Lesson 4 — Congruence Proofs
Big idea: Prove two triangles congruent first. Then every pair of corresponding parts is equal.
Words to know
| Word | Meaning |
|---|---|
| Two-column proof | A proof with statements on the left and reasons on the right. |
| Given | The information you start with. |
| Prove | The statement you must establish. |
| Statements | The steps of the argument. |
| Reasons | The definition, postulate, theorem, or given fact that justifies each step. |
| Corresponding parts | Matching sides and angles of congruent triangles. |
| CPCTC | Corresponding Parts of Congruent Triangles are Equal. |
Concepts to know
- The brief restatement of the definition: Corresponding parts of congruent triangles are equal.
- A proof is a chain: each statement must be justified by a reason.
- To use CPCTC, you must first prove the triangles congruent.
- A shared side is equal by the reflexive property.
- Common sources of equal parts:
- midpoints → two equal segments
- angle bisectors → two equal angles
- vertical angles → equal
- perpendicular lines → right angles → all right angles equal
- shared side → reflexive
The proof recipe
- Copy the figure and mark every given.
- Write Given and Prove.
- Find the three pairs you need.
- State the triangles congruent, in the correct order.
- Use CPCTC for the final side or angle.
Reason bank
| Reason | Use it when… |
|---|---|
| Given | The information is stated in the problem. |
| Definition of midpoint | A point divides a segment into two equal parts. |
| Definition of angle bisector | A ray divides an angle into two equal angles. |
| Vertical angles are equal | Two angles are across from each other at an intersection. |
| All right angles are equal | Two angles are both right angles. |
| Reflexive property | The same side belongs to both triangles. |
| ASA | Two angles and the included side match. |
| SAS | Two sides and the included angle match. |
| CPCTC | The triangles are already congruent. |
Worked example
Given: D is the midpoint of AC; BD ⊥ AC.
Prove: ∠BAD = ∠BCD.
| Statement | Reason |
|---|---|
| 1. D is the midpoint of AC. | Given |
| 2. AD = DC. | Definition of midpoint |
| 3. BD ⊥ AC. | Given |
| 4. ∠ADB and ∠CDB are right angles. | Perpendicular lines form right angles |
| 5. ∠ADB = ∠CDB. | All right angles are equal |
| 6. BD = BD. | Reflexive property |
| 7. ΔABD ≅ ΔCBD. | SAS |
| 8. ∠BAD = ∠BCD. | CPCTC |
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Write a proof | Use the five-step recipe above. |
| Prove a side or angle equal | Prove the triangles congruent, then use CPCTC. |
| Use a shared side | Cite the reflexive property. |
| Use a midpoint | Write the two equal segments first. |
| Use a bisector | Write the two equal angles first. |
| Choose ASA or SAS | Check whether the known side is included (ASA) or the known angle is included (SAS). |
Watch out
- Do not use CPCTC before proving the triangles congruent.
- Do not skip reasons.
- Do not assume a side or angle is equal because it looks equal.
- Keep the correspondence order correct.
- “Reflexive” is for a shared side, not for two different sides.