Geometry Regents Exam Study Guide

Flashcards covering polygons, coordinate geometry, triangles, congruence, similarity, trigonometry, factoring, and transformational geometry for the Geometry Regents Exam.

Sum of Interior Angles of a Polygon

180(n-2)

Each Interior Angle of a Regular Polygon

180(n-2)/n

Sum of Exterior Angles of a Polygon

360°

Each Exterior Angle of a Regular Polygon

360/n

Standard Form of a Line

y = mx + b, where 'm' is the slope and 'b' is the y-intercept.

Slope Formula

m = (y2-y1)/(x2-x1)

Slopes of Parallel Lines

Parallel lines have the SAME slope.

Slopes of Perpendicular Lines

Perpendicular lines have NEGATIVE RECIPROCAL slopes (flip & change the sign).

Collinear Points

Points that lie on the same line.

Midpoint Formula

M = ((x1+x2)/2, (y1+y2)/2)

Distance Formula

d = √((x2-x1)² + (y2-y1)²)

Scalene Triangle

A triangle with no congruent sides.

Isosceles Triangle

A triangle with 2 congruent sides.

Equilateral Triangle

A triangle with 3 congruent sides.

Acute Triangle

A triangle where all angles are < 90°.

Right Triangle

A triangle with one angle that is 90°.

Obtuse Triangle

A triangle with one angle that is > 90°.

Equiangular Triangle

A triangle with 3 congruent angles (each 60°).

Sum of Angles in a Triangle

All triangles have 180°.

Exterior Angle Theorem

The exterior angle is equal to the sum of the two non-adjacent interior angles.

Isosceles Triangle Properties

Has 2 congruent sides and 2 congruent base angles. The altitude drawn from the vertex is also the median and angle bisector.

Midsegment of a Triangle

A segment joining the midpoints of two sides. It is always parallel to the third side and half its length. It splits the triangle into two similar triangles.

Triangle Inequality Theorem (Sides)

The sum of any 2 sides must be greater than the third side; the difference of any 2 sides must be less than the third side.

Triangle Inequality Theorem (Side-Angle Relationship)

The longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.

Pythagorean Theorem

To find the missing side of any right triangle, use a² + b² = c² where 'a' and 'b' are the legs, and 'c' is the hypotenuse.

Alternate Interior Angles

Are congruent when formed by parallel lines cut by a transversal.

Alternate Exterior Angles

Are congruent when formed by parallel lines cut by a transversal.

Corresponding Angles

Are congruent when formed by parallel lines cut by a transversal.

Same-Side Interior Angles

Are supplementary when formed by parallel lines cut by a transversal.

Side-Splitter Theorem

If a line is parallel to a side of a triangle and intersects the other two sides, then this line divides those two sides proportionally.

Side-Side-Side (SSS) Congruence

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Side-Angle-Side (SAS) Congruence

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Angle-Side-Angle (ASA) Congruence

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Angle-Angle-Side (AAS) Congruence

If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangles are congruent.

Hypotenuse-Leg (HL) Congruence

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.

CPCTC

Corresponding Parts of Congruent Triangles are Congruent.

Angle-Angle (AA) Similarity

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Side-Angle-Side (SAS) Similarity

If an angle of one triangle is congruent to an angle of a second triangle and the sides including these angles are proportional, then the triangles are similar.

Side-Side-Side (SSS) Similarity

If the corresponding side lengths of two triangles are proportional, then the triangles are similar.

Similar Figures

Figures that have congruent angles and proportional sides.

CSSTP

Corresponding Sides of Similar Triangles are in Proportion.

Property of a Proportion

In a proportion, the product of the means equals the product of the extremes.

Altitude Theorem (Geometric Mean)

The altitude in a right triangle is the geometric mean between the two segments of the hypotenuse (SAAS / Heartbeat Method).

Leg Theorem (Geometric Mean)

The leg in a right triangle is the geometric mean between the segment of the hypotenuse it touches and the whole hypotenuse (HYLLS / PSSW).

Sine (SOH)

Opposite / Hypotenuse

Cosine (CAH)

Adjacent / Hypotenuse

Tangent (TOA)

Opposite / Adjacent

Solving for a Side (Trigonometry)

Use the sin, cos, and tan buttons.

Solving for an Angle (Trigonometry)

Use the sin⁻¹, cos⁻¹, and tan⁻¹ buttons.

Cofunctions (Trigonometry)

Sine and Cosine are cofunctions, which means they are complementary (e.g., sin(A) = cos(90°-A)).

Greatest Common Factor (GCF)

A factoring method where ab + ac = a(b+c).

Difference of Two Perfect Squares (DOTS)

A factoring method where x² - y² = (x + y)(x - y).

Trinomial Factoring (TRI)

A factoring method for expressions with three terms (e.g., x² - x + 6 can factor into (x+2)(x-3)).

Reflection

A rigid motion transformation that FLIPS a figure across a line of reflection.

Rotation

A rigid motion transformation that TURNS a figure around a center of rotation.

Translation

A rigid motion transformation that SHIFTS or MOVES a figure without changing its orientation.

Dilation

A transformation that ENLARGES or REDUCES a figure, creating similar figures.

Rigid Motion

Transformations that preserve distance, congruency, angle measure, and shape (Reflection, Rotation, Translation).

Dilation (Properties)

Creates similar figures but is NOT a rigid motion because it does NOT preserve distance.

Reflection across x-axis

(x, y) = (x, -y).

Reflection across y-axis

(x, y) = (-x, y).

Reflection across y=x

(x, y) = (y, x).

Reflection across y=-x

(x, y) = (-y, -x).

Rotation 90° (Counter-clockwise)

R90(x, y) = (-y, x).

Rotation 180°

R180(x, y) = (-x, -y).

Rotation 270° (Counter-clockwise)

R270(x, y) = (y, -x).

Translation Rule

Tab(x, y) = (x + a, y + b).

Dilation Rule from Origin

Dk(x, y) = (kx, ky).

Composition of Transformations Rule

When you see 'o', work from right to left (e.g., R90° o T3,-4 means Translation, then Rotation).

Composition of 2 Reflections over 2 Parallel Lines

Is equivalent to a TRANSLATION.

Composition of 2 Reflections over 2 Intersecting Lines

Is equivalent to a ROTATION.

Rotational Symmetry Theorem (Regular Polygon)

A regular polygon with 'n' sides always has rotational symmetry, with rotations in increments equal to its central angle of 360°/n.