Proofs with Triangle Congruence

Writing a geometric proof involves using a logical sequence of statements and reasons to show that two triangles are exactly the same size and shape. To establish this, we rely on five main congruence postulates and theorems.

The Five Congruence Criteria

To prove two triangles are congruent, you generally need to show that three specific parts of one triangle are congruent to three corresponding parts of the other:

• SSS (Side-Side-Side): All three corresponding sides are congruent.
• SAS (Side-Angle-Side): Two sides and the included angle (the angle exactly between the two sides) are congruent.
• ASA (Angle-Side-Angle): Two angles and the included side are congruent.
• AAS (Angle-Angle-Side): Two angles and a non-included side are congruent.
• HL (Hypotenuse-Leg): In a right triangle, the hypotenuse and one leg are congruent.

Key Intermediate Steps

Often, the given information won't explicitly hand you three congruent parts. You will need to read the diagram and use geometric relationships to uncover them:

• Reflexive Property: A side or angle shared by two triangles is congruent to itself (e.g., $AC \cong AC$).
• Vertical Angles: When two straight lines intersect, the angles opposite each other are congruent.
• Parallel Lines: If parallel lines are cut by a transversal, alternate interior angles are congruent.
• Midpoints and Bisectors: A midpoint divides a segment into two equal segments. An angle bisector cuts an angle into two equal, adjacent angles.

Example 1: Using Vertical Angles and Midpoints

Given: $M$ is the midpoint of both $AB$ and $CD$. Prove: $\triangle ACM \cong \triangle BDM$

Proof Logic:

1. $M$ is the midpoint of $AB$ and $CD$. (Given)
2. $AM \cong BM$ and $CM \cong DM$. (Definition of a midpoint)
3. $\angle AMC \cong \angle BMD$. (Vertical angles are congruent)
4. $\triangle ACM \cong \triangle BDM$. (SAS Congruence, using the two pairs of sides from Step 2 and the included angles from Step 3)

Example 2: The Isosceles Triangle Theorem

Given: Isosceles $\triangle ABC$ with $AB \cong AC$. Prove: $\angle B \cong \angle C$ (The base angles are congruent).

Hint: To prove this, we can draw an auxiliary line. Let $AD$ be the angle bisector of $\angle A$, intersecting $BC$ at point $D$.

Proof Logic:

1. $AB \cong AC$. (Given)
2. Let $AD$ bisect $\angle BAC$. (By construction)
3. $\angle BAD \cong \angle CAD$. (Definition of an angle bisector)
4. $AD \cong AD$. (Reflexive Property)
5. $\triangle ABD \cong \triangle ACD$. (SAS Congruence, using Steps 1, 3, and 4)
6. $\angle B \cong \angle C$. (CPCTC - Corresponding Parts of Congruent Triangles are Congruent)