Similar Figures - Tenth Grade Geometry
Introduction to Similar Figures
Similar Figures: Two figures that have the same shape but not necessarily the same size
Symbol: $\sim$ (is similar to)
Key Properties: Corresponding angles are equal AND corresponding sides are proportional
Scale Factor (k): The ratio of corresponding side lengths
Difference from Congruent: Congruent = same size and shape; Similar = same shape, different size
Symbol: $\sim$ (is similar to)
Key Properties: Corresponding angles are equal AND corresponding sides are proportional
Scale Factor (k): The ratio of corresponding side lengths
Difference from Congruent: Congruent = same size and shape; Similar = same shape, different size
Properties of Similar Figures:
What is PRESERVED (Equal):
• All corresponding angle measures
• Shape
• Ratios of corresponding sides
• Proportion relationships
What CHANGES:
• Size (unless scale factor = 1)
• Actual side lengths (multiplied by scale factor)
• Perimeter (multiplied by k)
• Area (multiplied by k²)
• Volume in 3D (multiplied by k³)
Transformations that Create Similar Figures:
• Dilation (resizing)
• Dilation + translation, reflection, or rotation
What is PRESERVED (Equal):
• All corresponding angle measures
• Shape
• Ratios of corresponding sides
• Proportion relationships
What CHANGES:
• Size (unless scale factor = 1)
• Actual side lengths (multiplied by scale factor)
• Perimeter (multiplied by k)
• Area (multiplied by k²)
• Volume in 3D (multiplied by k³)
Transformations that Create Similar Figures:
• Dilation (resizing)
• Dilation + translation, reflection, or rotation
1. Ratios in Similar Figures
Ratio: Comparison of two quantities
Proportion: Statement that two ratios are equal
Scale Factor: Ratio of corresponding side lengths in similar figures
Key Concept: In similar figures, ALL corresponding sides have the SAME ratio
Proportion: Statement that two ratios are equal
Scale Factor: Ratio of corresponding side lengths in similar figures
Key Concept: In similar figures, ALL corresponding sides have the SAME ratio
Scale Factor Formula:
$$k = \frac{\text{Length in Image}}{\text{Length in Pre-image}}$$
Where:
• $k$ = scale factor
• If $k > 1$: enlargement (image larger than original)
• If $k = 1$: same size (congruent)
• If $0 < k < 1$: reduction (image smaller than original)
For Similar Figures with scale factor k:
$$\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'} = k$$
Where $a, b, c$ are sides of one figure and $a', b', c'$ are corresponding sides of similar figure
$$k = \frac{\text{Length in Image}}{\text{Length in Pre-image}}$$
Where:
• $k$ = scale factor
• If $k > 1$: enlargement (image larger than original)
• If $k = 1$: same size (congruent)
• If $0 < k < 1$: reduction (image smaller than original)
For Similar Figures with scale factor k:
$$\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'} = k$$
Where $a, b, c$ are sides of one figure and $a', b', c'$ are corresponding sides of similar figure
Example 1: Find scale factor
Triangle 1 has sides 3, 4, 5
Triangle 2 has sides 6, 8, 10
Find the scale factor
$$k = \frac{6}{3} = 2$$
$$k = \frac{8}{4} = 2$$
$$k = \frac{10}{5} = 2$$
Scale factor k = 2 (Triangle 2 is twice as large as Triangle 1)
Triangle 1 has sides 3, 4, 5
Triangle 2 has sides 6, 8, 10
Find the scale factor
$$k = \frac{6}{3} = 2$$
$$k = \frac{8}{4} = 2$$
$$k = \frac{10}{5} = 2$$
Scale factor k = 2 (Triangle 2 is twice as large as Triangle 1)
2. Similarity Statements
Similarity Statement: A statement declaring two figures are similar
Notation: $\triangle ABC \sim \triangle DEF$
Order Matters: Order of vertices shows which parts correspond
Read as: "Triangle ABC is similar to triangle DEF"
Notation: $\triangle ABC \sim \triangle DEF$
Order Matters: Order of vertices shows which parts correspond
Read as: "Triangle ABC is similar to triangle DEF"
How to Read a Similarity Statement:
When we write $\triangle ABC \sim \triangle DEF$:
Corresponding Vertices:
• A corresponds to D
• B corresponds to E
• C corresponds to F
Corresponding Angles (Equal):
• $\angle A = \angle D$
• $\angle B = \angle E$
• $\angle C = \angle F$
Corresponding Sides (Proportional):
• $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k$ (scale factor)
When we write $\triangle ABC \sim \triangle DEF$:
Corresponding Vertices:
• A corresponds to D
• B corresponds to E
• C corresponds to F
Corresponding Angles (Equal):
• $\angle A = \angle D$
• $\angle B = \angle E$
• $\angle C = \angle F$
Corresponding Sides (Proportional):
• $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k$ (scale factor)
Example: Write similarity statement
Given: Triangle PQR with sides 5, 7, 9 and angles 40°, 60°, 80°
Triangle XYZ with sides 10, 14, 18 and angles 40°, 60°, 80°
Matching angles:
∠P = ∠X = 40°
∠Q = ∠Y = 60°
∠R = ∠Z = 80°
Similarity Statement: $\triangle PQR \sim \triangle XYZ$
Given: Triangle PQR with sides 5, 7, 9 and angles 40°, 60°, 80°
Triangle XYZ with sides 10, 14, 18 and angles 40°, 60°, 80°
Matching angles:
∠P = ∠X = 40°
∠Q = ∠Y = 60°
∠R = ∠Z = 80°
Similarity Statement: $\triangle PQR \sim \triangle XYZ$
3. Identify Similar Figures
Steps to Identify Similar Figures:
Step 1: Check Angles
All corresponding angles must be equal
Step 2: Check Side Ratios
Calculate ratios of all corresponding sides
Step 3: Verify Proportionality
All ratios must be equal (same scale factor)
Step 4: Conclusion
If angles equal AND sides proportional → figures are similar
Step 1: Check Angles
All corresponding angles must be equal
Step 2: Check Side Ratios
Calculate ratios of all corresponding sides
Step 3: Verify Proportionality
All ratios must be equal (same scale factor)
Step 4: Conclusion
If angles equal AND sides proportional → figures are similar
Example: Are these figures similar?
Rectangle 1: length = 6, width = 4
Rectangle 2: length = 9, width = 6
Check angles: Both rectangles have all 90° angles ✓
Check side ratios:
Length ratio: $\frac{9}{6} = \frac{3}{2}$
Width ratio: $\frac{6}{4} = \frac{3}{2}$
Both ratios equal $\frac{3}{2}$ ✓
Conclusion: YES, rectangles are similar with scale factor k = 1.5
Rectangle 1: length = 6, width = 4
Rectangle 2: length = 9, width = 6
Check angles: Both rectangles have all 90° angles ✓
Check side ratios:
Length ratio: $\frac{9}{6} = \frac{3}{2}$
Width ratio: $\frac{6}{4} = \frac{3}{2}$
Both ratios equal $\frac{3}{2}$ ✓
Conclusion: YES, rectangles are similar with scale factor k = 1.5
4. Side Lengths and Angle Measures in Similar Figures
Key Relationships:
For Angles:
$$\angle A = \angle A', \quad \angle B = \angle B', \quad \angle C = \angle C'$$
All corresponding angles are EQUAL
For Sides:
$$\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'} = k$$
All corresponding sides are PROPORTIONAL
Finding Unknown Sides:
If $\triangle ABC \sim \triangle DEF$ with scale factor k:
$$DE = k \times AB$$
$$EF = k \times BC$$
$$DF = k \times AC$$
For Angles:
$$\angle A = \angle A', \quad \angle B = \angle B', \quad \angle C = \angle C'$$
All corresponding angles are EQUAL
For Sides:
$$\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'} = k$$
All corresponding sides are PROPORTIONAL
Finding Unknown Sides:
If $\triangle ABC \sim \triangle DEF$ with scale factor k:
$$DE = k \times AB$$
$$EF = k \times BC$$
$$DF = k \times AC$$
Example: Find unknown sides
Given: $\triangle ABC \sim \triangle PQR$
AB = 4, BC = 6, AC = 8
PQ = 10, find QR and PR
Find scale factor:
$k = \frac{PQ}{AB} = \frac{10}{4} = 2.5$
Find QR:
$QR = k \times BC = 2.5 \times 6 = 15$
Find PR:
$PR = k \times AC = 2.5 \times 8 = 20$
Answer: QR = 15, PR = 20
Given: $\triangle ABC \sim \triangle PQR$
AB = 4, BC = 6, AC = 8
PQ = 10, find QR and PR
Find scale factor:
$k = \frac{PQ}{AB} = \frac{10}{4} = 2.5$
Find QR:
$QR = k \times BC = 2.5 \times 6 = 15$
Find PR:
$PR = k \times AC = 2.5 \times 8 = 20$
Answer: QR = 15, PR = 20
5. Similar Triangles and Indirect Measurement
Indirect Measurement: Using similar triangles to measure distances or heights that cannot be measured directly
Common Applications: Heights of buildings, trees, mountains
Method: Set up proportion using known measurements
Common Applications: Heights of buildings, trees, mountains
Method: Set up proportion using known measurements
Example: Find height of tree using shadow
Given:
• A person 6 feet tall casts a shadow 4 feet long
• A tree casts a shadow 20 feet long
• Find the height of the tree
Set up proportion:
$$\frac{\text{Person's height}}{\text{Person's shadow}} = \frac{\text{Tree's height}}{\text{Tree's shadow}}$$
$$\frac{6}{4} = \frac{h}{20}$$
$$4h = 120$$
$$h = 30$$
Answer: Tree is 30 feet tall
Given:
• A person 6 feet tall casts a shadow 4 feet long
• A tree casts a shadow 20 feet long
• Find the height of the tree
Set up proportion:
$$\frac{\text{Person's height}}{\text{Person's shadow}} = \frac{\text{Tree's height}}{\text{Tree's shadow}}$$
$$\frac{6}{4} = \frac{h}{20}$$
$$4h = 120$$
$$h = 30$$
Answer: Tree is 30 feet tall
6. Perimeters of Similar Figures
Perimeter Ratio Formula:
If two figures are similar with scale factor $k$, then:
$$\frac{\text{Perimeter of Image}}{\text{Perimeter of Pre-image}} = k$$
In other words:
$$P' = k \times P$$
Where:
• $P$ = perimeter of original figure
• $P'$ = perimeter of similar figure
• $k$ = scale factor
Key Concept: Perimeter ratio = Side length ratio = Scale factor
If two figures are similar with scale factor $k$, then:
$$\frac{\text{Perimeter of Image}}{\text{Perimeter of Pre-image}} = k$$
In other words:
$$P' = k \times P$$
Where:
• $P$ = perimeter of original figure
• $P'$ = perimeter of similar figure
• $k$ = scale factor
Key Concept: Perimeter ratio = Side length ratio = Scale factor
Example: Find perimeter
Triangle 1 has perimeter 30 cm and side 6 cm
Triangle 2 is similar with corresponding side 9 cm
Find perimeter of Triangle 2
Find scale factor:
$k = \frac{9}{6} = 1.5$
Find perimeter:
$P' = k \times P = 1.5 \times 30 = 45$ cm
Answer: Perimeter of Triangle 2 = 45 cm
Triangle 1 has perimeter 30 cm and side 6 cm
Triangle 2 is similar with corresponding side 9 cm
Find perimeter of Triangle 2
Find scale factor:
$k = \frac{9}{6} = 1.5$
Find perimeter:
$P' = k \times P = 1.5 \times 30 = 45$ cm
Answer: Perimeter of Triangle 2 = 45 cm
7. Angle-Angle Criterion for Similar Triangles
AA Similarity Theorem: If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar
Also Called: AA Similarity Postulate or AAA (since third angle must also be equal)
Easiest Method: Only need to check two angles!
Also Called: AA Similarity Postulate or AAA (since third angle must also be equal)
Easiest Method: Only need to check two angles!
AA Similarity Theorem:
$$\text{If } \angle A = \angle D \text{ and } \angle B = \angle E$$
$$\text{Then } \triangle ABC \sim \triangle DEF$$
Why it works:
By Triangle Angle Sum Theorem, if two angles are equal, the third must also be equal:
$$\angle C = 180° - \angle A - \angle B$$
$$\angle F = 180° - \angle D - \angle E$$
Since $\angle A = \angle D$ and $\angle B = \angle E$, then $\angle C = \angle F$
$$\text{If } \angle A = \angle D \text{ and } \angle B = \angle E$$
$$\text{Then } \triangle ABC \sim \triangle DEF$$
Why it works:
By Triangle Angle Sum Theorem, if two angles are equal, the third must also be equal:
$$\angle C = 180° - \angle A - \angle B$$
$$\angle F = 180° - \angle D - \angle E$$
Since $\angle A = \angle D$ and $\angle B = \angle E$, then $\angle C = \angle F$
Example: Prove using AA
Given: ∠P = 50°, ∠Q = 70° in △PQR
∠X = 50°, ∠Y = 70° in △XYZ
Prove triangles are similar
∠P = ∠X = 50°
∠Q = ∠Y = 70°
Two pairs of corresponding angles are equal.
Conclusion: △PQR ∼ △XYZ by AA Similarity
Given: ∠P = 50°, ∠Q = 70° in △PQR
∠X = 50°, ∠Y = 70° in △XYZ
Prove triangles are similar
∠P = ∠X = 50°
∠Q = ∠Y = 70°
Two pairs of corresponding angles are equal.
Conclusion: △PQR ∼ △XYZ by AA Similarity
8. Similarity Rules for Triangles (AA, SAS, SSS)
Three Methods to Prove Triangle Similarity
1. AA (Angle-Angle) Similarity:
Two pairs of corresponding angles are equal
$$\angle A = \angle D \text{ and } \angle B = \angle E \implies \triangle ABC \sim \triangle DEF$$
2. SAS (Side-Angle-Side) Similarity:
Two pairs of corresponding sides are proportional AND the included angles are equal
$$\frac{AB}{DE} = \frac{BC}{EF} \text{ and } \angle B = \angle E \implies \triangle ABC \sim \triangle DEF$$
3. SSS (Side-Side-Side) Similarity:
All three pairs of corresponding sides are proportional
$$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} \implies \triangle ABC \sim \triangle DEF$$
Two pairs of corresponding angles are equal
$$\angle A = \angle D \text{ and } \angle B = \angle E \implies \triangle ABC \sim \triangle DEF$$
2. SAS (Side-Angle-Side) Similarity:
Two pairs of corresponding sides are proportional AND the included angles are equal
$$\frac{AB}{DE} = \frac{BC}{EF} \text{ and } \angle B = \angle E \implies \triangle ABC \sim \triangle DEF$$
3. SSS (Side-Side-Side) Similarity:
All three pairs of corresponding sides are proportional
$$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} \implies \triangle ABC \sim \triangle DEF$$
Difference from Congruence:
Congruence (≅):
• Sides must be EQUAL
• SSS, SAS, ASA, AAS, HL
Similarity (∼):
• Sides must be PROPORTIONAL
• AA, SAS, SSS (different from congruence criteria!)
Congruence (≅):
• Sides must be EQUAL
• SSS, SAS, ASA, AAS, HL
Similarity (∼):
• Sides must be PROPORTIONAL
• AA, SAS, SSS (different from congruence criteria!)
Example 1: SAS Similarity
Given: In △ABC and △DEF:
AB = 6, BC = 9, ∠B = 50°
DE = 4, EF = 6, ∠E = 50°
Check proportions:
$\frac{AB}{DE} = \frac{6}{4} = 1.5$
$\frac{BC}{EF} = \frac{9}{6} = 1.5$
∠B = ∠E = 50°
Conclusion: △ABC ∼ △DEF by SAS Similarity
Given: In △ABC and △DEF:
AB = 6, BC = 9, ∠B = 50°
DE = 4, EF = 6, ∠E = 50°
Check proportions:
$\frac{AB}{DE} = \frac{6}{4} = 1.5$
$\frac{BC}{EF} = \frac{9}{6} = 1.5$
∠B = ∠E = 50°
Conclusion: △ABC ∼ △DEF by SAS Similarity
Example 2: SSS Similarity
Triangle 1: sides 4, 6, 8
Triangle 2: sides 6, 9, 12
Check all ratios:
$\frac{6}{4} = 1.5$
$\frac{9}{6} = 1.5$
$\frac{12}{8} = 1.5$
All ratios equal 1.5
Conclusion: Triangles are similar by SSS Similarity
Triangle 1: sides 4, 6, 8
Triangle 2: sides 6, 9, 12
Check all ratios:
$\frac{6}{4} = 1.5$
$\frac{9}{6} = 1.5$
$\frac{12}{8} = 1.5$
All ratios equal 1.5
Conclusion: Triangles are similar by SSS Similarity
9. Similar Triangles and Similarity Transformations
Similarity Transformation: A transformation that creates similar figures
Main Transformation: Dilation (resizing)
Can Include: Dilation combined with rigid motions (translation, reflection, rotation)
Result: Pre-image and image are similar (not necessarily congruent)
Main Transformation: Dilation (resizing)
Can Include: Dilation combined with rigid motions (translation, reflection, rotation)
Result: Pre-image and image are similar (not necessarily congruent)
Types of Transformations:
Rigid Motions (Preserve Size and Shape):
• Translation
• Reflection
• Rotation
→ Create CONGRUENT figures
Similarity Transformations (Preserve Shape):
• Dilation (changes size)
• Dilation + rigid motions
→ Create SIMILAR figures
Rigid Motions (Preserve Size and Shape):
• Translation
• Reflection
• Rotation
→ Create CONGRUENT figures
Similarity Transformations (Preserve Shape):
• Dilation (changes size)
• Dilation + rigid motions
→ Create SIMILAR figures
10. Similarity of Circles
Important Property: ALL circles are similar to each other
Reason: All circles have the same shape (perfectly round)
Scale Factor: Ratio of radii or ratio of diameters
Key Concept: Only the size differs, not the shape
Reason: All circles have the same shape (perfectly round)
Scale Factor: Ratio of radii or ratio of diameters
Key Concept: Only the size differs, not the shape
Circle Similarity:
For two circles with radii $r_1$ and $r_2$:
$$\text{Scale factor } k = \frac{r_2}{r_1}$$
Or using diameters:
$$k = \frac{d_2}{d_1}$$
Properties:
• All circles are similar (∼)
• Ratio of circumferences = k
• Ratio of areas = k²
For two circles with radii $r_1$ and $r_2$:
$$\text{Scale factor } k = \frac{r_2}{r_1}$$
Or using diameters:
$$k = \frac{d_2}{d_1}$$
Properties:
• All circles are similar (∼)
• Ratio of circumferences = k
• Ratio of areas = k²
11. Triangle Proportionality Theorem
Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally
Also Called: Side-Splitter Theorem
Key Word: Parallel line creates proportional segments
Also Called: Side-Splitter Theorem
Key Word: Parallel line creates proportional segments
Triangle Proportionality Theorem:
If line DE is parallel to side BC in △ABC, then:
$$\frac{AD}{DB} = \frac{AE}{EC}$$
Converse:
If $\frac{AD}{DB} = \frac{AE}{EC}$, then DE ∥ BC
Alternative Form:
$$\frac{AD}{AB} = \frac{AE}{AC}$$
If line DE is parallel to side BC in △ABC, then:
$$\frac{AD}{DB} = \frac{AE}{EC}$$
Converse:
If $\frac{AD}{DB} = \frac{AE}{EC}$, then DE ∥ BC
Alternative Form:
$$\frac{AD}{AB} = \frac{AE}{AC}$$
Example: Triangle Proportionality
Given: In △ABC, DE ∥ BC
AD = 4, DB = 6, AE = 6, find EC
Using Triangle Proportionality Theorem:
$$\frac{AD}{DB} = \frac{AE}{EC}$$
$$\frac{4}{6} = \frac{6}{EC}$$
$$4 \cdot EC = 36$$
$$EC = 9$$
Answer: EC = 9
Given: In △ABC, DE ∥ BC
AD = 4, DB = 6, AE = 6, find EC
Using Triangle Proportionality Theorem:
$$\frac{AD}{DB} = \frac{AE}{EC}$$
$$\frac{4}{6} = \frac{6}{EC}$$
$$4 \cdot EC = 36$$
$$EC = 9$$
Answer: EC = 9
12. Similarity and Altitudes in Right Triangles
Right Triangle Altitude Theorem: When altitude is drawn to the hypotenuse of a right triangle, it creates three similar triangles
Three Similar Triangles: The original triangle and two smaller triangles
Geometric Mean: The altitude is the geometric mean of the two segments of the hypotenuse
Three Similar Triangles: The original triangle and two smaller triangles
Geometric Mean: The altitude is the geometric mean of the two segments of the hypotenuse
Right Triangle Altitude Theorem:
In right triangle ABC with altitude CD to hypotenuse AB:
Three similar triangles:
$$\triangle ABC \sim \triangle ACD \sim \triangle CBD$$
Geometric Mean (Altitude):
$$h^2 = p \cdot q$$
or
$$h = \sqrt{pq}$$
Where:
• $h$ = altitude to hypotenuse
• $p$ and $q$ = segments of hypotenuse
Leg-Segment Relationships:
$$a^2 = p \cdot c \quad \text{and} \quad b^2 = q \cdot c$$
Where $a, b$ are legs and $c$ is hypotenuse
In right triangle ABC with altitude CD to hypotenuse AB:
Three similar triangles:
$$\triangle ABC \sim \triangle ACD \sim \triangle CBD$$
Geometric Mean (Altitude):
$$h^2 = p \cdot q$$
or
$$h = \sqrt{pq}$$
Where:
• $h$ = altitude to hypotenuse
• $p$ and $q$ = segments of hypotenuse
Leg-Segment Relationships:
$$a^2 = p \cdot c \quad \text{and} \quad b^2 = q \cdot c$$
Where $a, b$ are legs and $c$ is hypotenuse
Example: Find altitude
Given: Right triangle with hypotenuse segments p = 4 and q = 9
Find the altitude h
Using geometric mean formula:
$$h = \sqrt{pq}$$
$$h = \sqrt{4 \times 9}$$
$$h = \sqrt{36}$$
$$h = 6$$
Answer: Altitude = 6 units
Given: Right triangle with hypotenuse segments p = 4 and q = 9
Find the altitude h
Using geometric mean formula:
$$h = \sqrt{pq}$$
$$h = \sqrt{4 \times 9}$$
$$h = \sqrt{36}$$
$$h = 6$$
Answer: Altitude = 6 units
13. Areas of Similar Figures
Area Ratio Formula:
If two figures are similar with scale factor $k$, then:
$$\frac{\text{Area of Image}}{\text{Area of Pre-image}} = k^2$$
In other words:
$$A' = k^2 \times A$$
Where:
• $A$ = area of original figure
• $A'$ = area of similar figure
• $k$ = scale factor
Key Concept: Area ratio = (Scale factor)² = (Side ratio)²
If two figures are similar with scale factor $k$, then:
$$\frac{\text{Area of Image}}{\text{Area of Pre-image}} = k^2$$
In other words:
$$A' = k^2 \times A$$
Where:
• $A$ = area of original figure
• $A'$ = area of similar figure
• $k$ = scale factor
Key Concept: Area ratio = (Scale factor)² = (Side ratio)²
Example: Find area
Triangle 1 has area 20 cm² and side 5 cm
Triangle 2 is similar with corresponding side 10 cm
Find area of Triangle 2
Find scale factor:
$k = \frac{10}{5} = 2$
Find area:
$A' = k^2 \times A = 2^2 \times 20 = 4 \times 20 = 80$ cm²
Answer: Area of Triangle 2 = 80 cm²
Triangle 1 has area 20 cm² and side 5 cm
Triangle 2 is similar with corresponding side 10 cm
Find area of Triangle 2
Find scale factor:
$k = \frac{10}{5} = 2$
Find area:
$A' = k^2 \times A = 2^2 \times 20 = 4 \times 20 = 80$ cm²
Answer: Area of Triangle 2 = 80 cm²
14. Prove Similarity Statements
Steps to Prove Triangles are Similar:
Step 1: Given Information
List what is given
Step 2: What to Prove
State which triangles to prove similar
Step 3: Choose Method
Decide which similarity criterion to use (AA, SAS, or SSS)
Step 4: Show Requirements
• For AA: Show two pairs of angles equal
• For SAS: Show two pairs of sides proportional and included angles equal
• For SSS: Show all three pairs of sides proportional
Step 5: State Conclusion
Conclude triangles are similar by [criterion]
Step 1: Given Information
List what is given
Step 2: What to Prove
State which triangles to prove similar
Step 3: Choose Method
Decide which similarity criterion to use (AA, SAS, or SSS)
Step 4: Show Requirements
• For AA: Show two pairs of angles equal
• For SAS: Show two pairs of sides proportional and included angles equal
• For SSS: Show all three pairs of sides proportional
Step 5: State Conclusion
Conclude triangles are similar by [criterion]
Example: Two-column proof
Given: ∠A = ∠D, ∠B = ∠E
Prove: △ABC ∼ △DEF
Given: ∠A = ∠D, ∠B = ∠E
Prove: △ABC ∼ △DEF
| Statements | Reasons |
|---|---|
| 1. ∠A = ∠D | 1. Given |
| 2. ∠B = ∠E | 2. Given |
| 3. △ABC ∼ △DEF | 3. AA Similarity Theorem |
Triangle Similarity Criteria Summary
| Criterion | Requirements | What to Check | Notation |
|---|---|---|---|
| AA | 2 angles equal | Two pairs of corresponding angles | ∠A = ∠D, ∠B = ∠E |
| SAS | 2 sides proportional + included angle equal | Side-Angle-Side (angle between sides) | $\frac{AB}{DE} = \frac{BC}{EF}$, ∠B = ∠E |
| SSS | 3 sides proportional | All three pairs of sides | $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$ |
Scale Factor Effects
| Property | Effect of Scale Factor k | Formula | Example (k = 3) |
|---|---|---|---|
| Side Lengths | Multiplied by k | $\text{New} = k \times \text{Original}$ | 5 → 15 |
| Perimeter | Multiplied by k | $P' = k \times P$ | 20 → 60 |
| Area | Multiplied by k² | $A' = k^2 \times A$ | 10 → 90 |
| Volume (3D) | Multiplied by k³ | $V' = k^3 \times V$ | 8 → 216 |
| Angles | NO CHANGE | Angles stay same | 60° → 60° |
Similar vs. Congruent
| Property | Similar Figures | Congruent Figures |
|---|---|---|
| Shape | Same shape | Same shape |
| Size | Can be different | Same size |
| Angles | Corresponding angles equal | Corresponding angles equal |
| Sides | Corresponding sides PROPORTIONAL | Corresponding sides EQUAL |
| Symbol | ∼ | ≅ |
| Scale Factor | Any positive value | Always 1 |
| Transformations | Dilation (± rigid motions) | Rigid motions only |
Key Formulas Quick Reference
| Formula Name | Formula | Use |
|---|---|---|
| Scale Factor | $k = \frac{\text{Image}}{\text{Pre-image}}$ | Find size relationship |
| Side Proportion | $\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'} = k$ | Check if sides proportional |
| Perimeter Ratio | $\frac{P'}{P} = k$ | Find perimeter of similar figure |
| Area Ratio | $\frac{A'}{A} = k^2$ | Find area of similar figure |
| Geometric Mean (Altitude) | $h = \sqrt{pq}$ | Find altitude in right triangle |
| Triangle Proportionality | $\frac{AD}{DB} = \frac{AE}{EC}$ | Parallel line in triangle |
Similarity Theorems Summary
| Theorem | Statement | Application |
|---|---|---|
| AA Similarity | Two angles equal → triangles similar | Easiest method - only check 2 angles |
| SAS Similarity | Two sides proportional + included angle equal | When you know some sides and an angle |
| SSS Similarity | All three sides proportional → similar | When you know all side lengths |
| Triangle Proportionality | Parallel line divides sides proportionally | Find unknown segments |
| Right Triangle Altitude | Altitude to hypotenuse creates 3 similar triangles | Right triangles only |
| Circle Similarity | All circles are similar | Any two circles |
Success Tips for Similar Figures:
✓ Similar = same SHAPE, different size; Congruent = same shape AND size
✓ All corresponding angles EQUAL, all corresponding sides PROPORTIONAL
✓ Scale factor k = Image/Pre-image (for any corresponding sides)
✓ Three similarity criteria: AA (easiest!), SAS, SSS
✓ Perimeter ratio = k; Area ratio = k²; Volume ratio = k³
✓ ALL circles are similar to each other
✓ Triangle Proportionality: parallel line → proportional segments
✓ Right triangle altitude: creates 3 similar triangles, h = √(pq)
✓ For indirect measurement: set up proportion with similar triangles
✓ Symbol: ∼ (similar), ≅ (congruent) - don't confuse them!
✓ Similar = same SHAPE, different size; Congruent = same shape AND size
✓ All corresponding angles EQUAL, all corresponding sides PROPORTIONAL
✓ Scale factor k = Image/Pre-image (for any corresponding sides)
✓ Three similarity criteria: AA (easiest!), SAS, SSS
✓ Perimeter ratio = k; Area ratio = k²; Volume ratio = k³
✓ ALL circles are similar to each other
✓ Triangle Proportionality: parallel line → proportional segments
✓ Right triangle altitude: creates 3 similar triangles, h = √(pq)
✓ For indirect measurement: set up proportion with similar triangles
✓ Symbol: ∼ (similar), ≅ (congruent) - don't confuse them!