In this chapter, you will learn
- —Understand the concept of congruence in triangles
- —Learn and apply SSS (Side-Side-Side) criterion
- —Learn and apply SAS (Side-Angle-Side) criterion
- —Learn and apply ASA (Angle-Side-Angle) criterion
- —Learn and apply RHS (Right angle-Hypotenuse-Side) criterion
- —Apply CPCT (Corresponding Parts of Congruent Triangles) to find unknown measurements
What is Congruence?
Congruent triangles are triangles that have the same shape and size. All corresponding sides and angles are equal.
Definition: Two triangles are congruent if their corresponding sides are equal and corresponding angles are equal.
Notation: △ABC ≅ △DEF means triangle ABC is congruent to triangle DEF.
Key Point: Congruent triangles can be in different positions and orientations, but they have identical dimensions.
Real-Life Applications:
- Matching tiles or tiles patterns in construction
- Identical mechanical parts in machinery
- Photo frames with same dimensions
- Identical triangular supports in bridges
Exam Tip
Always write corresponding vertices in order when stating congruence: △ABC ≅ △DEF means A↔D, B↔E, C↔F.
Common Mistake
Don't confuse similar triangles (same shape, different size) with congruent triangles (same shape AND size).
SSS Criterion (Side-Side-Side)
SSS Criterion: If all three sides of one triangle are equal to the corresponding three sides of another triangle, then the triangles are congruent.
SSS Congruence Rule:
If AB = DE, BC = EF, and CA = FD, then △ABC ≅ △DEF
Example:
△PQR: PQ = 5 cm, QR = 6 cm, RP = 7 cm
△XYZ: XY = 5 cm, YZ = 6 cm, ZX = 7 cm
Since all three sides match, △PQR ≅ △XYZ (by SSS)
Why SSS works: Three sides uniquely determine the shape and size of a triangle. If all sides match, the triangles must be identical.
Exam Tip
SSS is the simplest criterion when you have all three side lengths. No need to check angles.
Common Mistake
Make sure you match corresponding sides correctly. AB must equal DE, not DF.
SAS Criterion (Side-Angle-Side)
SAS Criterion: If two sides and the included angle (angle between them) of one triangle equal the corresponding sides and angle of another triangle, they are congruent.
SAS Congruence Rule:
If AB = DE, ∠B = ∠E, and BC = EF, then △ABC ≅ △DEF
The angle must be BETWEEN the two sides.
Example:
△ABC: AB = 4 cm, ∠B = 60°, BC = 5 cm
△PQR: PQ = 4 cm, ∠Q = 60°, QR = 5 cm
Since two sides and included angle match, △ABC ≅ △PQR (by SAS)
Important: The angle must be INCLUDED between the two given sides, not at the end.
Exam Tip
Remember: SAS requires the angle to be BETWEEN the two sides. If angle is not between, it's not SAS.
Common Mistake
Confusing with SSA (Side-Side-Angle). SSA does NOT guarantee congruence. The angle must be INCLUDED.
ASA Criterion (Angle-Side-Angle)
ASA Criterion: If two angles and the included side (side between them) of one triangle equal the corresponding angles and side of another triangle, they are congruent.
ASA Congruence Rule:
If ∠A = ∠D, AB = DE, and ∠B = ∠E, then △ABC ≅ △DEF
The side must be BETWEEN the two angles.
Example:
△ABC: ∠A = 50°, AB = 6 cm, ∠B = 70°
△XYZ: ∠X = 50°, XY = 6 cm, ∠Y = 70°
Since two angles and included side match, △ABC ≅ △XYZ (by ASA)
Note: In ASA, the side must lie between the two given angles.
Exam Tip
ASA is useful when you know angles and can measure the side between them.
Common Mistake
Don't mix up ASA with AAS (Angle-Angle-Side). They are different congruence criteria.
RHS Criterion (Right angle-Hypotenuse-Side)
RHS Criterion: In right triangles, if the hypotenuse and one side of one right triangle equal the hypotenuse and corresponding side of another right triangle, they are congruent.
RHS Congruence Rule (for Right Triangles):
If ∠A = 90°, ∠D = 90°, BC = EF (hypotenuses), and AB = DE, then △ABC ≅ △DEF
Example:
Right △ABC: ∠A = 90°, BC (hypotenuse) = 10 cm, AB = 6 cm
Right △PQR: ∠P = 90°, QR (hypotenuse) = 10 cm, PQ = 6 cm
Since right angles, hypotenuses, and one side match, △ABC ≅ △PQR (by RHS)
Key Point: RHS ONLY applies to right triangles. Compare hypotenuse and one leg (not hypotenuse).
Exam Tip
RHS is the easiest when dealing with right triangles. Just compare hypotenuse and any one side.
Common Mistake
RHS only works for RIGHT triangles. Don't use it for acute or obtuse triangles.
CPCT - Corresponding Parts of Congruent Triangles
CPCT: If two triangles are congruent, then all corresponding parts (sides and angles) are equal.
CPCT Rule:
If △ABC ≅ △DEF, then:
- AB = DE, BC = EF, CA = FD (corresponding sides are equal)
- ∠A = ∠D, ∠B = ∠E, ∠C = ∠F (corresponding angles are equal)
Example Problem:
Given: △PQR ≅ △ABC with PQ = 5 cm, QR = 7 cm, PR = 6 cm
Find: Length of AB
Solution: Since △PQR ≅ △ABC, by CPCT, AB = PQ = 5 cm
Usage: CPCT is used to find unknown angles and sides after proving triangles are congruent.
Exam Tip
Always use CPCT after proving congruence. State the congruence first, then use CPCT to find unknowns.
Common Mistake
Don't apply CPCT without first proving the triangles are congruent using one of the four criteria.
Chapter Summary
Congruence of Triangles deals with identifying identical triangles. This chapter covers:
- Concept: Congruent triangles have equal corresponding sides and angles
- SSS Criterion: All three sides equal → triangles congruent
- SAS Criterion: Two sides + included angle equal → triangles congruent
- ASA Criterion: Two angles + included side equal → triangles congruent
- RHS Criterion: For right triangles: hypotenuse + one side equal → triangles congruent
- CPCT: Use after proving congruence to find unknown parts
Exam Focus: Proving triangle congruence, identifying correct criteria, using CPCT to find unknowns, solving geometry problems.