Geometry Regents Exam Notes

Angles

  • Angles Inside Triangles: Add up to 180^".

  • Angles Inside Quadrilaterals: Add up to 360^".

  • Angles Inside any Polygon with n sides: Add up to 180^"(n – 2)

  • Each Angle Inside a Regular Polygon: """!#(!&""" where n is the number of sides.

  • Angles Outside any Polygon: Add up to 360^". Each exterior angle in a regular polygon measures """!#(!&".

Polygons to Know

  • Triangle (3 sides)

  • Quadrilateral (4 sides)

  • Pentagon (5 sides)

  • Hexagon (6 sides)

  • Octagon (8 sides)

  • Decagon (10 sides)

Angle Pairs

  • Complementary Angles: Two angles that add to 90^".

  • Supplementary Angles: Two angles that add to 180^".

  • Linear Pair: Two angles that add to 180^" and are adjacent (form a line).

  • Vertical Angles: Angles opposite one another formed when two lines intersect. Vertical angles are congruent.

Area Formulas

  • Square: A = s^2

  • Rectangle: A = LW

  • Triangle: A = \frac{1}{2}bh

  • Circle: A = πr^2

  • Trapezoid: A = \frac{1}{2} h(b1 + b2)

Coordinate Geometry Formulas

  • Distance Formula: d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}

  • Midpoint Formula: M = (\frac{x1 + x2}{2}, \frac{y1 + y2}{2})

  • Slope Formula: m = \frac{y2 - y1}{x2 - x1}

Circles (Equations)

  • Centered at Origin: x^2 + y^2 = r^2

  • Centered at (h, k): (x – h)^2 + (y – k)^2 = r^2

  • Example: (x – 3)^2 + (y + 5)^2 = 16 has center (3, -5) and radius \sqrt{16} = 4.

  • Note: Change the signs of x and y to find the center. If no number is written (as in x2), then use zero. The number after the equal sign is the radius squared.

Proofs Using Coordinate Geometry

  • Distance: Used to prove congruence.

  • Midpoint: Used to prove bisecting.

  • Slope: Used to prove parallel (equal slopes) or perpendicular (negative reciprocal slopes).

Angles in Circles

  • Central Angle: Equal to the arc.

  • Inscribed Angle: Half the arc.

  • Angle Inside Circle: Add the arcs, then divide by 2.

  • Angle Outside Circle: Subtract the arcs, then divide by 2.

  • Tangent/Chord Angle: Half the arc.

Segments in Circles

  • Intersecting Chords: (LEFT)(RIGHT) = (LEFT)(RIGHT)

  • Two Secants: (WHOLE) (OUTER) = (WHOLE) (OUTER)

  • Secant/Tangent: (WHOLE)(OUTER) = (TANGENT)^2

  • Two Tangents: Are congruent to one another.

Tangents and Chords

  • Tangent/Diameter: Are Perpendicular.

  • Chord ⊥ Diameter: Will bisect the chord.

  • Parallel Segments: If 2 segments are parallel, then the arcs between are congruent.

Quadrilaterals

  • Parallelogram:

    • Opposite sides congruent and parallel.

    • Opposite angles congruent.

    • Consecutive angles supplementary.

    • Diagonals bisect each other.

  • Rectangle: All properties of a parallelogram, plus:

    • All 90^" angles.

    • Diagonals congruent.

  • Rhombus: All properties of a parallelogram, plus:

    • All sides congruent.

    • Diagonals perpendicular.

    • Diagonals bisect angles.

  • Square: All properties of a parallelogram, rectangle, and rhombus.

Trapezoids

  • Only ONE pair of opposite sides are PARALLEL.

  • Angles: a + b = 180^", c + d = 180^"

  • Isosceles Trapezoid: Non-parallel sides are congruent.

    • Upper Base Angles congruent.

    • Lower Base Angles congruent.

    • 1 Upper + 1 Lower = 180^"

    • Diagonals congruent.

Proving Quadrilaterals (Coordinate Geometry)

  • Parallelogram: Find the distance of all 4 sides and show opposite sides are congruent.

  • Rectangle: Find the distance of all 4 sides AND the 2 diagonals and show that opposite sides are congruent and the diagonals are also congruent.

  • Rhombus: Find the distance of all 4 sides and show that ALL sides are congruent.

  • Square: Find the distance of all 4 sides AND the 2 diagonals and show that ALL sides are congruent and the diagonals are also congruent.

  • Trapezoid: Find the slope of all 4 sides and show that one pair of opposite sides is PARALLEL and the other pair is NOT PARALLEL.

  • Isosceles Trapezoid: First, prove it’s a trapezoid, then find the distance of the non-parallel sides and show they are congruent.
    *Note: Use the Midpoint Formula only if asked to prove that segments BISECT each other (same midpoint → bisect).

Types of Triangles (by Sides)

  • Scalene: No congruent sides.

  • Isosceles: 2 congruent sides.

  • Equilateral: 3 congruent sides.

Types of Triangles (by Angles)

  • Acute: All 3 acute angles.

  • Right: 1 right angle (2 acute).

  • Obtuse: 1 obtuse angle (2 acute).

Isosceles Triangle Properties

  • 2 congruent sides called LEGS; other side is BASE.

  • Angles opposite legs are congruent (BASE ANGLES); other angle is VERTEX.

Equilateral Triangle Properties

  • All sides congruent, all angles congruent (each angle measures 60^").

Triangle Segments

  • Median: BISECTS the opposite SIDE (intersects at midpoint of opp. side).

  • Altitude: Meets the opposite side and forms a right angle.

  • Angle Bisector: BISECTS the ANGLE from where it was drawn.

  • Perpendicular Bisector: (1) BISECTS the opposite SIDE and (2) forms a right angle with opposite side. Note: It does NOT have to come from opposite angle.

Points of Concurrence in Triangles

  • Centroid: Intersection of the three Medians. Will always be located inside the triangle. Divides into 2:1 ratio (section near vertex is twice as long as section near midpoint).

  • Circumcenter: Intersection of the three Perpendicular Bisectors. Will be inside if triangle is ACUTE. Will be outside if triangle if OBTUSE. Will be on triangle if triangle is RIGHT.

  • Incenter: Intersection of the three Angle Bisectors. Will always be located inside the triangle. Incenter will also be the center of the circle inscribed in the triangle.

  • Orthocenter: Intersection of the three Altitudes. Will be inside if triangle is ACUTE. Will be outside if triangle if OBTUSE. Will be on triangle if triangle is RIGHT.

Pairs:

  • MEDIAN -- CENTROID

  • PERPENDICULAR BISECTOR -- CIRCUMCENTER

  • ANGLE BISECTOR -- INCENTER

  • ALTITUDE -- ORTHOCENTER

Euler Line

  • The three points (CENTROID, CIRCUMCENTER, ORTHOCENTER) will always lie on the same line called the Euler Line. The centroid will be between the other 2, but twice as close to circumcenter.

Mid-Segment (Midline)

  • Formed when MIDPOINTS of two sides of a triangle are connected. A mid-segment will be…

    • HALF the length of the 3rd side.

    • PARALLEL to the 3rd side.
      *Note: If all three midpoints are connected, a triangle will be formed. This “smaller” triangle will have exactly HALF the perimeter of the big triangle.

Right Triangles

  • Pythagorean Theorem: Used to find the missing side of a right triangle: a^2 + b^2 = c^2

  • Altitude drawn to Hypotenuse:

Congruent Triangles Proofs

Step 1: Name the right angles that are formed (Reason: Perpendicular lines form right angles).
Step 2: State that the right angles are congruent (Reason: All right angles are congruent).

Similar Triangles

  • 2 triangles that have all of their angles congruent and sides are in proportion (same shape but different size).

  • The ratio of the sides is the same as the ratio the PERIMETERS, ALTITUDES, MEDIANS, and ANGLE BISECTORS. However, the ratio of the AREAS of the triangles will be the square of the ratio of the sides.

*Side-Splitter Theorem: If a segment is parallel to one side of a triangle, then the sides of the triangle are in proportion because the two triangles are similar.

*Proportions using SIDES:
$$""!!"!! = ""!!"!! or "