Congruent Figures Review - Tenth Grade Geometry
Introduction to Congruent Figures
Congruent Figures: Two figures that have exactly the same size and shape
Symbol: $\cong$ (is congruent to)
Key Property: All corresponding sides are equal in length AND all corresponding angles are equal in measure
Test: If one figure can be placed exactly on top of another (through rigid motions), they are congruent
Difference from Similar: Similar figures have the same shape but not necessarily the same size
Symbol: $\cong$ (is congruent to)
Key Property: All corresponding sides are equal in length AND all corresponding angles are equal in measure
Test: If one figure can be placed exactly on top of another (through rigid motions), they are congruent
Difference from Similar: Similar figures have the same shape but not necessarily the same size
Properties of Congruent Figures:
What is PRESERVED (Equal):
• All corresponding side lengths
• All corresponding angle measures
• Perimeter
• Area
• Shape and size
Transformations that Create Congruent Figures:
• Translation (slide)
• Reflection (flip)
• Rotation (turn)
• Any combination of these (rigid motions)
NOT Congruence Transformation:
• Dilation (changes size, creates similar but not congruent figures)
What is PRESERVED (Equal):
• All corresponding side lengths
• All corresponding angle measures
• Perimeter
• Area
• Shape and size
Transformations that Create Congruent Figures:
• Translation (slide)
• Reflection (flip)
• Rotation (turn)
• Any combination of these (rigid motions)
NOT Congruence Transformation:
• Dilation (changes size, creates similar but not congruent figures)
Three Properties of Congruence:
1. Reflexive Property:
$$\text{Any figure} \cong \text{itself}$$
Example: $\triangle ABC \cong \triangle ABC$
2. Symmetric Property:
$$\text{If } A \cong B, \text{ then } B \cong A$$
Example: If $\triangle ABC \cong \triangle DEF$, then $\triangle DEF \cong \triangle ABC$
3. Transitive Property:
$$\text{If } A \cong B \text{ and } B \cong C, \text{ then } A \cong C$$
Example: If $\triangle ABC \cong \triangle DEF$ and $\triangle DEF \cong \triangle GHI$, then $\triangle ABC \cong \triangle GHI$
1. Reflexive Property:
$$\text{Any figure} \cong \text{itself}$$
Example: $\triangle ABC \cong \triangle ABC$
2. Symmetric Property:
$$\text{If } A \cong B, \text{ then } B \cong A$$
Example: If $\triangle ABC \cong \triangle DEF$, then $\triangle DEF \cong \triangle ABC$
3. Transitive Property:
$$\text{If } A \cong B \text{ and } B \cong C, \text{ then } A \cong C$$
Example: If $\triangle ABC \cong \triangle DEF$ and $\triangle DEF \cong \triangle GHI$, then $\triangle ABC \cong \triangle GHI$
1. Congruence Statements and Corresponding Parts
Congruence Statement: A statement that declares two figures are congruent
Notation: $\triangle ABC \cong \triangle DEF$
Order Matters: The order of vertices shows which parts correspond
Corresponding Parts: Parts that match in congruent figures (same position)
Notation: $\triangle ABC \cong \triangle DEF$
Order Matters: The order of vertices shows which parts correspond
Corresponding Parts: Parts that match in congruent figures (same position)
Understanding Congruence Statements
How to Read a Congruence Statement:
When we write $\triangle ABC \cong \triangle DEF$, it means:
Corresponding Vertices:
• A corresponds to D
• B corresponds to E
• C corresponds to F
Corresponding Sides:
• $\overline{AB} \cong \overline{DE}$ (first two vertices)
• $\overline{BC} \cong \overline{EF}$ (second and third vertices)
• $\overline{AC} \cong \overline{DF}$ (first and third vertices)
Corresponding Angles:
• $\angle A \cong \angle D$
• $\angle B \cong \angle E$
• $\angle C \cong \angle F$
Important: The order of letters in the congruence statement tells you which parts correspond!
When we write $\triangle ABC \cong \triangle DEF$, it means:
Corresponding Vertices:
• A corresponds to D
• B corresponds to E
• C corresponds to F
Corresponding Sides:
• $\overline{AB} \cong \overline{DE}$ (first two vertices)
• $\overline{BC} \cong \overline{EF}$ (second and third vertices)
• $\overline{AC} \cong \overline{DF}$ (first and third vertices)
Corresponding Angles:
• $\angle A \cong \angle D$
• $\angle B \cong \angle E$
• $\angle C \cong \angle F$
Important: The order of letters in the congruence statement tells you which parts correspond!
Example 1: Identify corresponding parts
Given: $\triangle PQR \cong \triangle XYZ$
Corresponding Sides:
• $PQ = XY$
• $QR = YZ$
• $PR = XZ$
Corresponding Angles:
• $\angle P = \angle X$
• $\angle Q = \angle Y$
• $\angle R = \angle Z$
Given: $\triangle PQR \cong \triangle XYZ$
Corresponding Sides:
• $PQ = XY$
• $QR = YZ$
• $PR = XZ$
Corresponding Angles:
• $\angle P = \angle X$
• $\angle Q = \angle Y$
• $\angle R = \angle Z$
Example 2: Write correct congruence statement
Given: $\triangle ABC$ with sides 3, 4, 5 and $\triangle DEF$ with sides 3, 4, 5
If AB = 3, BC = 4, AC = 5 and DE = 3, EF = 4, DF = 5
Correct statement: $\triangle ABC \cong \triangle DEF$
This tells us:
• AB corresponds to DE (both = 3)
• BC corresponds to EF (both = 4)
• AC corresponds to DF (both = 5)
Given: $\triangle ABC$ with sides 3, 4, 5 and $\triangle DEF$ with sides 3, 4, 5
If AB = 3, BC = 4, AC = 5 and DE = 3, EF = 4, DF = 5
Correct statement: $\triangle ABC \cong \triangle DEF$
This tells us:
• AB corresponds to DE (both = 3)
• BC corresponds to EF (both = 4)
• AC corresponds to DF (both = 5)
Example 3: Wrong order matters!
If $\triangle ABC \cong \triangle DEF$, then:
Correct corresponding parts:
$\angle A \cong \angle D$, $\angle B \cong \angle E$, $\angle C \cong \angle F$
INCORRECT to say:
$\angle A \cong \angle E$ (Wrong! A corresponds to D, not E)
The order in the congruence statement determines correspondence!
If $\triangle ABC \cong \triangle DEF$, then:
Correct corresponding parts:
$\angle A \cong \angle D$, $\angle B \cong \angle E$, $\angle C \cong \angle F$
INCORRECT to say:
$\angle A \cong \angle E$ (Wrong! A corresponds to D, not E)
The order in the congruence statement determines correspondence!
CPCTC - Corresponding Parts of Congruent Triangles are Congruent
CPCTC: Abbreviation for "Corresponding Parts of Congruent Triangles are Congruent"
Meaning: Once you prove two triangles are congruent, you automatically know ALL corresponding parts are congruent
Use: Used as a reason in proofs after establishing triangle congruence
Applies to: All corresponding sides AND angles
Meaning: Once you prove two triangles are congruent, you automatically know ALL corresponding parts are congruent
Use: Used as a reason in proofs after establishing triangle congruence
Applies to: All corresponding sides AND angles
CPCTC Theorem:
$$\text{If } \triangle ABC \cong \triangle DEF, \text{ then:}$$
All corresponding sides are congruent:
$$\overline{AB} \cong \overline{DE}$$
$$\overline{BC} \cong \overline{EF}$$
$$\overline{AC} \cong \overline{DF}$$
All corresponding angles are congruent:
$$\angle A \cong \angle D$$
$$\angle B \cong \angle E$$
$$\angle C \cong \angle F$$
$$\text{If } \triangle ABC \cong \triangle DEF, \text{ then:}$$
All corresponding sides are congruent:
$$\overline{AB} \cong \overline{DE}$$
$$\overline{BC} \cong \overline{EF}$$
$$\overline{AC} \cong \overline{DF}$$
All corresponding angles are congruent:
$$\angle A \cong \angle D$$
$$\angle B \cong \angle E$$
$$\angle C \cong \angle F$$
Using CPCTC in Proofs:
Step 1: First, prove the triangles are congruent using one of the triangle congruence criteria:
• SSS (Side-Side-Side)
• SAS (Side-Angle-Side)
• ASA (Angle-Side-Angle)
• AAS (Angle-Angle-Side)
• HL (Hypotenuse-Leg for right triangles)
Step 2: State "By CPCTC" to conclude that any corresponding parts are congruent
Important: You MUST prove congruence BEFORE using CPCTC
Step 1: First, prove the triangles are congruent using one of the triangle congruence criteria:
• SSS (Side-Side-Side)
• SAS (Side-Angle-Side)
• ASA (Angle-Side-Angle)
• AAS (Angle-Angle-Side)
• HL (Hypotenuse-Leg for right triangles)
Step 2: State "By CPCTC" to conclude that any corresponding parts are congruent
Important: You MUST prove congruence BEFORE using CPCTC
2. Solve Problems Involving Corresponding Parts
Goal: Use congruence to find unknown side lengths, angle measures, or prove statements
Strategy: Identify which parts correspond, then use congruence to establish equality
Common Problems: Finding missing measurements, proving segments or angles equal
Strategy: Identify which parts correspond, then use congruence to establish equality
Common Problems: Finding missing measurements, proving segments or angles equal
Finding Missing Measurements
Steps to Solve Corresponding Parts Problems:
Step 1: Read the congruence statement carefully
Step 2: Identify which parts correspond based on order of vertices
Step 3: Write equations showing corresponding parts are equal
Step 4: Solve for unknown values
Step 5: Check your answer makes sense
Step 1: Read the congruence statement carefully
Step 2: Identify which parts correspond based on order of vertices
Step 3: Write equations showing corresponding parts are equal
Step 4: Solve for unknown values
Step 5: Check your answer makes sense
Example 1: Find side lengths
Given: $\triangle ABC \cong \triangle PQR$
If AB = 5, BC = 7, AC = 9, find PQ, QR, and PR
Solution:
Since $\triangle ABC \cong \triangle PQR$:
AB corresponds to PQ → $PQ = AB = 5$
BC corresponds to QR → $QR = BC = 7$
AC corresponds to PR → $PR = AC = 9$
Answer: PQ = 5, QR = 7, PR = 9
Given: $\triangle ABC \cong \triangle PQR$
If AB = 5, BC = 7, AC = 9, find PQ, QR, and PR
Solution:
Since $\triangle ABC \cong \triangle PQR$:
AB corresponds to PQ → $PQ = AB = 5$
BC corresponds to QR → $QR = BC = 7$
AC corresponds to PR → $PR = AC = 9$
Answer: PQ = 5, QR = 7, PR = 9
Example 2: Find angle measures
Given: $\triangle DEF \cong \triangle XYZ$
If $\angle D = 40°$, $\angle E = 60°$, $\angle F = 80°$, find angles of $\triangle XYZ$
Solution:
By CPCTC:
$\angle D = \angle X = 40°$
$\angle E = \angle Y = 60°$
$\angle F = \angle Z = 80°$
Answer: $\angle X = 40°$, $\angle Y = 60°$, $\angle Z = 80°$
Given: $\triangle DEF \cong \triangle XYZ$
If $\angle D = 40°$, $\angle E = 60°$, $\angle F = 80°$, find angles of $\triangle XYZ$
Solution:
By CPCTC:
$\angle D = \angle X = 40°$
$\angle E = \angle Y = 60°$
$\angle F = \angle Z = 80°$
Answer: $\angle X = 40°$, $\angle Y = 60°$, $\angle Z = 80°$
Example 3: Algebraic problems
Given: $\triangle ABC \cong \triangle DEF$
AB = 2x + 3, DE = 15. Find x.
Solution:
Since AB corresponds to DE and triangles are congruent:
$AB = DE$
$2x + 3 = 15$
$2x = 12$
$x = 6$
Answer: x = 6
Given: $\triangle ABC \cong \triangle DEF$
AB = 2x + 3, DE = 15. Find x.
Solution:
Since AB corresponds to DE and triangles are congruent:
$AB = DE$
$2x + 3 = 15$
$2x = 12$
$x = 6$
Answer: x = 6
Example 4: Multiple unknowns
Given: $\triangle PQR \cong \triangle STU$
PQ = 3a, ST = 12, QR = 2b + 1, TU = 9
Find a and b
Solution:
PQ corresponds to ST:
$3a = 12$
$a = 4$
QR corresponds to TU:
$2b + 1 = 9$
$2b = 8$
$b = 4$
Answer: a = 4, b = 4
Given: $\triangle PQR \cong \triangle STU$
PQ = 3a, ST = 12, QR = 2b + 1, TU = 9
Find a and b
Solution:
PQ corresponds to ST:
$3a = 12$
$a = 4$
QR corresponds to TU:
$2b + 1 = 9$
$2b = 8$
$b = 4$
Answer: a = 4, b = 4
Example 5: Using angle relationships
Given: $\triangle ABC \cong \triangle DEF$
$\angle A = 3x + 10$, $\angle D = 2x + 30$. Find x and $\angle A$
Solution:
Since $\angle A$ corresponds to $\angle D$:
$\angle A = \angle D$
$3x + 10 = 2x + 30$
$3x - 2x = 30 - 10$
$x = 20$
$\angle A = 3(20) + 10 = 70°$
Answer: x = 20, $\angle A = 70°$
Given: $\triangle ABC \cong \triangle DEF$
$\angle A = 3x + 10$, $\angle D = 2x + 30$. Find x and $\angle A$
Solution:
Since $\angle A$ corresponds to $\angle D$:
$\angle A = \angle D$
$3x + 10 = 2x + 30$
$3x - 2x = 30 - 10$
$x = 20$
$\angle A = 3(20) + 10 = 70°$
Answer: x = 20, $\angle A = 70°$
3. Identify Congruent Figures
Identifying Congruent Figures: Determining if two figures have the same size and shape
Methods: Visual inspection, measurement comparison, transformation analysis
Key Question: Can one figure be mapped onto the other using only rigid motions?
Methods: Visual inspection, measurement comparison, transformation analysis
Key Question: Can one figure be mapped onto the other using only rigid motions?
Methods to Identify Congruent Figures
Method 1: Measurement Check
For Triangles:
• Check if all three pairs of corresponding sides are equal
• Check if all three pairs of corresponding angles are equal
• If YES to both, triangles are congruent
For Other Polygons:
• Check all corresponding sides
• Check all corresponding angles
• Both must match for congruence
Method 2: Transformation Check
Can you map one figure onto the other using:
• Translation (slide)?
• Reflection (flip)?
• Rotation (turn)?
• Combination of these?
If YES, figures are congruent
If you need dilation (resize), they are NOT congruent (only similar)
For Triangles:
• Check if all three pairs of corresponding sides are equal
• Check if all three pairs of corresponding angles are equal
• If YES to both, triangles are congruent
For Other Polygons:
• Check all corresponding sides
• Check all corresponding angles
• Both must match for congruence
Method 2: Transformation Check
Can you map one figure onto the other using:
• Translation (slide)?
• Reflection (flip)?
• Rotation (turn)?
• Combination of these?
If YES, figures are congruent
If you need dilation (resize), they are NOT congruent (only similar)
Example 1: Identify congruent triangles
Triangle 1: sides 3, 4, 5
Triangle 2: sides 3, 4, 5
Triangle 3: sides 6, 8, 10
Analysis:
• Triangle 1 and Triangle 2: All sides equal → CONGRUENT
• Triangle 1 and Triangle 3: Sides different (even though proportional) → NOT congruent (similar only)
Note: Triangle 3 is similar to triangles 1 and 2, but NOT congruent
Triangle 1: sides 3, 4, 5
Triangle 2: sides 3, 4, 5
Triangle 3: sides 6, 8, 10
Analysis:
• Triangle 1 and Triangle 2: All sides equal → CONGRUENT
• Triangle 1 and Triangle 3: Sides different (even though proportional) → NOT congruent (similar only)
Note: Triangle 3 is similar to triangles 1 and 2, but NOT congruent
Example 2: Using transformations
Figure A is at coordinates (1,2), (3,2), (3,4)
Figure B is at coordinates (5,2), (7,2), (7,4)
Analysis:
Measure sides:
• Both have base = 2 units
• Both have height = 2 units
• Both have hypotenuse = $\sqrt{8}$ units
Transformation: Translate Figure A right 4 units → matches Figure B exactly
Conclusion: Figures ARE congruent (translation maps one to other)
Figure A is at coordinates (1,2), (3,2), (3,4)
Figure B is at coordinates (5,2), (7,2), (7,4)
Analysis:
Measure sides:
• Both have base = 2 units
• Both have height = 2 units
• Both have hypotenuse = $\sqrt{8}$ units
Transformation: Translate Figure A right 4 units → matches Figure B exactly
Conclusion: Figures ARE congruent (translation maps one to other)
Example 3: NOT congruent
Rectangle 1: 4 × 6
Rectangle 2: 3 × 8
Analysis:
• Both rectangles have same perimeter (20)
• Both rectangles have same area (24)
• BUT dimensions are different
Conclusion: NOT congruent
(Having same perimeter or area doesn't guarantee congruence!)
Rectangle 1: 4 × 6
Rectangle 2: 3 × 8
Analysis:
• Both rectangles have same perimeter (20)
• Both rectangles have same area (24)
• BUT dimensions are different
Conclusion: NOT congruent
(Having same perimeter or area doesn't guarantee congruence!)
4. Determine if Two Figures are Congruent: Justify Your Answer
Justification: Providing mathematical reasoning for why figures are or aren't congruent
Requirements: Evidence (measurements, transformations) + logical explanation
Key Skills: Comparing measurements, identifying transformations, writing clear explanations
Requirements: Evidence (measurements, transformations) + logical explanation
Key Skills: Comparing measurements, identifying transformations, writing clear explanations
Triangle Congruence Criteria
Five Ways to Prove Triangles Congruent:
1. SSS (Side-Side-Side):
All three pairs of corresponding sides are equal
$$\overline{AB} \cong \overline{DE}, \overline{BC} \cong \overline{EF}, \overline{AC} \cong \overline{DF}$$
2. SAS (Side-Angle-Side):
Two pairs of sides and the included angle are equal
$$\overline{AB} \cong \overline{DE}, \angle B \cong \angle E, \overline{BC} \cong \overline{EF}$$
3. ASA (Angle-Side-Angle):
Two pairs of angles and the included side are equal
$$\angle A \cong \angle D, \overline{AB} \cong \overline{DE}, \angle B \cong \angle E$$
4. AAS (Angle-Angle-Side):
Two pairs of angles and a non-included side are equal
$$\angle A \cong \angle D, \angle B \cong \angle E, \overline{BC} \cong \overline{EF}$$
5. HL (Hypotenuse-Leg) - Right Triangles Only:
Hypotenuse and one leg of right triangles are equal
$$\overline{AC} \cong \overline{DF}, \overline{AB} \cong \overline{DE}, \angle B = \angle E = 90°$$
1. SSS (Side-Side-Side):
All three pairs of corresponding sides are equal
$$\overline{AB} \cong \overline{DE}, \overline{BC} \cong \overline{EF}, \overline{AC} \cong \overline{DF}$$
2. SAS (Side-Angle-Side):
Two pairs of sides and the included angle are equal
$$\overline{AB} \cong \overline{DE}, \angle B \cong \angle E, \overline{BC} \cong \overline{EF}$$
3. ASA (Angle-Side-Angle):
Two pairs of angles and the included side are equal
$$\angle A \cong \angle D, \overline{AB} \cong \overline{DE}, \angle B \cong \angle E$$
4. AAS (Angle-Angle-Side):
Two pairs of angles and a non-included side are equal
$$\angle A \cong \angle D, \angle B \cong \angle E, \overline{BC} \cong \overline{EF}$$
5. HL (Hypotenuse-Leg) - Right Triangles Only:
Hypotenuse and one leg of right triangles are equal
$$\overline{AC} \cong \overline{DF}, \overline{AB} \cong \overline{DE}, \angle B = \angle E = 90°$$
What is NOT sufficient for triangle congruence:
• AAA (Angle-Angle-Angle): Creates similar triangles, NOT necessarily congruent
• SSA (Side-Side-Angle): Ambiguous case - can create two different triangles
• Just perimeter or area: Not enough information
These criteria only prove SIMILARITY, not CONGRUENCE!
• AAA (Angle-Angle-Angle): Creates similar triangles, NOT necessarily congruent
• SSA (Side-Side-Angle): Ambiguous case - can create two different triangles
• Just perimeter or area: Not enough information
These criteria only prove SIMILARITY, not CONGRUENCE!
Writing Justifications
Steps to Write a Complete Justification:
Step 1: State what you're determining
"Determine if $\triangle ABC \cong \triangle DEF$"
Step 2: Provide evidence
• List all given information
• Compare measurements
• Identify transformations
Step 3: Apply criteria or theorem
• For triangles: Name the congruence criterion (SSS, SAS, etc.)
• For transformations: Name the transformation(s)
Step 4: State conclusion
"Therefore, $\triangle ABC \cong \triangle DEF$ by [criterion/method]"
or "Therefore, the figures are NOT congruent because..."
Step 5: Explain why (if not congruent)
If not congruent, explain what's different
Step 1: State what you're determining
"Determine if $\triangle ABC \cong \triangle DEF$"
Step 2: Provide evidence
• List all given information
• Compare measurements
• Identify transformations
Step 3: Apply criteria or theorem
• For triangles: Name the congruence criterion (SSS, SAS, etc.)
• For transformations: Name the transformation(s)
Step 4: State conclusion
"Therefore, $\triangle ABC \cong \triangle DEF$ by [criterion/method]"
or "Therefore, the figures are NOT congruent because..."
Step 5: Explain why (if not congruent)
If not congruent, explain what's different
Example 1: Justify using SSS
Given: Triangle ABC with sides 5, 7, 9
Triangle DEF with sides 5, 7, 9
Determine if congruent and justify
Justification:
Given information:
• AB = 5, DE = 5
• BC = 7, EF = 7
• AC = 9, DF = 9
All three pairs of corresponding sides are equal.
Conclusion: $\triangle ABC \cong \triangle DEF$ by SSS (Side-Side-Side) criterion.
Therefore, the triangles ARE congruent.
Given: Triangle ABC with sides 5, 7, 9
Triangle DEF with sides 5, 7, 9
Determine if congruent and justify
Justification:
Given information:
• AB = 5, DE = 5
• BC = 7, EF = 7
• AC = 9, DF = 9
All three pairs of corresponding sides are equal.
Conclusion: $\triangle ABC \cong \triangle DEF$ by SSS (Side-Side-Side) criterion.
Therefore, the triangles ARE congruent.
Example 2: Justify using transformation
Given: Square ABCD with vertices at (0,0), (2,0), (2,2), (0,2)
Square EFGH with vertices at (4,1), (6,1), (6,3), (4,3)
Determine if congruent and justify
Justification:
Evidence:
• Both are squares with side length 2
• All sides equal: AB = BC = CD = DA = EF = FG = GH = HE = 2
• All angles are 90°
Transformation:
Square ABCD can be translated right 4 units and up 1 unit to map exactly onto Square EFGH.
Conclusion: Squares ABCD and EFGH ARE congruent because a translation (rigid motion) maps one onto the other perfectly.
Therefore, the figures ARE congruent.
Given: Square ABCD with vertices at (0,0), (2,0), (2,2), (0,2)
Square EFGH with vertices at (4,1), (6,1), (6,3), (4,3)
Determine if congruent and justify
Justification:
Evidence:
• Both are squares with side length 2
• All sides equal: AB = BC = CD = DA = EF = FG = GH = HE = 2
• All angles are 90°
Transformation:
Square ABCD can be translated right 4 units and up 1 unit to map exactly onto Square EFGH.
Conclusion: Squares ABCD and EFGH ARE congruent because a translation (rigid motion) maps one onto the other perfectly.
Therefore, the figures ARE congruent.
Example 3: Justify NOT congruent
Given: Triangle ABC with angles 30°, 60°, 90° and sides 5, 10, 8.66
Triangle DEF with angles 30°, 60°, 90° and sides 10, 20, 17.32
Determine if congruent and justify
Justification:
Evidence:
• Both triangles have the same angles (30°, 60°, 90°)
• However, the corresponding sides are NOT equal:
- ABC has sides 5, 10, 8.66
- DEF has sides 10, 20, 17.32
• Triangle DEF is exactly twice the size of Triangle ABC
Analysis:
While the triangles have the same shape (same angles), they have different sizes. Triangle DEF has been dilated by a scale factor of 2 from Triangle ABC.
Conclusion: Triangles ABC and DEF are NOT congruent. They are SIMILAR (same shape, different size).
Reason: For congruence, corresponding sides must be equal, not just proportional. A dilation is NOT a rigid motion.
Given: Triangle ABC with angles 30°, 60°, 90° and sides 5, 10, 8.66
Triangle DEF with angles 30°, 60°, 90° and sides 10, 20, 17.32
Determine if congruent and justify
Justification:
Evidence:
• Both triangles have the same angles (30°, 60°, 90°)
• However, the corresponding sides are NOT equal:
- ABC has sides 5, 10, 8.66
- DEF has sides 10, 20, 17.32
• Triangle DEF is exactly twice the size of Triangle ABC
Analysis:
While the triangles have the same shape (same angles), they have different sizes. Triangle DEF has been dilated by a scale factor of 2 from Triangle ABC.
Conclusion: Triangles ABC and DEF are NOT congruent. They are SIMILAR (same shape, different size).
Reason: For congruence, corresponding sides must be equal, not just proportional. A dilation is NOT a rigid motion.
Example 4: Justify using SAS
Given: In $\triangle ABC$ and $\triangle DEF$:
AB = 6, DE = 6, BC = 8, EF = 8, $\angle B = 50°$, $\angle E = 50°$
Determine if congruent and justify
Justification:
Evidence:
• Side AB = DE = 6 (corresponding sides equal)
• Angle B = Angle E = 50° (corresponding angles equal)
• Side BC = EF = 8 (corresponding sides equal)
Analysis:
Two pairs of corresponding sides are equal, and the included angle (the angle between those two sides) is also equal.
Conclusion: $\triangle ABC \cong \triangle DEF$ by SAS (Side-Angle-Side) criterion.
Therefore, the triangles ARE congruent.
Given: In $\triangle ABC$ and $\triangle DEF$:
AB = 6, DE = 6, BC = 8, EF = 8, $\angle B = 50°$, $\angle E = 50°$
Determine if congruent and justify
Justification:
Evidence:
• Side AB = DE = 6 (corresponding sides equal)
• Angle B = Angle E = 50° (corresponding angles equal)
• Side BC = EF = 8 (corresponding sides equal)
Analysis:
Two pairs of corresponding sides are equal, and the included angle (the angle between those two sides) is also equal.
Conclusion: $\triangle ABC \cong \triangle DEF$ by SAS (Side-Angle-Side) criterion.
Therefore, the triangles ARE congruent.
Triangle Congruence Criteria Summary
| Criterion | What Must Be Equal | Minimum Information Needed | Diagram Pattern |
|---|---|---|---|
| SSS | All 3 pairs of sides | 3 sides | All sides marked equal |
| SAS | 2 sides + included angle | Side-Angle-Side | Two sides and angle between them |
| ASA | 2 angles + included side | Angle-Side-Angle | Two angles and side between them |
| AAS | 2 angles + non-included side | Angle-Angle-Side | Two angles and a side not between them |
| HL | Hypotenuse + leg (right triangles) | Hypotenuse-Leg | Right angle + hypotenuse + one leg |
Congruence vs. Similarity
| Property | Congruent Figures | Similar Figures |
|---|---|---|
| Size | Same size | Can be different sizes |
| Shape | Same shape | Same shape |
| Corresponding Sides | Equal lengths | Proportional lengths |
| Corresponding Angles | Equal measures | Equal measures |
| Symbol | $\cong$ | $\sim$ |
| Transformations | Translation, reflection, rotation | Above + dilation |
| Example | Two identical triangles | Triangle and its enlargement |
Corresponding Parts Quick Reference
| If Statement Says | Then Corresponding Parts Are | Example |
|---|---|---|
| $\triangle ABC \cong \triangle DEF$ | A↔D, B↔E, C↔F | AB = DE, BC = EF, AC = DF |
| $\triangle PQR \cong \triangle XYZ$ | P↔X, Q↔Y, R↔Z | ∠P = ∠X, ∠Q = ∠Y, ∠R = ∠Z |
| $\triangle ABC \cong \triangle FED$ | A↔F, B↔E, C↔D | AB = FE, BC = ED, AC = FD |
Congruence Properties
| Property | Statement | Example |
|---|---|---|
| Reflexive | Any figure is congruent to itself | $\triangle ABC \cong \triangle ABC$ |
| Symmetric | If A ≅ B, then B ≅ A | If $\triangle ABC \cong \triangle DEF$, then $\triangle DEF \cong \triangle ABC$ |
| Transitive | If A ≅ B and B ≅ C, then A ≅ C | If $\triangle ABC \cong \triangle DEF$ and $\triangle DEF \cong \triangle GHI$, then $\triangle ABC \cong \triangle GHI$ |
Key Formulas and Theorems
| Name | Statement | Use |
|---|---|---|
| CPCTC | Corresponding Parts of Congruent Triangles are Congruent | After proving congruence, conclude all parts equal |
| Congruence Definition | Same size AND same shape | Determine if figures are congruent |
| Rigid Motion | Translation, reflection, rotation preserve congruence | Justify congruence using transformations |
| Third Angle Theorem | If two angles of triangles are equal, third angles are equal | Find missing angle in congruent triangles |
Success Tips for Congruent Figures:
✓ Congruent = same size AND same shape (all parts equal)
✓ Order in congruence statement shows which parts correspond
✓ CPCTC: Use ONLY after proving congruence
✓ Triangle congruence: SSS, SAS, ASA, AAS, HL (NOT AAA or SSA)
✓ Rigid motions (translation, reflection, rotation) create congruent figures
✓ Dilation creates SIMILAR figures, NOT congruent
✓ To justify: provide evidence + apply criterion + state conclusion
✓ Similar ≠ Congruent: Similar has same shape, congruent has same size too
✓ Check ALL corresponding parts when identifying congruence
✓ Write clear statements: $\triangle ABC \cong \triangle DEF$ by [criterion]
✓ Congruent = same size AND same shape (all parts equal)
✓ Order in congruence statement shows which parts correspond
✓ CPCTC: Use ONLY after proving congruence
✓ Triangle congruence: SSS, SAS, ASA, AAS, HL (NOT AAA or SSA)
✓ Rigid motions (translation, reflection, rotation) create congruent figures
✓ Dilation creates SIMILAR figures, NOT congruent
✓ To justify: provide evidence + apply criterion + state conclusion
✓ Similar ≠ Congruent: Similar has same shape, congruent has same size too
✓ Check ALL corresponding parts when identifying congruence
✓ Write clear statements: $\triangle ABC \cong \triangle DEF$ by [criterion]