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New YorkMaths

Geometry: Congruence and Proof

5 dot points across 5 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do you carry out the classic compass-and-straightedge constructions, and why do they work?

Perform compass-and-straightedge constructions: copy a segment and an angle; bisect a segment (perpendicular bisector) and an angle; construct a perpendicular and a parallel line; construct an equilateral triangle; and explain why each construction produces the intended figure.
A NY Regents Geometry answer on compass-and-straightedge constructions: copying segments and angles, perpendicular and angle bisectors, perpendicular and parallel lines, the equilateral triangle, and why each one works.
9 min answer →

How do you prove a quadrilateral is a parallelogram, rectangle, rhombus, or square?

Prove theorems about parallelograms (opposite sides and angles congruent, diagonals bisect each other) and prove that a given quadrilateral is a parallelogram, rectangle, rhombus, or square using side, angle, and diagonal properties.
A NY Regents Geometry answer on quadrilateral proofs: the parallelogram properties, the ways to prove a parallelogram, and how the added conditions distinguish a rectangle, rhombus, and square.
9 min answer →

How do you prove theorems about angles formed by lines and about the angles and segments of triangles?

Prove theorems about lines and angles (vertical angles, the angle relationships from parallel lines cut by a transversal) and about triangles (the angle sum is 180 degrees, the exterior angle theorem, the isosceles triangle base angles, and the midsegment theorem).
A NY Regents Geometry answer on proving angle and triangle theorems: vertical angles, parallel-line angle pairs, the triangle angle sum, the exterior angle theorem, isosceles base angles, and the midsegment theorem.
9 min answer →

How do rigid motions move a figure without changing its size or shape, and how do they define congruence?

Represent and perform reflections, rotations, and translations using rules and the coordinate plane; recognize that rigid motions preserve distance and angle; and define congruence of two figures as the existence of a sequence of rigid motions mapping one onto the other.
A NY Regents Geometry answer on rigid motions: performing reflections, rotations, and translations on the coordinate plane, why they preserve distance and angle, and how a sequence of rigid motions defines congruence.
9 min answer →

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.
9 min answer →