Unit 1.6

TRIANGLE
CONGRUENCE

When are two shapes truly identical? In geometry, we don't guess. We prove it using rigorous shortcuts like SSS and SAS. This is the heart of geometric proof.

STRUCTURAL INTEGRITY

Triangles are the only rigid
polygon. They cannot flex.

The Proof Builder

The most common proof involves Adjacent Triangles (sharing a side). We use the Reflexive Property to claim that the shared side is equal to itself, giving us a free "Side" for our SSS or SAS proof.

Given:

AD ≅ CD
BD bisects ∠ADC

Prove:

ΔABD ≅ ΔCBD

Proof Logic

Statements

Reasons

1. AD ≅ CD

Given

2. BD bisects ∠ADC

Given

3. ∠ADB ≅ ∠CDB

Drop Reason Here

4. BD ≅ BD

Drop Reason Here

5. ΔABD ≅ ΔCBD

Drop Reason Here

Reflexive Property

SAS Congruence

Definition of Bisector

SSS Congruence

Vertical Angles

The 5 Postulates

SSS

Side-Side-Side

If 3 sides of one triangle are congruent to 3 sides of another, the triangles are congruent.

SAS

Side-Angle-Side

If 2 sides and the included angle are congruent, the triangles are congruent.

ASA

Angle-Side-Angle

If 2 angles and the included side are congruent, the triangles are congruent.

AAS

Angle-Angle-Side

If 2 angles and a non-included side are congruent, the triangles are congruent.

HL

Hypotenuse-Leg

If the hypotenuse and a leg of a right triangle are congruent, the triangles are congruent.

Toolbox

CPCTCCorresponding Parts of Congruent Triangles are Congruent. (Used AFTER proving congruence).

Reflexive PropertyA quantity is equal to itself. (Used when triangles share a common side).

Vertical AnglesAngles opposite each other at an intersection. They are always equal.

MidpointA point that divides a segment into two congruent segments.

BisectorA line or ray that divides an angle or segment into two equal parts.