Comprehensive Study Guide for Geometry, Trigonometry, and Triangle Relationships

Special Right Triangles

The $45^\circ-45^\circ-90^\circ$ Triangle: - This is an isosceles right triangle where the two legs are congruent. - The relationship between the legs and the hypotenuse is defined by the formula: $\text{hypotenuse} = \text{leg} \times \sqrt{2}$ . - If the leg length is denoted as $x$ , the sides follow the ratio $x : x : x\sqrt{2}$ .
The $30^\circ-60^\circ-90^\circ$ Triangle: - This triangle features three distinct side lengths relative to the angles. - The shorter leg (opposite the $30^\circ$ angle) is denoted as $x$ . - The longer leg (opposite the $60^\circ$ angle) is calculated as $x\sqrt{3}$ . - The hypotenuse (opposite the $90^\circ$ angle) is always twice the length of the shorter leg, or $2x$ . - The ratio of the sides is $x : x\sqrt{3} : 2x$ .

Probability

Basic Probability Concepts: - Probability measures the likelihood of an event occurring, ranging from $0$ (impossible) to $1$ (certain). - Theoretical Probability is calculated using the formula: $P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}$ .
Sample Space and Events: - The sample space is the set of all possible outcomes of an experiment. - Complementary events are events where the probability of an event happening plus the probability of it not happening equals $1$ . $P(A) + P(A^c) = 1$ .

Circles: Tangents, Curves, and Inscribed Angles

Tangent Lines: - A tangent is a line in the plane of a circle that intersects the circle at exactly one point, known as the point of tangency. - Tangent Theorem: A line is tangent to a circle if and only if it is perpendicular to the radius drawn to the point of tangency. This creates a $90^\circ$ angle at the intersection point.
Curves and Arcs: - An arc is a portion of the circumference of a circle. - Minor Arcs: Arcs with a measure less than $180^\circ$ . - Major Arcs: Arcs with a measure greater than $180^\circ$ . - Semicircles: Arcs with a measure exactly equal to $180^\circ$ .
Inscribed Angles: - An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. - Inscribed Angle Theorem: The measure of an inscribed angle is exactly half the measure of its intercepted arc. $\text{Inscribed Angle} = \frac{1}{2} \times \text{Measure of Intercepted Arc}$ . - If two inscribed angles intercept the same arc, then the angles are congruent.

Trigonometry and the Pythagorean Theorem

The Pythagorean Theorem: - Applicable only to right triangles, stating that the square of the hypotenuse is equal to the sum of the squares of the legs. - Formula: $a^2 + b^2 = c^2$ , where $c$ is the hypotenuse and $a$ and $b$ are the legs.
Trigonometric Ratios: - These ratios relate the angles of a right triangle to the lengths of its sides. - Sine (sin): $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ - Cosine (cos): $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ - Tangent (tan): $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
Finding a Missing Side: - If an angle and one side length are known, use the appropriate trigonometric ratio and solve for the unknown variable.
Finding a Missing Angle: - If two sides are known, use inverse trigonometric functions ( $\sin^{-1}$ , $\cos^{-1}$ , or $\tan^{-1}$ ) to calculate the measure of the angle.
Angles of Evolution and Depression: - Angle of Evolution: The angle formed by a horizontal line and a line of sight to a point above the horizontal line. - Angle of Depression: The angle formed by a horizontal line and a line of sight to a point below the horizontal line. - These two angles are congruent when they involve the same two points of interest (due to alternating interior angles between parallel horizontal lines).

Similar Figures and Transformations

Dilation: - A transformation that produces an image that is the same shape as the original but a different size. - It is described by a center of dilation and a scale factor $k$ . - If $k > 1$ , the figure is an enlargement. If $0 < k < 1$ , the figure is a reduction.
Similarity Transformation: - A transformation (or a sequence of transformations) that includes a dilation. The resulting figures are similar, meaning they have congruent corresponding angles and proportional corresponding sides.
Proven Triangle Similarity: - AA (Angle-Angle) Similarity: If two angles of one triangle are congruent to two angles of another, the triangles are similar. - SSS (Side-Side-Side) Similarity: If the corresponding side lengths of two triangles are proportional, the triangles are similar. - SAS (Side-Angle-Side) Similarity: If an angle of one triangle is congruent to an angle of another, and the sides including those angles are proportional, the triangles are similar.
Similar Perforations and Proportions: - Similar figures imply that all corresponding lengths reflect the same scale factor ratio. This is expressed as $\frac{a}{b} = \frac{c}{d}$ .

Quadrilaterals and Other Polygons

Properties of Parallelograms: - Opposite sides are parallel and congruent. - Opposite angles are congruent. - Consecutive angles are supplementary ( $\text{Sum} = 180^\circ$ ). - Diagonals bisect each other.
Parallelogram Proof: - To prove a quadrilateral is a parallelogram, one must show one of the following: both pairs of opposite sides are parallel; both pairs of opposite sides are congruent; one pair of opposite sides is both parallel and congruent; or diagonals bisect each other.
Properties of Special Parallelograms: - Rectangles: A parallelogram with four right angles and congruent diagonals. - Rhombuses: A parallelogram with four congruent sides and perpendicular diagonals that bisect opposite angles. - Squares: A parallelogram that is both a rectangle and a rhombus (four right angles and four congruent sides).
Kites: - A quadrilateral with two pairs of consecutive congruent sides. - Diagonals are perpendicular. - Exactly one pair of opposite angles is congruent.
Trapezoids: - A quadrilateral with exactly one pair of parallel sides (the bases). - Isosceles Trapezoid: A trapezoid where the legs are congruent, base angles are congruent, and diagonals are congruent.
Parallelograms on the Coordinate Grid: - Used to verify properties using coordinates. - Distance Formula: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ to check side congruence. - Slope Formula: $m = \frac{y_2-y_1}{x_2-x_1}$ to check for parallel (same slope) or perpendicular (negative reciprocal) lines. - Midpoint Formula: $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$ to check if diagonals bisect each other.

Relationships in Triangles

Perpendicular and Angle Bisectors: - Perpendicular Bisector Theorem: Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. The intersection of all three perpendicular bisectors is called the Circumcenter. - Angle Bisector Theorem: Any point on the bisector of an angle is equidistant from the sides of the angle. The intersection of all three angle bisectors is called the Incenter.
Medians and Altitudes: - Medians: A segment from a vertex to the midpoint of the opposite side. The intersection point is the Centroid, which acts as the center of gravity and is located two-thirds of the distance from each vertex to the midpoint of the opposite side. - Altitudes: A perpendicular segment from a vertex to the opposite side. The intersection point is the Orthocenter.
Inequalities in Triangles: - Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side ( $a + b > c$ ). - Angle-Side Relationship: In a triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.