geometry

Conditional Statement

A statement of the form 'If p, then q' where p is the hypothesis and q is the conclusion.

Hypothesis

The part of a conditional statement that follows 'If'.

Conclusion

The part of a conditional statement that follows 'then'.

Converse

The statement formed by reversing the hypothesis and conclusion of a conditional statement.

Inverse

The statement formed by negating both the hypothesis and conclusion of a conditional statement.

Contrapositive

The statement formed by negating and reversing the hypothesis and conclusion of a conditional statement.

Biconditional Statement

A statement in the form 'p if and only if q', true when both p and q are either true or false.

Truth Values

Indicates whether a statement is true (T) or false (F).

Inductive Reasoning

Reasoning that involves making generalizations based on specific evidence.

Deductive Reasoning

Reasoning that uses known facts, definitions, and theorems to draw conclusions.

Counterexample

A specific case that proves a statement false.

Law of Detachment

If 'p implies q' is true and p is true, then q must also be true.

Law of Syllogism

If 'p implies q' and 'q implies r' are both true, then 'p implies r' is also true.

Corresponding Angles

Angles that are in the same position on parallel lines when crossed by a transversal.

Alternate Interior Angles

Angles that are on opposite sides of the transversal and inside the two parallel lines.

Alternate Exterior Angles

Angles that are on opposite sides of the transversal and outside the two parallel lines.

Same-Side Interior Angles

Angles that are on the same side of the transversal and inside the parallel lines.

Vertical Angles

Angles opposite each other when two lines cross; they are congruent.

Linear Pairs

Two adjacent angles that form a straight line; they are supplementary.

Complementary Angles

Two angles that sum to 90 degrees.

Supplementary Angles

Two angles that sum to 180 degrees.

Solving for x in Angle Equations

Set the angle equations equal to each other or to known values to find x.

Identifying Parallel Lines from Angle Relationships

If pairs of corresponding, alternate interior, or alternate exterior angles are congruent, the lines are parallel.

Congruent Angles

Angles that have the same measure.

Supplementary Angle Pairs

Angle pairs that sum to 180 degrees.

Slope Formula

$m = \frac{y_2 - y_1}{x_2 - x_1}$.

Midpoint Formula

Midpoint M is given by $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$.

Distance Formula

Distance d between two points is given by $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Parallel Slopes

Lines are parallel if they have the same slope (m).

Perpendicular Slopes

Lines are perpendicular if the product of their slopes is -1.

Graphing Lines

Use the slope-intercept form or point-slope form to graph lines.

Equation of a Line (Slope-Intercept Form)

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

Equation of a Line (Point-Slope Form)

$y - y_1 = m(x - x_1)$ where (x_1, y_1) is a point on the line.

Triangle Angle Sum Theorem

The sum of the angles in a triangle is 180 degrees.

Exterior Angle Theorem

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

Isosceles Triangle Theorem

In an isosceles triangle, the angles opposite the equal sides are equal.

Equilateral Triangles

All sides and angles are equal (each angle is 60 degrees).

Scalene Triangles

All sides and angles are different.

Acute Triangle

All angles are less than 90 degrees.

Right Triangle

One angle is exactly 90 degrees.

Obtuse Triangle

One angle is greater than 90 degrees.

Inequalities in Triangles

The longest side is opposite the largest angle; the shortest side is opposite the smallest angle.

Triangle Congruence (SSS)

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

Triangle Congruence (SAS)

If two sides and the included angle of one triangle are equal to the corresponding parts of another triangle, they are congruent.

Triangle Congruence (ASA)

If two angles and the included side of one triangle are equal to the corresponding parts of another triangle, they are congruent.

Triangle Congruence (AAS)

If two angles and a non-included side of one triangle are equal to the corresponding parts of another triangle, they are congruent.

Triangle Congruence (HL)

If the hypotenuse and one leg are equal in two right triangles, they are congruent.

CPCTC

Corresponding Parts of Congruent Triangles are Congruent.

Two-Column Proofs

Proofs that have statements and reasons in two columns.

Flow Proofs

Proofs that use diagrams to show the logical connections.

Methods that DO Prove Congruence

SSS, SAS, ASA, AAS, HL.

Methods that DO NOT Prove Congruence

AAA, SSA.

AA Similarity

If two angles of one triangle are equal to two angles of another triangle, they are similar.

SAS Similarity

If two sides of one triangle are in proportion to two sides of another triangle, and their included angles are equal, they are similar.

SSS Similarity

If three sides of one triangle are in proportion to three sides of another triangle, they are similar.

Proportions

A statement that two ratios are equal.

Corresponding Sides

Sides that are in the same relative position in similar figures.

Scale Factors

The ratio of the lengths of corresponding sides of two similar figures.

Perimeter Scale Factors

Perimeter scales by the same factor as the side lengths.

Area Scale Factors

Area scales by the square of the scale factor (x²).

Right Triangle Hypotenuse Identification

The hypotenuse is the longest side opposite the right angle.

Pythagorean Theorem

In a right triangle: $a^2 + b^2 = c^2$ where c is the hypotenuse.

Finding Missing Sides

Use the Pythagorean theorem or trigonometric ratios to find missing sides.

45-45-90 Triangle Ratios

The sides are in the ratio 1:1:$\sqrt{2}$.

30-60-90 Triangle Ratios

The sides are in the ratio 1:$\sqrt{3}$:2.

Geometric Mean

The positive square root of the product of two numbers.

Altitude to Hypotenuse

The altitude divides the triangle into two smaller triangles that are similar to the original triangle.

Trigonometric Ratios (Sine, Cosine, Tangent)

Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

SOHCAHTOA

A mnemonic for remembering sine, cosine, and tangent: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Special Triangle Side Ratios

In a 45-45-90 triangle: 1:1:$\sqrt{2}$; in a 30-60-90 triangle: 1:$\sqrt{3}$:2.

Trigonometric Ratios Summary

Sine, Cosine, and Tangent ratios for angles.

Parallelogram Properties

Opposite sides are equal; opposite angles are equal; adjacent angles are supplementary.

Rectangle Properties

All angles are right angles; diagonals are equal.

Rhombus Properties

All sides are equal; diagonals bisect each other at right angles.

Square Properties

All sides and angles are equal; diagonals are equal and bisect each other at right angles.

Trapezoid Properties

At least one pair of parallel sides.

Isosceles Trapezoid Properties

Base angles are equal, legs are equal in length.

Kite Properties

Two pairs of adjacent sides are equal; one pair of opposite angles are equal.

Diagonal Properties of Quadrilaterals

The properties of diagonals vary among different quadrilaterals.

Comparing Quadrilateral Properties

Identify properties to determine which quadrilateral has certain attributes (e.g., 'Which quadrilateral has perpendicular diagonals?').

Interior Angle Sum (Polygons)

The sum of the interior angles of a polygon is $(n-2) \times 180$, where n is the number of sides.

Exterior Angle Sum (Polygons)

The sum of the exterior angles of any polygon is always 360 degrees.

Finding Missing Angles in Polygons

Use the interior or exterior angle sum to find missing angles.

Identifying Polygon Names

A polygon is named based on the number of sides (e.g., triangle, quadrilateral, pentagon).

Circle Central Angle

An angle whose vertex is at the center of the circle.

Inscribed Angle

An angle whose vertex is on the circle and whose sides are chords.

Arcs

A portion of the circumference of a circle.

Chords

A segment whose endpoints lie on the circle.

Tangents to Circles

A line that touches the circle at exactly one point.

Radius and Tangent Relationship

A radius drawn to the point of tangency is perpendicular to the tangent line.

Arc Length Formula

$L = r\theta$, where r is the radius and $\theta$ is in radians.

Circumference Formula

$C = 2\pi r$ or $C = \pi d$.

Sector Area Formula

$A = \frac{1}{2} r^2 \theta$, where $\theta$ is in radians.

Inscribed Angle Theorem

The inscribed angle is half the measure of the central angle that subtends the same arc.

Tangent Properties

The tangent line is perpendicular to the radius at the point of tangency.

Intercepted Arcs

The arc that lies between the lines that intersect the circle.

Standard Form of Circle Equation

$(x-h)^2 + (y-k)^2 = r^2$, where (h,k) is the center and r is the radius.

Identifying Circle Center

The center is the point (h, k) in the standard form of the circle.

Identifying Circle Radius

The radius is the distance from the center to any point on the circle.

Graphing Circles

Use the center and radius from the standard form to plot the circle.