

Tiles/Patterns: Repeating designs

• Floor tiles are congruent
• Wallpaper patterns

Money: Same denomination bills

• All $20 bills are congruent

Similarity:

Maps: Scale drawings

• Map is similar to actual area
• 1 inch = 10 miles

Models: Miniatures

• Model cars similar to real cars
• Architectural models

Photography: Different print sizes

• 4×6 photo similar to 8×10 photo

Shadows: Similar to object

• Your shadow is similar to you

Photocopies: Enlarged/reduced

• 150% enlargement creates similar image

---

Indirect Measurement

Use similarity to measure hard-to-reach distances!

Example: A tree casts a 30 ft shadow. A 6 ft person casts an 8 ft shadow. How tall is the tree?

Solution: Triangles are similar (sun creates same angle)

Tree height/Tree shadow = Person height/Person shadow h/30 = 6/8

Cross multiply: 8h = 180 h = 22.5 ft

Answer: Tree is 22.5 ft tall

---

Common Mistakes to Avoid

❌ Mistake 1: Thinking AAA proves congruence

• Wrong: Same angles → congruent
• Right: Same angles → similar (might not be congruent!)

❌ Mistake 2: Forgetting to square for area

• Wrong: k = 2, so area doubles
• Right: k = 2, so area multiplies by 4

❌ Mistake 3: Using wrong corresponding sides

• Make sure you match corresponding parts!
• Order matters: ABC ~ DEF means A↔D, B↔E, C↔F

❌ Mistake 4: Mixing up congruent and similar symbols

• ≅ is congruent
• ~ is similar

❌ Mistake 5: Not simplifying scale factor

• Express 6/9 as 2/3 (simplified)

---

Problem-Solving Strategy

To prove congruence:

1. Identify corresponding parts
2. Check if sides and angles match
3. Use SSS, SAS, ASA, AAS, or HL
4. State conclusion

To prove similarity:

1. Check if angles are equal (AA)
2. Or check if sides are proportional (SSS or SAS)
3. Find scale factor if needed
4. State conclusion

To find missing measures:

1. Set up proportion using corresponding sides
2. Cross multiply
3. Solve for unknown
4. Check reasonableness

---

Quick Reference

Congruence (≅):

• Same size AND shape
• Rigid transformations
• Tests: SSS, SAS, ASA, AAS, HL
• Perimeter ratio: 1:1
• Area ratio: 1:1

Similarity (~):

• Same shape, different size
• Dilations (+ rigid transformations)
• Tests: AA, SSS, SAS
• Perimeter ratio: k:1
• Area ratio: k²:1

Scale Factor: k = (new length)/(original length)

Remember:

• All congruent figures are similar
• Not all similar figures are congruent

---

Practice Tips

Tip 1: Draw and label carefully

• Mark equal angles with arcs
• Mark equal sides with tick marks
• Helps visualize corresponding parts

Tip 2: Check all the conditions

• For congruence: need size AND shape
• For similarity: shape is enough

Tip 3: Set up proportions carefully

• Make sure ratios use corresponding sides
• Keep order consistent

Tip 4: Use AA when possible

• Easiest similarity test
• Just need two angles!

---

Summary

Congruent figures are identical:

• Same size and shape
• Created by rigid transformations
• All corresponding parts equal
• Symbol: ≅

Similar figures have same shape:

• Different sizes (usually)
• Created by dilations
• Corresponding angles equal
• Corresponding sides proportional
• Symbol: ~

Key concepts:

• Scale factor relates similar figures
• Triangle congruence tests: SSS, SAS, ASA, AAS, HL
• Triangle similarity tests: AA, SSS, SAS
• Perimeter scales by k, area scales by k²

Applications:

• Manufacturing (congruence)
• Maps and models (similarity)
• Indirect measurement (similarity)
• Design and architecture (both)

Understanding congruence and similarity helps you analyze shapes, solve problems, and see mathematical relationships in the real world!