Perpendicular Bisector Theorem and Proof Skills

Deck 2 of 2: Triangle Congruence Meets Classical Geometry

In this deck:

• Prove the Perpendicular Bisector Theorem (both directions)
• Understand biconditional theorems
• Connect all four angle and line theorems

What You Will Learn This Deck

By the end of this deck, you should be able to:

4. Prove the Perpendicular Bisector Theorem — both directions
5. Construct logical proofs citing definitions, postulates, and theorems

From Deck 1 (as tools): Vertical Angles Theorem, Corresponding Angles Postulate, Alternate Interior Angles Theorem

Observing Equidistance on a Perpendicular Bisector

Pick any point P on the bisector. Measure PA and PB. What do you find?

Biconditional Theorems: Two Proofs Required

The Perpendicular Bisector Theorem is a biconditional:

"A point lies on the perpendicular bisector of if and only if it is equidistant from A and B."

"If and only if" = two separate statements:

• Forward: If P is on the perpendicular bisector →
• Converse: If → P is on the perpendicular bisector

Each direction requires its own complete proof.

Setup: Segment, Midpoint, and Bisector

Definitions established for both proofs:

• : given segment, endpoints A and B
• M: midpoint of , so
• : perpendicular bisector — line through M with
• P: any point on , so
• PM: shared side (reflexive property)

Three key facts emerge: , ,

Forward Proof: On Bisector → Equidistant (SAS)

Check: Identify the Three SAS Congruences

Try this before the next slide.

In the forward proof, SAS requires three congruences. List them:

1. First side: ____
2. Included angle: ____
3. Second side: ____

Name the reason (definition, postulate, or property) for each.

Answer: The Three Congruences for SAS

• Side: (midpoint)
• Angle: (perpendicular)
• Side: (reflexive)

Together: SAS → → .

Converse Proof: Part 1 — Establish SSS Congruence

Given: ; M is the midpoint of
To Prove: P lies on the perpendicular bisector of

Converse Proof: Deriving the Right Angle

From Part 1: , so

Why SSS Here, Not SAS?

The two proofs use different congruence criteria. Here is why:

The given information determines which congruence criterion to use.

Check: Why SSS, Not SAS, for the Converse?

Think before reading the answer below.

Question: In the converse proof, why can't we use SAS?

Answer: PA = PB gives sides only — no angle at M. Determining that right angle is the goal; using it as a premise would be circular. SSS avoids needing any angle upfront.

Four Theorems: How They Connect

Each theorem becomes a tool for the next.

Multi-Step Proof: Co-Interior Angles Are Supplementary

Given: , ∠3 and ∠5 are co-interior (same-side interior)
To Prove: ∠3 and ∠5 are supplementary

This proof uses results from Deck 1 as tools.

The Equidistant Far Point: Not Just Nearby

, , on the bisector :

The theorem holds for ALL points on the bisector — even very far ones.

Proof-Writing Strategy: A Five-Step Framework

When approaching any proof:

1. Diagram and label — draw and name all parts
2. State given and to prove — before writing anything
3. Inventory tools — which definitions and theorems apply?
4. Forward and backward — from given; from conclusion
5. Connect — find where the chains meet

Key Takeaways: All Four Theorems

✓ Vertical Angles: congruent — linear pairs + algebra
✓ Corresponding Angles: congruent — postulate (parallel required)
✓ Alternate Interior: congruent — corresponding + vertical
✓ Perpendicular Bisector: biconditional — SAS forward, SSS converse

Two proofs required · No circular reasoning · Parallel lines required

What Comes Next: Triangle Theorems

In HSG.CO.C.10 you will:

• Prove the Triangle Angle Sum Theorem using alternate interior angles
• Prove the Isosceles Triangle Theorem using the Perpendicular Bisector Theorem
• Extend to parallelogram properties in HSG.CO.C.11