What are the valid triangle congruence criteria on the Regents?

There are five: SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), and HL (hypotenuse and a leg, for right triangles). SSA and AAA are not valid: AAA only proves similarity, and SSA can produce two different triangles. The word included is decisive in SAS and ASA.

What is CPCTC and when do you use it?

CPCTC stands for corresponding parts of congruent triangles are congruent. You use it after you have proven two triangles congruent, to conclude that a remaining pair of corresponding sides or angles is equal. It is the final step of most proofs. You cannot use CPCTC to prove the triangles congruent in the first place; it is the consequence, not the cause.

How do you define congruence using rigid motions?

Two figures are congruent if and only if a sequence of rigid motions (reflections, rotations, and translations) maps one exactly onto the other. Because rigid motions preserve distance and angle, corresponding parts of the two figures must be equal. A dilation is not a rigid motion because it changes size, so it produces a similar figure, not a congruent one.

What tools are allowed in a geometric construction?

Only a compass, to copy distances by drawing arcs, and a straightedge, to draw straight lines. You may not use a ruler to measure lengths or a protractor to measure angles. Leave all construction arcs visible, since they are the evidence the construction was done correctly. Most constructions are justified with congruent triangles by SSS.

How do you prove a quadrilateral is a parallelogram on the coordinate plane?

Show any one of: both pairs of opposite sides parallel (equal slopes), both pairs congruent (equal distances), diagonals bisecting each other (shared midpoint), or one pair both parallel and congruent. Slopes test parallel, the distance formula tests congruent sides, and the midpoint formula tests bisecting diagonals. State clearly which condition you used.

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NY Regents Geometry: a complete guide to congruence and proof on the exam

A deep-dive NY Regents Geometry guide to the congruence-and-proof strand. Covers rigid motions and the transformational definition of congruence, the triangle congruence criteria and CPCTC, compass-and-straightedge constructions, angle and triangle theorems, and parallelogram and quadrilateral proofs, with the credit-based proof technique the Regents rewards.

Generated by Claude Opus 4.816 min readG-COUpdated 2026-06-02

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

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What this strand demands
Rigid motions and congruence
Triangle congruence and CPCTC
Geometric constructions
Angle and triangle theorems
Parallelogram and quadrilateral proofs
How this strand is examined
Check your knowledge

What this strand demands

The congruence-and-proof strand is the heart of NY Regents Geometry, and it supplies the proof questions in Parts III and IV. The exam rewards a transformational understanding of congruence, fluent use of the triangle congruence criteria and CPCTC, accurate compass-and-straightedge constructions with justification, and clean, fully reasoned proofs. This guide ties together the dot-point pages, each with its own practice: rigid motions and transformations, triangle congruence and CPCTC, geometric constructions, proofs about lines, angles, and triangles, and parallelogram and quadrilateral proofs.

Rigid motions and congruence

The three rigid motions, reflections, rotations, and translations, move a figure without changing its size or shape, so they preserve distance and angle. On the coordinate plane they become rules: reflect over the $x$ -axis sends $(x, y) \to (x, -y)$ , over the $y$ -axis sends $(x, y) \to (-x, y)$ , and a 180-degree rotation about the origin sends $(x, y) \to (-x, -y)$ . Two figures are congruent if and only if a sequence of rigid motions maps one onto the other. A dilation changes size, so it is not a rigid motion.

Triangle congruence and CPCTC

Prove triangles congruent with SSS, SAS, ASA, AAS, or HL; reject SSA and AAA. The included part matters: SAS needs the angle between the two sides, ASA the side between the two angles. After a congruence is established, CPCTC concludes any further pair of corresponding parts.

Geometric constructions

Use a compass and straightedge only. The perpendicular bisector comes from equal-radius arcs at each endpoint of a segment; the angle bisector from an arc across both sides then equal arcs from those points. Perpendiculars, parallels (by copying an angle), and the equilateral triangle build on these. The justification is almost always congruent triangles by SSS, with the equal compass radii as the equal sides, then CPCTC.

Angle and triangle theorems

Vertical angles are congruent; parallel lines cut by a transversal give congruent corresponding and alternate interior angles, and supplementary co-interior angles. A triangle's angles sum to 180 degrees, and an exterior angle equals the sum of the two remote interior angles. Isosceles base angles are congruent (proved by an auxiliary bisector and SAS), and a midsegment is parallel to the third side and half its length.

Parallelogram and quadrilateral proofs

A parallelogram has both pairs of opposite sides parallel, so opposite sides and angles are congruent and the diagonals bisect each other. Prove a parallelogram by any one of five conditions. Then an extra condition specializes it: congruent diagonals give a rectangle, perpendicular diagonals a rhombus, both a square.

A full triangle-congruence proof

Given: $\overline{PR}$ and $\overline{QS}$ bisect each other at $M$ . Prove: $\overline{PQ} \cong \overline{RS}$ .

step 1 State the given as congruent segments

Because the segments bisect each other at $M$ : $\overline{PM} \cong \overline{RM}$ and $\overline{QM} \cong \overline{SM}$ .

step 2 Identify the vertical angles

$\angle PMQ$ and $\angle RMS$ are vertical angles, so $\angle PMQ \cong \angle RMS$ .

step 3 Apply SAS

Two sides and the included angle match: $\overline{PM} \cong \overline{RM}$ , $\angle PMQ \cong \angle RMS$ , $\overline{QM} \cong \overline{SM}$ . By SAS, triangle $PMQ \cong$ triangle $RMS$ .

step 4 Conclude with CPCTC

Since the triangles are congruent, $\overline{PQ} \cong \overline{RS}$ by CPCTC.

How this strand is examined

Part I (2 credits). Apply a transformation rule, identify a congruence criterion, recognize a construction, or solve an angle equation from parallel lines.
Part II (2 credits). A short construction with justification, or a brief congruence/angle explanation. Show the key reason.
Part III and IV (4 to 6 credits). A full triangle-congruence proof, an isosceles or midsegment proof, or a coordinate proof that a quadrilateral is a parallelogram or a special type. Every statement needs a reason, ending in CPCTC or a clear classification.

Check your knowledge

Work these as you would for credit.

Find the image of $(3, -7)$ under a reflection over the $y$ -axis. (1 credit)
Name the congruence criterion for two sides and the included angle. (1 credit)
Why is SSA not a valid congruence rule? (2 credits)
Describe the perpendicular-bisector construction. (2 credits)
Two parallel lines cut by a transversal give alternate interior angles $2x + 15$ and $4x - 25$ . Find $x$ . (2 credits)
A triangle has interior angles 50 and 60 degrees. Find the exterior angle at the third vertex. (2 credits)
Given $\overline{AB} \cong \overline{AC}$ and $\overline{AD}$ bisects $\angle A$ , prove $\angle B \cong \angle C$ . (4 credits)
Show that the quadrilateral with vertices $(0,0), (3,1), (4,4), (1,3)$ is a parallelogram. (4 credits)

Solutions

Q1: Reflect over the $y$ -axis negates $x$ : $(-3, -7)$ .
Q2: SAS (side-angle-side).
Q3: SSA fixes two sides and a non-included angle, which can produce two different triangles (the ambiguous case), so it does not guarantee congruence.
Q4: Equal-radius arcs from each endpoint (radius more than half the segment) intersect at two points equidistant from both ends; the line through them is the perpendicular bisector.
Q5: Alternate interior angles are equal: $2x + 15 = 4x - 25$ , so $40 = 2x$ and $x = 20$ .
Q6: Exterior angle equals the sum of the remote interior angles: $50 + 60 = 110$ degrees.
Q7: $\angle BAD \cong \angle CAD$ (bisector), $\overline{AB} \cong \overline{AC}$ (given), $\overline{AD} \cong \overline{AD}$ (reflexive). By SAS, triangle $ABD \cong$ triangle $ACD$ , so $\angle B \cong \angle C$ by CPCTC.
Q8: Slope of $(0,0)$ to $(3,1)$ is $\frac{1}{3}$ ; slope of $(1,3)$ to $(4,4)$ is $\frac{1}{3}$ (one pair parallel). Length of each is $\sqrt{10}$ (congruent). One pair both parallel and congruent makes it a parallelogram.

Sources & how we know this

Regents Examination in Geometry — NYSED (2024)