Unit 3 Vocabulary and Postulates

Unit 3: Triangle Congruency

Vocabulary and Postulates

• SSS (Side-Side-Side): Two triangles are congruent if all three sides of one triangle are congruent to the corresponding three sides of the other triangle.

• SAS (Side-Angle-Side): Two triangles are congruent if two sides and the included angle (the angle between those two sides) of one triangle are congruent to the corresponding two sides and included angle of the other triangle.

• ASA (Angle-Side-Angle): Two triangles are congruent if two angles and the included side (the side between those two angles) of one triangle are congruent to the corresponding two angles and included side of the other triangle.

• AAS (Angle-Angle-Side): Two triangles are congruent if two angles and a non-included side (a side not between the two angles) of one triangle are congruent to the corresponding two angles and non-included side of the other triangle.

• HL (Hypotenuse-Leg): This postulate applies only to right triangles. Two right triangles are congruent if the hypotenuse and one leg of one triangle are congruent to the corresponding hypotenuse and leg of the other triangle.

Triangle Congruency Statements

• Triangle congruency postulates require congruency statements to support them. The names of the postulates (SSS, SAS, ASA, AAS, HL) indicate which congruencies must be established before a triangle congruency statement can be made.

• If two triangles are congruent, such as $\triangle ABC \cong \triangle DEF$, then all corresponding sides and angles are also congruent. This gives us:

  • $AB \cong DE$

  • $BC \cong EF$

  • $CA \cong FD$

  • $\angle ABC \cong \angle DEF$

  • $\angle BCA \cong \angle EFD$

  • $\angle CAB \cong \angle FDE$

Congruent Angles from Parallel and Transversal Lines

• Congruent angle statements can arise from the relationships formed when parallel lines are intersected by transversal lines.

• Transitive Property: If $A = B$ and $B = C$, then $A = C$.

• Reflexive Property: $A = A$. A figure or part of a shape is congruent to itself.

Angle Relationships

• Given the diagram with lines a, b, c, d, e, f, g, and h:

  • $\angle a \cong \angle d$ - Vertical Angles are Congruent

  • $\angle b \cong \angle g$ - Alternate Exterior Angles are Congruent

  • $\angle c \cong \angle f$ - Alternate Interior Angles are Congruent

  • $\angle d \cong \angle h$ - Corresponding Angles are Congruent

  • $\angle a$ & $\angle b$ are supplemental (Linear Pairs are Supplemental)

  • $\angle b$ & $\angle h$ are supplemental (Same Side Exterior Angles are Supplemental)

  • $\angle c$ & $\angle e$ are supplemental (Same Side Interior Angles are Supplemental)