A proof is three parts and a reason for each. The hunt is for the third part.
Find the third part
Two triangles share a common side. You are given one pair of equal sides and one pair of equal angles. What gives you the third part for free?
A side belonging to both triangles is equal to itself, and the reflexive property is the reason you cite for it. It is a genuine statement in the proof, not an assumption. A puts the cart first: CPCTC is what you use after congruence is established, never to establish it. Shared sides and vertical angles are the two free parts worth looking for every time.
| Statement | Reason |
|---|---|
| AB = DE | Given |
| ∠A = ∠D | Given |
| AC = DF | Given |
| △ABC ≅ △DEF | SAS |
| BC = EF | CPCTC |
Every line needs a reason, and the only ones allowed are given, a definition, a postulate, or a theorem already proved. "It looks like it" is never a reason.
Look for a shared side and for vertical angles before anything else.
Given: M is the midpoint of AB and of CD
prove △AMC ≅ △BMD
| Statement | Reason |
|---|---|
| M is the midpoint of AB | Given |
| AM = MB | Definition of midpoint |
| M is the midpoint of CD | Given |
| CM = MD | Definition of midpoint |
| ∠AMC = ∠BMD | Vertical angles are equal |
| △AMC ≅ △BMD | SAS |
Notice the order: side, angle, side — and the angle really is between the two sides at M, so SAS genuinely applies.
Two sides and an angle that is not between them is SSA, which proves nothing. Look at the diagram and confirm the angle really does sit between the two sides before naming the reason.
△AMC ≅ △BMD by SAS
established
therefore AC = BD
CPCTC
and ∠A = ∠B
CPCTC again
Three parts prove the congruence; CPCTC then hands you the other three. Using it before the congruence line is circular — you would be assuming what you set out to prove.
Each line is a statement with its reason, exactly as in a two-column proof.
AD and BC cross at E, AB ∥ CD, and E is the midpoint of AD. Prove △ABE ≅ △DCE.
∠BAE = ∠CDE
alternate interior angles, AB ∥ CD
AE = DE
definition of midpoint
∠AEB = ∠DEC
vertical angles are equal
△ABE ≅ △DCE
ASA: AE lies between the two angles
Nothing here was a given angle: both angles came free, from parallel lines and from crossing lines.
Given AB = CD and CD = EF. What can you write about AB and EF?
AB = CD, CD = EF
given
AB = EF
transitive property
Reflexive: AB = AB. Symmetric: if AB = CD then CD = AB. Transitive: two equalities sharing a middle term chain together.
Continue the first proof: show E is the midpoint of BC.
△ABE ≅ △DCE
proved above
BE = CE
CPCTC
E is the midpoint of BC
definition of midpoint
CPCTC gives a pair of equal parts; a definition can then turn that pair into a new fact.
Expect a two-column proof with one reason missing, a figure asking which test proves the triangles congruent, and the name of a property (reflexive, transitive, vertical angles, definition of midpoint). The fastest route is to list the given parts, then hunt for the free ones — a shared side, vertical angles, angles from parallel lines — until there are three. The 10-second check: confirm the angle (or side) you call "included" really sits between the other two parts, and that CPCTC appears only after the congruence line.
Name the reason
In a proof, a student writes "∠AMC = ∠BMD" for two angles formed by two intersecting lines. What reason should be given?
Angles opposite each other where two lines cross are vertical angles, and that theorem is the reason. C is for a shared side, not an angle. D would only apply if the problem stated it, and here it follows from the diagram. A comes after congruence — using it here would assume the conclusion.
Hint: ask where the fact comes from. Something stated in the problem is "given"; something that follows from the diagram needs its own theorem.
