New YorkMathsSyllabus dot point

Which combinations of sides and angles prove two triangles congruent, and what can you conclude afterward?

Use the triangle congruence criteria (SSS, SAS, ASA, AAS, HL) to prove two triangles congruent, and use CPCTC (corresponding parts of congruent triangles are congruent) to justify further equal sides or angles.

A NY Regents Geometry answer on triangle congruence: the SSS, SAS, ASA, AAS, and HL criteria, why SSA and AAA fail, and using CPCTC to conclude further equal parts once triangles are proven congruent.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this topic is asking
The five valid criteria
Using CPCTC
Setting up a proof for credit
Try this

What this topic is asking

The Regents Geometry exam (the Congruence, G-CO, and Similarity, Right Triangles, and Trigonometry, G-SRT, clusters) wants you to prove two triangles congruent using the five valid criteria, SSS, SAS, ASA, AAS, HL, and then to use CPCTC to conclude that any further corresponding parts are equal. Triangle congruence is the backbone of the proof questions in Parts III and IV.

The five valid criteria

A triangle is determined by the right three measurements. The valid shortcuts are:

The word included is decisive: SAS needs the angle between the two sides, and ASA needs the side between the two angles. Two criteria look valid but are not: AAA fixes only the shape, not the size, so it proves similarity, not congruence; and SSA (two sides and a non-included angle) can yield two different triangles, the "ambiguous case", so it is not a congruence rule.

Using CPCTC

Once triangles are proven congruent, every pair of corresponding parts is equal. This principle, CPCTC (corresponding parts of congruent triangles are congruent), is how you reach the actual goal of most proofs: prove a pair of triangles congruent, then cite CPCTC to conclude the specific sides or angles the question asks about.

The logic runs in one direction: you must establish the triangle congruence first, then CPCTC follows. You cannot use CPCTC to prove the triangles congruent; it is the consequence, not the cause.

A congruence proof ending in CPCTC

In a figure, $\overline{AB} \cong \overline{CB}$ and $\overline{BD}$ bisects $\angle ABC$ . Prove that $\angle A \cong \angle C$ .

step 1 List the given information

$\overline{AB} \cong \overline{CB}$ (given), and $\overline{BD}$ bisects $\angle ABC$ , so $\angle ABD \cong \angle CBD$ (definition of angle bisector).

step 2 Find the shared side

$\overline{BD} \cong \overline{BD}$ by the reflexive property, since it is shared by both triangles $ABD$ and $CBD$ .

step 3 Apply a congruence criterion

Two sides and the included angle match: $\overline{AB} \cong \overline{CB}$ , $\angle ABD \cong \angle CBD$ , $\overline{BD} \cong \overline{BD}$ . By SAS, triangle $ABD \cong$ triangle $CBD$ .

step 4 Conclude with CPCTC

Since the triangles are congruent, the corresponding angles $\angle A$ and $\angle C$ are congruent by CPCTC.

Setting up a proof for credit

A Regents proof, whether two-column or paragraph, is scored on a complete chain of reasoning. Every statement needs a reason: a given, a definition (midpoint, bisector, perpendicular), a property (reflexive, vertical angles), or a theorem. A clarifying habit that protects credit is to always look for the hidden congruent part: a shared side (reflexive property) or a pair of vertical angles is very often the third piece you need, and it is the step students most often forget to state. Naming it explicitly turns an incomplete proof into a full-credit one.

Try this

Q1. Which criterion uses two angles and the side between them? [1 credit]

Cue. ASA (angle-side-angle).

Q2. Why is AAA not a congruence criterion? [2 credits]

Cue. Equal angles fix the shape but not the size, so AAA proves only similarity.

Exam-style practice questions

Practice questions written in the style of NYSED exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Regents (style)2 marksPart I (multiple choice). In triangles

ABC

and

DEF

AB = DE

\angle A = \angle D

, and

AC = DF

. Which criterion proves the triangles congruent? (1) SSS (2) SAS (3) ASA (4) AAA

Show worked answer →

The correct answer is (2).

Two pairs of sides ( $AB = DE$ and $AC = DF$ ) and the included angle between them ( $\angle A = \angle D$ ) are given. The angle is included because it sits between the two named sides, so this is SAS (side-angle-side). It is not ASA, which needs two angles and the included side, and AAA (choice 4) is not a valid congruence criterion at all because it only proves similarity.

Regents (style)4 marksPart III (constructed response). Given that

M

is the midpoint of both

\overline{AC}

and

\overline{BD}

, prove that

\overline{AB} \cong \overline{CD}

Show worked answer →

A 4-credit proof. Award credit for a correct congruence pathway and a correct CPCTC conclusion.

Since $M$ is the midpoint of $\overline{AC}$ , $\overline{AM} \cong \overline{CM}$ . Since $M$ is the midpoint of $\overline{BD}$ , $\overline{BM} \cong \overline{DM}$ . The angles $\angle AMB$ and $\angle CMD$ are vertical angles, so $\angle AMB \cong \angle CMD$ . By SAS, triangle $AMB \cong$ triangle $CMD$ . Therefore $\overline{AB} \cong \overline{CD}$ by CPCTC. Omitting the vertical-angle statement, or stating congruence without invoking CPCTC for the final step, loses credit.

Related dot points

Sources & how we know this

Regents Examination in Geometry — NYSED (2024)
New York State Next Generation Mathematics Learning Standards — NYSED (2017)