Congruent and Similar Figures

What are Congruent Figures?

Congruent figures have the same size and shape.

All corresponding parts are equal:

• Same side lengths
• Same angle measures

Symbol: ≅ (congruent to)

Think: Exact copies!

Example: Congruent Triangles

Triangle ABC ≅ Triangle DEF

Means:

• Side AB = Side DE
• Side BC = Side EF
• Side CA = Side FD
• Angle A = Angle D
• Angle B = Angle E
• Angle C = Angle F

Identifying Congruent Figures

Check:

1. All corresponding sides equal
2. All corresponding angles equal

Example 1: Are They Congruent?

Rectangle 1: 5 cm × 3 cm Rectangle 2: 5 cm × 3 cm

All sides match, all angles 90°

Yes, congruent! (Rectangle 1 ≅ Rectangle 2)

Example 2: NOT Congruent

Square 1: 4 cm sides Square 2: 6 cm sides

Same shape, different size

Not congruent (but they are similar!)

What are Similar Figures?

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

All corresponding angles are equal All corresponding sides are proportional

Symbol: ~ (similar to)

Think: Enlargements or reductions!

Example: Similar Triangles

Triangle ABC ~ Triangle XYZ

Means:

• Angle A = Angle X
• Angle B = Angle Y
• Angle C = Angle Z
• AB/XY = BC/YZ = CA/ZX (same ratio)

Scale Factor

Scale factor is the ratio of corresponding side lengths.

Scale Factor = New Length / Original Length

Example 1: Find Scale Factor

Original triangle: Sides 3, 4, 5 Enlarged triangle: Sides 6, 8, 10

Scale factor: 6/3 = 8/4 = 10/5 = 2

Answer: Scale factor is 2 (doubled)

Example 2: Reduction

Original square: Side 12 cm Reduced square: Side 4 cm

Scale factor: 4/12 = 1/3

Answer: Scale factor is 1/3 (reduced to 1/3 size)

Using Scale Factor

Example 1: Find Missing Side

Two similar rectangles. Scale factor = 3.

Original: Length 5 cm, Width 2 cm Enlarged: Length ? cm, Width ? cm

Multiply by scale factor:

• Length: 5 × 3 = 15 cm
• Width: 2 × 3 = 6 cm

Answer: 15 cm × 6 cm

Example 2: Find Unknown

Similar triangles. Scale factor = 1/2.

Large triangle: Base 10 m Small triangle: Base ? m

Calculate: 10 × (1/2) = 5 m

Answer: 5 m

Finding Corresponding Parts

Corresponding parts match up between similar or congruent figures.

Tips:

• Same position in the figure
• Listed in same order in names
• Match angles first, then sides

Example: Corresponding Parts

Triangle ABC ~ Triangle DEF

Corresponding angles:

• A ↔ D
• B ↔ E
• C ↔ F

Corresponding sides:

• AB ↔ DE (opposite angles C and F)
• BC ↔ EF (opposite angles A and D)
• CA ↔ FD (opposite angles B and E)

Proportional Sides

For similar figures: Ratios of corresponding sides are equal

Set up proportion: a/b = c/d

Example: Find Missing Side

Similar triangles:

Triangle 1: Sides 4, 6, x Triangle 2: Sides 8, 12, 18

Set up proportion:

4/8 = 6/12 = x/18

Use any pair:

4/8 = x/18
8x = 72
x = 9

Answer: x = 9

Check: 4/8 = 6/12 = 9/18 = 1/2 ✓

Congruent vs. Similar

Key: Congruent figures are always similar (scale factor = 1)!

Real-World Applications

Maps: Similar to real area (scale factor like 1:1000)

• 1 inch on map = 1000 inches in reality

Architecture: Scale models of buildings

• Model is similar to actual building

Photography: Enlarging/reducing images

• Same shape, different sizes

Art: Creating proportional drawings

• Grid method uses similar rectangles

Manufacturing: Making different sizes of same product

• Small, medium, large shirts (similar shapes)

Solving Problems with Similar Figures

Example: Shadow Problem

Tree casts 30 ft shadow Person (6 ft tall) casts 5 ft shadow

How tall is the tree?

Set up proportion:

tree height / tree shadow = person height / person shadow
x / 30 = 6 / 5
5x = 180
x = 36 ft

Answer: Tree is 36 ft tall

Area and Volume Relationships

For similar figures with scale factor k:

Linear measures: Multiply by k Area: Multiply by k² Volume: Multiply by k³

Example: Area of Similar Figures

Two similar rectangles. Scale factor = 3.

Small rectangle: Area = 4 cm²

Large rectangle area:

• 4 × 3² = 4 × 9 = 36 cm²

Answer: 36 cm²

NOT 4 × 3 = 12! Must square the scale factor for area.

Identifying Similar Triangles

Three ways to prove triangles similar:

AA (Angle-Angle): Two pairs of corresponding angles equal SSS (Side-Side-Side): All three side ratios equal SAS (Side-Angle-Side): Two side ratios and included angle equal

Example: AA Similarity

Triangle 1: Angles 50°, 60°, 70° Triangle 2: Angles 50°, 60°, 70°

Two angles match → Similar! (AA)

Practice