Congruence and Similarity - Seventh Grade
Understanding Congruent and Similar Figures
1. Congruent vs. Similar Figures
Congruent Figures
Congruent Figures:
SAME SHAPE and SAME SIZE
Symbol: ≅
Properties of Congruent Figures:
• All corresponding sides are EQUAL in length
• All corresponding angles are EQUAL in measure
• One figure can be placed exactly on top of the other (superimposed)
• Figures may be rotated or flipped but remain congruent
Similar Figures
Similar Figures:
SAME SHAPE but DIFFERENT SIZE
Symbol: ~
Properties of Similar Figures:
• All corresponding angles are EQUAL
• All corresponding sides are PROPORTIONAL (same ratio)
• Figures are the same shape but different sizes
• One is an enlargement or reduction of the other
Key Difference
All CONGRUENT figures are SIMILAR
But NOT all similar figures are congruent!
2. Congruence Statements and Corresponding Parts
Writing Congruence Statements
Important Rule:
When writing congruence statements,
CORRESPONDING vertices must be in the SAME ORDER
△ABC ≅ △DEF
This means:
A corresponds to D
B corresponds to E
C corresponds to F
Corresponding Parts
Corresponding Angles:
∠A = ∠D (first letters)
∠B = ∠E (second letters)
∠C = ∠F (third letters)
Corresponding Sides:
AB = DE (first two letters)
BC = EF (last two letters)
AC = DF (first and last letters)
Example
Given: △PQR ≅ △STU. Find all corresponding parts.
Corresponding Angles:
∠P = ∠S
∠Q = ∠T
∠R = ∠U
Corresponding Sides:
PQ = ST
QR = TU
PR = SU
3. Side Lengths and Angle Measures of Congruent Figures
The Rule
If figures are CONGRUENT:
Corresponding Sides = EQUAL
Corresponding Angles = EQUAL
Example
Given: △ABC ≅ △XYZ. If AB = 5 cm, ∠B = 60°, and BC = 7 cm, find the corresponding measurements in △XYZ.
Since the triangles are congruent:
AB = XY, so XY = 5 cm
∠B = ∠Y, so ∠Y = 60°
BC = YZ, so YZ = 7 cm
All corresponding parts are exactly equal!
4. Side Lengths and Angle Measures of Similar Figures
The Rule
If figures are SIMILAR:
Corresponding Angles = EQUAL
Corresponding Sides = PROPORTIONAL
Proportion Rule for Similar Figures
a/b = c/d = e/f
Where a, c, e are sides of one figure
and b, d, f are corresponding sides of the other
Scale Factor:
The constant ratio between corresponding sides is called the scale factor
Scale factor = Side in Figure 2 / Corresponding Side in Figure 1
Example
Given: △ABC ~ △XYZ. If AB = 4 cm, BC = 6 cm, XY = 8 cm, and ∠B = 50°, find YZ and ∠Y.
Step 1: Find scale factor
Scale factor = XY/AB = 8/4 = 2
Step 2: Find YZ using proportion
BC/YZ = AB/XY
6/YZ = 4/8
4 × YZ = 6 × 8
YZ = 48/4 = 12 cm
Step 3: Find ∠Y
Since triangles are similar, corresponding angles are equal
∠Y = ∠B = 50°
Answer: YZ = 12 cm, ∠Y = 50°
5. Similar Figures and Indirect Measurement
What is Indirect Measurement?
Indirect measurement uses SIMILAR FIGURES
to find measurements that are
difficult or impossible to measure directly
Examples: Height of a building, width of a river, distance across a lake
Steps for Indirect Measurement
Step 1: Identify the similar figures (usually triangles)
Step 2: Find corresponding sides
Step 3: Set up a proportion
Step 4: Cross-multiply and solve
Classic Example: Shadow Method
Problem: A 6-foot tall person casts a 4-foot shadow. At the same time, a tree casts a 20-foot shadow. How tall is the tree?
Step 1: The person and their shadow form a triangle similar to the tree and its shadow
Step 2: Identify corresponding sides
Person's height ↔ Tree's height
Person's shadow ↔ Tree's shadow
Step 3: Set up proportion
Person's height / Person's shadow = Tree's height / Tree's shadow
6/4 = h/20
Step 4: Cross-multiply
4 × h = 6 × 20
4h = 120
h = 30 feet
Answer: The tree is 30 feet tall
Another Example
Problem: Two similar triangles. First triangle has sides 3 cm and 5 cm. The corresponding side to the 3 cm side in the second triangle is 9 cm. Find the side corresponding to the 5 cm side.
Set up proportion:
3/9 = 5/x
Cross-multiply:
3x = 5 × 9
3x = 45
x = 15 cm
Answer: The corresponding side is 15 cm
Quick Reference: Congruent vs. Similar
| Property | Congruent Figures | Similar Figures |
|---|---|---|
| Symbol | ≅ | ~ |
| Shape | Same | Same |
| Size | Same | Different |
| Corresponding Angles | Equal | Equal |
| Corresponding Sides | Equal | Proportional |
💡 Important Tips to Remember
✓ Congruent: Same shape AND same size (≅)
✓ Similar: Same shape but different size (~)
✓ Corresponding parts: Match vertices in order
✓ Congruent figures: All sides equal, all angles equal
✓ Similar figures: All angles equal, sides proportional
✓ Scale factor: Constant ratio between corresponding sides
✓ Proportions: Set up ratios and cross-multiply to solve
✓ Indirect measurement: Uses similar figures and proportions
✓ All congruent figures are similar but not vice versa!
✓ Order matters: △ABC ≅ △DEF means specific correspondence
🧠 Memory Tricks & Strategies
Congruent:
"Same shape, same size - congruent is the prize!"
Similar:
"Same shape but size can vary - similar figures carry!"
Corresponding Parts:
"Match the letters in the same position - that's the corresponding condition!"
Similar Figures Sides:
"Angles equal, sides in ratio - that's similar, don't you know!"
Indirect Measurement:
"Set up proportion, cross-multiply fast - find the height you can't measure at last!"
Symbols:
"≅ for congruent, ~ for similar too - know your symbols through and through!"
Master Congruence and Similarity! ≅ ~ 📐
Remember: All congruent figures are similar, but not all similar figures are congruent!