Geometry Regents Review Sheet

Every proof is a chain of 'if that, then this.' The trick is knowing which move unlocks the next.

Practice Geometry free Reference & formulas

Big ideas

Two triangles are congruent if you can match 3 pairs of parts in an approved pattern (SSS, SAS, ASA, AAS, HL).
'Similar' means same shape different size: corresponding angles equal, sides proportional.
Every transformation preserves something. Translations, reflections, rotations preserve size AND shape (rigid). Dilations preserve shape only (angles same, sides scaled).

Angles & parallel lines

When a line crosses two parallel lines, all the angle relationships lock in.

Vertical angles

Two lines crossing → opposite angles are equal.

Example: One angle is 40° → Vertical angle also 40°; adjacent angles = 140°

Parallel + transversal

Corresponding, alternate interior, alternate exterior angles are all EQUAL. Same-side interior angles SUM to 180°.

Example: Two parallel lines cut by transversal. One angle is 65° → Corresponding angle = 65°, same-side interior = 115°

Triangle angle sum

\text{angles sum to } 180°

Always. Any triangle. Any type.

Example: Two angles 40° and 60° → Third = 180 − 100 = 80°

Exterior angle

Exterior angle = sum of the two non-adjacent interior angles.

Example: Exterior of 130°, one remote interior 70° → Other interior = 60°

Trap: 'Corresponding' vs 'alternate interior' — memorize which position is which so you don't guess.

Tip: Mark all the given information on the diagram FIRST. Congruent segments get tick marks; congruent angles get arc marks.

Triangles — congruence & similarity

Same size, same shape = congruent. Same shape, any size = similar.

Congruence shortcuts

SSS, SAS, ASA, AAS, HL (Hypotenuse-Leg, right triangles only). SSA is NOT a valid shortcut.

Example: Two triangles share a side, and two other pairs of sides match → SSS → congruent

Similarity shortcuts

AA (two pairs of equal angles), SAS~ (sides proportional with equal included angle), SSS~ (all three sides proportional).

Example: Two angles are 40° and 60° in each → AA → similar

CPCTC

'Corresponding Parts of Congruent Triangles are Congruent.' Once triangles are congruent, ALL their matching parts are equal.

Example: Prove △ABC ≅ △DEF. What about AB and DE? → AB ≅ DE by CPCTC

Similar → proportional sides

In similar triangles, side ratios are equal.

Example: △ABC ~ △DEF, AB = 6, DE = 9, BC = 10 → EF = 10 × 9/6 = 15

Trap: SSA is famously ambiguous — don't use it as a congruence shortcut.

Tip: Look for shared sides or vertical angles — they're 'free' congruent parts often needed to finish a proof.

Right-triangle trig (SOH-CAH-TOA)

In a right triangle, the ratios of sides depend only on the acute angle.

Three ratios

sin = opp/hyp, cos = adj/hyp, tan = opp/adj. Relative to the acute angle you're using.

Example: Opposite 3, hypotenuse 5, find sin → sin = 3/5

Pythagorean theorem

a^2 + b^2 = c^2

Only in right triangles. c is the hypotenuse.

Example: Legs 3 and 4 → Hypotenuse = √(9+16) = 5

Special right triangles

45-45-90: sides 1, 1, √2. 30-60-90: sides 1, √3, 2 (opposite 30, 60, 90).

Example: 45-45-90 with legs 5 → Hypotenuse = 5√2

Inverse trig

sin⁻¹, cos⁻¹, tan⁻¹ take a ratio and give you back the angle.

Example: tan θ = 3/4, find θ → θ = tan⁻¹(3/4) ≈ 36.87°

Trap: The 'opposite' side depends on which angle you're standing at. Re-orient every time.

Tip: For 'find the angle of elevation/depression,' draw a right triangle and label opposite/adjacent/hyp — the trig ratio falls out.

Circles

Radii, chords, arcs, and inscribed angles — with a small toolkit of angle rules.

Central vs inscribed angle

Central angle = arc measure. Inscribed angle = HALF the arc.

Example: Central angle 80°, inscribed on same arc? → Inscribed = 40°

Chord-chord angle

Two chords meeting inside → angle = average of intercepted arcs.

Example: Arcs 60° and 100° → Angle = 80°

Tangent-secant angle

From outside the circle → angle = half the DIFFERENCE of intercepted arcs.

Example: Arcs 120° and 40° → Angle = (120 − 40)/2 = 40°

Arc length & sector area

\text{arc} = r\theta,\ \text{sector} = \tfrac{1}{2}r^2\theta

θ in radians. Or use (arc/circumference) = (central angle / 360°).

Example: Radius 6, central angle 60° → Arc = 6 × π/3 = 2π

Trap: 'Inscribed in a semicircle' — the inscribed angle is ALWAYS 90°. Regents loves this one.

Tip: For any chord, the perpendicular from the center bisects the chord. Great for constructions and coordinate problems.

Area, volume, transformations

Reference sheet lives here — memorize a couple, look up the rest.

Common areas

Triangle ½bh. Trapezoid ½(b₁+b₂)h. Circle πr². Sector (fraction) of πr².

Example: Trapezoid, bases 4 and 6, height 5 → ½(4+6)(5) = 25

Common volumes

Prism/cylinder Bh (base area × height). Pyramid/cone ⅓Bh. Sphere ⁴⁄₃πr³.

Example: Cone radius 3, height 4 → V = ⅓ × π9 × 4 = 12π

Transformations

Translation slides. Reflection flips. Rotation spins. Dilation scales.

Example: Dilate (2, 4) by scale 3 from origin → (6, 12)

Rigid vs non-rigid

Rigid transformations preserve distance AND angle (translations, reflections, rotations). Dilations preserve angle only.

Example: Which produces similar (not congruent)? → Dilation

Trap: Volume scales by k³ under a dilation of scale k. Area scales by k². Length scales by k.

Tip: For coordinate proofs, use distance formula for length, midpoint formula for midpoint, slope for parallel/perpendicular.

Frequently asked questions

How should I study for the Geometry Regents?

Work real past-exam questions by topic, review this one-page summary of the must-know concepts, and read the worked solution every time you miss one. The exam reuses the same ideas year after year.