Geometry EOC Review Study Guide

Foundations of Geometry

• Inductive reasoning: making a conjecture (guess) based on observation of patterns.
• Deductive reasoning: proving a statement based on facts (definitions, theorems, postulates,…).
• Counterexample: an example that disproves a statement.
• Undefined terms: point, line, plane.
• Collinear: on the same line.
• Coplanar: in the same plane.
• Skew lines: Non-coplanar and never intersect.
• Postulate: a statement that is assumed to be true (also called an axiom).
• Theorem: a statement that must be proven true.

Reasoning and Proof

• Hypothesis: the "if" part of a conditional statement (p).
• Conclusion: the "then" part of a conditional statement (q).
• Conditional statement: an "if-then" statement.
• Converse: switch the "if" and the "then" parts of the conditional.
• Inverse: Negate both the "if" and the "then" parts.
• Contrapositive: Switch and negate both parts.
• Biconditional: a conditional and its converse are both true and combined into one statement with "if and only if".
• Counterexample: a specific example where the hypothesis of a conditional is true but the conclusion is false.

Angle Relationships

• Angle Bisector: any figure that divides an angle into two congruent angles.
• Midpoint of a Segment: A point that divides the segment into two congruent segments.
• Segment Addition Postulate: If B is between A and C, then $AB + BC = AC$.
• Angle Addition Postulate: If B is in the interior of $\angle AOC$, then $m\angle AOB + m\angle BOC = m\angle AOC$.
• Adjacent angles: angles that are next to each other (e.g., angles 3 and 4).
• Vertical angles: angles that are opposite each other and congruent (e.g., angles 2 and 3).
• Linear pair: a pair of adjacent angles that form a straight line (sum of 180 degrees) (e.g., angles 1 and 3).
• Complementary Angles: two angles whose sum is 90 degrees (e.g., angles 2 and 5).
• Supplementary Angles: two angles whose sum is 180 degrees (e.g., angles 1 and 3).

Properties of Equality and Congruence

• Reflexive Property of Equality: $A = A$
• Symmetric Property of Equality: If $a = b$, then $b = a$
• Transitive Property of Equality: If $a = b$ and $b = c$, then $a = c$.
• Substitution Property of Equality: If $a = b$, then a can be substituted for b.
• Reflexive Property of Congruence: $\angle A \cong \angle A$
• Symmetric Property of Congruence: If $\angle A \cong \angle B$, then $\angle B \cong \angle A$.
• Transitive Property of Congruence: If $\angle A \cong \angle B$ and $\angle B \cong \angle C$, then $\angle A \cong \angle C$.

Parallel and Perpendicular Lines

• Parallel Lines: If a transversal intersects parallel lines:
  • Corresponding Angles are congruent.
  • Alternate Interior Angles are congruent.
  • Alternate Exterior Angles are congruent.
  • Consecutive Interior Angles are supplementary (sum of 180).
  • Consecutive Exterior Angles are supplementary (sum of 180).
• Use Properties of Parallel Lines to prove angle congruence.
• Use Converses to prove lines are parallel.
• If two lines are parallel to a third line, then they are parallel to each other.
• In a plane, if two lines are perpendicular to a third line, they are parallel to each other.

Triangle Angle Sum

• Triangle angle sum: the angles in a triangle add up to 180 degrees.
• Triangle exterior angles: each exterior angle is the sum of the two remote interior angles.

Polygon Angle Sum

• Polygon angle sum: for a polygon with n sides, the angles add up to $(n-2)180$.
• Measure of an interior angle of a regular polygon: $\frac{(n-2)180}{n}$.
• Sum of the measures of exterior angles is 360.
• Measure of a single exterior angle is $\frac{360}{n}$.

Coordinate Geometry

• Slope formula: $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$.
• Parallel lines have equal slopes.
• Perpendicular lines have negative reciprocal slopes, and the product of their slopes equals -1 ($m<em>1 * m</em>2 = -1$).
• Distance Formula: $d = \sqrt{(x<em>2 - x</em>1)^2 + (y<em>2 - y</em>1)^2}$.
• Midpoint Formula: $m = (\frac{x<em>1 + x</em>2}{2}, \frac{y<em>1 + y</em>2}{2})$.

Transformations

• A transformation is an operation that moves or changes a geometric figure in some way to produce a new figure. The new figure is called the image. Another name for the original image is the preimage.
• A translation moves every point of a figure the same distance in the same direction. $(x, y) \rightarrow (x+a, y+b)$

Reflections

• Reflect over x-axis: $(x, y) \rightarrow (x, -y)$
• Reflect over y-axis: $(x, y) \rightarrow (-x, y)$
• Reflect over both axes: $(x, y) \rightarrow (-x, -y)$
• Reflect over line y=x: $(x, y) \rightarrow (y, x)$
• Reflect over line y=-x: $(x, y) \rightarrow (-y, -x)$

Rotations

• A rotation turns a figure about a fixed point, called the center of rotation.
  • 90° rotation (counter-clockwise): $(x, y) \rightarrow (-y, x)$
  • 180° rotation: $(x, y) \rightarrow (-x, -y)$
  • 270° rotation (counter-clockwise): $(x, y) \rightarrow (y, -x)$
  • 360° rotation: $(x, y) \rightarrow (x, y)$

Equation of a Circle

• The standard equation of a circle with center $(h, k)$ and radius r is: $(x-h)^2 + (y-k)^2 = r^2$.

Constructions

• How to construct a Perpendicular Bisector.
• How to construct an Angle Bisector.
• How to construct a Line Parallel to a Given Line.
• How to copy an Angle.

Parallelograms

• If a quadrilateral is a parallelogram, then its opposite sides and its opposite angles are congruent.
• If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
• If a quadrilateral is a parallelogram, then its diagonals bisect each other.
• If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is a parallelogram.

Special Parallelograms

• A quadrilateral is a rectangle if and only if it has four right angles.
• A quadrilateral is a rhombus if and only if it has four congruent sides.
• A quadrilateral is a square if and only if it is a rhombus and a rectangle.
• A parallelogram is a rectangle if and only if its diagonals are congruent.
• A parallelogram is a rhombus if and only if its diagonals are perpendicular.
• A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.

Trapezoids and Kites

• If a trapezoid is isosceles, then each pair of base angles is congruent.
• A trapezoid is isosceles if and only if its diagonals are congruent.
• The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases.
• If a quadrilateral is a kite, then its diagonals are perpendicular.
• If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

Right Triangle Trigonometry

• Pythagorean Theorem: $a^2 + b^2 = c^2$
  • In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
• Converse of the Pythagorean Theorem: If $c^2 = a^2 + b^2$, then it is a right triangle.
• Acute triangle: If c^2 < a^2 + b^2, then it is an acute triangle.
• Obtuse triangle: If c^2 > a^2 + b^2, then it is an obtuse triangle.

Special right triangles:

• 30-60-90 Triangle
  • hypotenuse = 2 * shorter leg
  • longer leg = shorter leg * $\sqrt{3}$
• 45-45-90 Triangle
  • hypotenuse = leg * $\sqrt{2}$

Trigonometry

• Is used to find the lengths of sides in a right triangle when Pythagorean Theorem or Special Right Triangles won't work.
  • Sine: $sin \theta = \frac{opposite}{hypotenuse}$ (SOH)
  • Cosine: $cos \theta = \frac{adjacent}{hypotenuse}$ (CAH)
  • Tangent: $tan \theta = \frac{opposite}{adjacent}$ (TOA)

Corresponding Parts

• In two congruent figures, all the parts of one figure are congruent to the corresponding parts of the other figure.
• When you write a congruence statement always list the corresponding vertices in the same order.

Congruent Triangles

*Third Angle Theorem: If two angles of two triangles are congruent then the third angles are also congruent.

Triangle Congruence Postulates/Theorems:

• SSS (Side-Side-Side)

• SAS (Side-Angle-Side)

• ASA (Angle-Side-Angle)

• AAS (Angle-Angle-Side)

• HL (Hypotenuse-Leg)

• Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent) after proving triangles are congruent, to prove that parts of the triangles are congruent.

Relationships Within Triangles

• Point of concurrency: the point where 3 or more lines intersect.
• Circumcenter (of a triangle): the point of concurrency of the perpendicular bisectors of a triangle.
  • The circumcenter of a triangle is equidistant from the vertices.
• Incenter: the point of concurrency of the angle bisectors.
  • The incenter of a triangle is equidistant from the sides.
• Centroid: the point of concurrency of the medians.
  • The centroid is at a point on each median two-thirds of the distance from the vertex to the midpoint of the opposite side.
• Orthocenter: the point of concurrency of the altitudes.
• Midsegment: the segment that connects the midpoints of two sides of a triangle.
  • The midsegment is $\frac{1}{2}$ the length of the 3rd side and is parallel to it.
• Perpendicular Bisector: If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints.
• Angle Bisector: If a point lies on the angle bisector of an angle, then it is equidistant from the sides of the angle.

Triangle Inequality

• The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
• The measure of the third side of a triangle must be less than the sum of the other two sides and greater than their difference.
• Longest side of a triangle is opposite the largest angle.
• Smallest side of a triangle is opposite the smallest angle.

Isosceles and Equilateral Triangles

• Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those two sides are congruent.
• In an isosceles triangle, the bisector of the vertex angle is the perpendicular bisector of the base.
• If a triangle is equilateral, then the triangle is equiangular.

Similarity

• Angle-Angle (AA) Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
• Side-Side-Side (SSS) Similarity Theorem: If the corresponding side lengths of two triangles are proportional, then the triangles are similar.
• Side-Angle-Side (SAS) Similarity Theorem: If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.
• If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides proportionally.
• The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle.
• The geometric mean of two positive numbers is the positive square root of their product.

Surface Area and Volume

Perimeter and Area

• Perimeter and Circumference:
  • $C = \pi d = 2 \pi r$
  • $P = 2l + 2w$
  • $P = 4s$
• Area:
  • Triangle: $A = \frac{1}{2}bh$
  • Parallelogram: $A = bh$
  • Circle: $A = \pi r^2$

3D Shapes

*Prism
* $L.A. = ph$
* $S.A. = L.A. + 2B$
* $V = Bh$
*Cylinder
* $L.A. = 2 \pi rh$
* $S.A. = 2 \pi rh + 2B$
* $V = \pi r^2 h$
*Cone
* $L.A. = \pi rl$
* $S.A. = \pi rl + B$
* $V = \frac{1}{3} \pi r^2 h$
*Pyramid
* $L.A. = \frac{1}{2} pl$
* $S.A. = L.A. + B$
* $V = \frac{1}{3}Bh$
*Rectangular Prism
* $L.A. = 2wh + 2lh$
* $S.A. = 2wh + 2lh + 2lw$
* $V = lwh$
*Sphere
* $S.A. = 4 \pi r^2$
* $V = \frac{4}{3} \pi r^3$

Circle Definitions

• TRS is a semicircle. $m\stackrel{\frown}{TRS} = 180$
• RS is a minor arc. $m\stackrel{\frown}{RS} = m \angle RPS$
• RTS is a major arc. $m\stackrel{\frown}{RTS} = 360 - m\stackrel{\frown}{RS}$
• A chord is a segment whose endpoints are on a circle.
• A diameter is a chord that contains the center of the circle.
• A secant is a line that intersects a circle in two points.
• A tangent is a line in the plane of a circle that intersects the circle in exactly one point, the point of tangency.
• An Inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle.

Intersecting Lines

Inscribed Angle

• \angle B : $m \angle B = \frac{1}{2} m \stackrel{\frown}{AC}$
• Tangent-Chord: $m \angle B = \frac{1}{2} m \stackrel{\frown}{AC}$
• Tangent-Radius: $\stackrel{\frown}{AB} \perp \stackrel{\frown}{OP}$

Intersecting Lines and Angle Measures

*Inside the Circle:
* $m \angle 1 = \frac{1}{2}(x + y)$
*Outside the Circle:
* $m \angle 1 = \frac{1}{2}(x - y)$

Segment Lengths

• For a given point and circle, the product of the lengths of the two segments from the point to the circle is constant along any line through the point and circle.
  • Segments of Chords: $a * b = c * d$
  • Segments of Secants: $(w + x)w = (y + z)y$
  • Segments of Tangent-Secant: $(y + z)y = t^2$