Geometry Exam Study Guide

Logic

• Conditional Statement: If P then Q
• Converse: Change the order of the conditional statement.
• Inverse: Negate (take the opposite) of the conditional statement.
• Contrapositive: Change the order and negate the conditional statement.
• Biconditional: "If and only if" (represented by double-sided arrows)

Symbols

• P: Hypothesis
• Q: Conclusion
• $P \rightarrow Q$: If P then Q
• $\sim$: Not / Negation
• $\wedge$: And
• $\vee$: Or
• $\therefore$: Therefore

Laws

• Law of Detachment: If $P \rightarrow Q$ and P is true, then Q is true.
• Law of Syllogism: If $P \rightarrow Q$ and $Q \rightarrow R$, then $P \rightarrow R$.
• The contrapositive is true only when the original statement is true.
• Converse and inverse have the same truth value.

Examples

• Example 1
  • If I practice my brain dump, then I will remember my formulas.
  • I practice my formulas.
  • Conclusion: I will remember my formulas.
• Example 2
  • If I do my practice questions, then I'll be prepared for my SOL.
  • If I'm prepared for my SOL, then I'll pass.
  • Conclusion: If I do my practice questions, then I'll pass.
  • Or: If I didn't pass, then I did not do my practice questions (contrapositive).

Venn Diagrams

• All: Smaller shape inside a larger shape.
• Some: Overlap between shapes.
• None: No overlap between shapes.

Tips for Complex Venn Diagrams

1. Label each section of the Venn diagram with a letter corresponding to the category (e.g., F for French, S for Spanish, G for German).
2. Use these labels to answer specific questions.
3. To find the total, add up all the numbers in the Venn diagram, including those outside any circles.

Formulas and Transformations

• $\sim R \rightarrow \sim P$
• Students taking French: Add up every section labeled with "F."
• Students taking both Spanish and German: Add up sections with "S" and "G."
• Students only taking German: Look for sections with only "G."

Slopes

• Parallel Lines: Equal slopes
• Perpendicular Lines: Negative reciprocals (flip the fraction, switch the sign). Their product is -1.
  • If $m<em>1$ and $m</em>2$ are perpendicular slopes, then $m<em>1 \times m</em>2 = -1$
• Slope: Rise over Run
• Distance: $\sqrt{rise^2 + run^2}$
• Midpoint: Rise divided by 2 and Run divided by 2 to find the midpoint between the given points

Lines

• Vertical Line: Undefined slope, equation is $x = number$
• Horizontal Line: Slope of zero, equation is $y = number$

Transformations

• Translation: Slide
• Reflection: Flip/fold
• Rotation: Spin/turn

Symmetry

• Line Symmetry: Fold across a line and it's the same.
• Rotation Symmetry: Rotate it and it looks the same.
• Point Symmetry: Rotation symmetry of 180 degrees.

Parallel Lines Cut by a Transversal

• Vertical Angles: Congruent (across from each other, like an X)
• Alternate Interior Angles: Congruent (on different sides of the transversal, inside the parallel lines)
• Alternate Exterior Angles: Congruent (on different sides of the transversal, outside the parallel lines)
• Corresponding Angles: Congruent

Tips for Identifying Angle Relationships

1. Put an O at each obtuse angle and an A at each acute angle.
2. Same letters are congruent.
3. Different letters add up to 180 (supplementary).

• Linear Pairs: Supplementary
• Consecutive Interior Angles: Supplementary

Proving Lines are Parallel

• Check if angle relationships verify (congruent or supplementary as expected).
• Parallelograms only have point symmetry.
• Rhombuses have line and point symmetry.

Triangle Basics

• Triangle angles add up to 180 degrees.
• Exterior angle = sum of the two remote interior angles.

Constructions

• Copy a Line Segment: Two lines look the same.
• Perpendicular Bisector: Line going through two X's (looks like a football).
• Perpendicular Through a Point Not on the Line: Point A is not on the line.
• Perpendicular Through a Point on the Line: Point A is on the line.
• Angle Bisector: Arc with an X in the middle of the angle.
• Copying an Angle: Two angles, or looks like there will be two angles.

Congruent and Similar Triangles

• Corresponding angles are congruent.
• Corresponding sides are congruent.
• Order matters in congruence statements.

Ways to Prove Triangles Congruent

• Side-Side-Side (SSS): All three sides are equal.
• Side-Angle-Side (SAS): Two sides and the included angle are congruent.
• Angle-Side-Angle (ASA): One side and two adjacent angles are congruent.
• Angle-Angle-Side (AAS): Two angles and a non-included side are congruent. The side is NOT between the two angles.
• Hypotenuse-Leg (HL): Applicable only to right triangles where the hypotenuse and one leg are congruent. This does NOT count as Angle-Side-Side.

Similar Triangles

• Proving similarity involves showing that corresponding sides are proportional.
• Set up proportions and use the "butterfly" method (cross-multiplication) to solve.

Triangle Inequality Theorem

• The sum of the two smaller sides must be greater than the largest side for a triangle to be valid.
• If the sum is equal to or less than the largest side, it is not a triangle.

Triangle Sides and Angles

• The largest angle is opposite the longest side.
• The smallest angle is opposite the shortest side.
• In an isosceles triangle, the base angles are congruent.
• If angles are congruent, the sides opposite those angles are congruent.
• Vertical angles are congruent.

Finding Possible Lengths for the Third Side of a Triangle

1. Subtract the two given side lengths.
2. Add the two given side lengths.
3. The third side must be greater than the difference and less than the sum:
   • a - b < x < a + b

Right Triangles

• Pythagorean Theorem:
  • $a^2 + b^2 = c^2$
  • (given on the formula sheet)
  • If $a^2 + b^2 = c^2$, it is a right triangle.
  • If a^2 + b^2 > c^2, it is an acute triangle.
  • If a^2 + b^2 < c^2, it is an obtuse triangle.
• Sohcahtoa:
  • Sine = Opposite / Hypotenuse
  • Cosine = Adjacent / Hypotenuse
  • Tangent = Opposite / Adjacent
    The placement of opposite and adjacent sides depends on the angle's position.
    To find an angle measure given two sides, use inverse trig functions (arcsin, arccos, arctan) and ensure your calculator is in degree mode.

Quadrilaterals

Parallelograms

• Opposite sides are parallel.
• Opposite sides are congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary (add up to 180 degrees).
• Diagonals bisect each other.

Rectangle

• A parallelogram with all angles equal to 90 degrees.
• Diagonals are congruent.

Rhombus

• A parallelogram with all four sides equal.
• Diagonals bisect each other.
• Diagonals intersect at a 90-degree angle.

Square

• Has all the properties of parallelograms, rectangles, and rhombuses (all sides equal, diagonals equal, 90-degree angles).

Kite

• Two pairs of adjacent sides are congruent.
• Opposite sides are not congruent.

Trapezoid

• Only the bases are parallel.

Isosceles Trapezoid

• Legs (non-parallel sides) are congruent.
  Diagonals are congruent.

Polygons

• Sum of Interior Angles: $(n - 2) \times 180$ (where n is the number of sides)
• Each Angle in a Regular Polygon: $\frac{(n - 2) \times 180}{n}$
• Sum of Exterior Angles: 360 degrees
• Each Exterior Angle in a Regular Polygon: $\frac{360}{n}$
• Interior and exterior angles next to each other are supplementary (add up to 180 degrees).

Circles

Arcs and Angles

• Central Angle: Angle at the center of the circle equals the measure of the intercepted arc (Pac-Man Formula):
  • $angle = arc$
• Inscribed Angle: Angle on the circle equals half the measure of the intercepted arc (Hungry Pac-Man Formula):
  • $angle = \frac{1}{2} \times arc$
• Angle Inside the Circle: (Super Smash Brothers Formula):
  • $angle = \frac{(arc<em>1 + arc</em>2)}{2}$
• Angle Outside the Circle: (Slingshot Formula):
  • $angle = \frac{(arc<em>1 - arc</em>2)}{2}$

Line Segment Formulas

• Two Chords:
  • $part \times part = part \times part$ or $A \times B = C \times D$
• Two Secants:
  • $outside \times whole = outside \times whole$
• Tangent and Secant:
  • $outside \times whole = tangent^2$

Other Circle Theorems

• Congruent chords have congruent arcs.
• A diameter perpendicular to a chord bisects the chord and the arc.
• Congruent chords are equidistant from the center.
• Tangents from the same point are congruent.
• A tangent is perpendicular to a radius at the point of tangency.

Circle Formulas

• Arc Length:
  • $\frac{angle}{360} \times circumference = \frac{angle}{360} \times 2 \pi r$
• Area of a Sector:
  • $\frac{angle}{360} \times area = \frac{angle}{360} \times \pi r^2$

3D Shapes

• Lateral Area: Area of the sides, not including the bases.
• Surface Area: Total area, including all faces and bases.
• Volume: Amount of space inside the 3D shape.

Similar Shapes

• Scale Factor: a : b
• Ratio of Perimeters/Lengths: a : b
• Ratio of Areas: $a^2 : b^2$
• Ratio of Volumes: $a^3 : b^3$