Geometry End of the Year Study Guide

Comprehensive vocabulary and foundational definitions for high school geometry, covering reasoning, angle relationships, triangle properties, similarity, and basic trigonometry.

Inductive reasoning

The process of making a conjecture (guess) based on the observation of patterns.

Deductive reasoning

The process of proving a statement based on facts such as definitions, theorems, or postulates.

Counterexample

A specific example that proves a statement or conjecture to be false.

Undefined terms

The three basic elements of geometry: point, line, and plane.

Collinear

Points that lie on the same line.

Coplanar

Points or lines that lie in the same plane.

Skew lines

Non-coplanar lines that never intersect.

Postulate

A statement that is assumed to be true, also known as an axiom.

Theorem

A geometric statement that must be proven true.

Hypothesis

The "if" part of a conditional statement, represented by the letter $p$.

Conclusion

The "then" part of a conditional statement, represented by the letter $q$.

Converse

A statement formed by switching the hypothesis and the conclusion of a conditional statement.

Inverse

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

Contrapositive

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

Biconditional

A statement that combines a conditional and its converse into one using the phrase "if and only if."

Angle Bisector

Any figure that divides an angle into two congruent angles.

Midpoint of a Segment

A point that divides a 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 $\text{AOC}$, then $m\angle AOB + m\angle BOC = m\angle AOC$.

Adjacent angles

Two angles that are located next to each other.

Vertical angles

Two angles that are across from each other at an intersection and are equal in measure.

Linear pair

A pair of adjacent angles whose sum is $180$ degrees.

Complementary Angles

Two angles whose measures have a sum of $90$ degrees.

Supplementary Angles

Two angles whose measures have a sum of $180$ degrees.

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 $b$.

Corresponding Angles

Angles in the same relative position when a transversal intersects parallel lines; they are congruent.

Consecutive Interior Angles

Interior angles on the same side of the transversal; they are supplementary (sum to $180$ degrees).

Triangle angle sum

The sum of the measures of the angles in any triangle is $180$ degrees.

Triangle exterior angles

The measure of each exterior angle is equal to the sum of the two remote interior angles.

Polygon angle sum

For a polygon with $n$ sides, the interior angles add up to $(n - 2) \times 180$.

Measure of a single interior angle

In a regular polygon with $n$ sides, the measure is calculated as $\frac{(n - 2) \times 180}{n}$.

Sum of exterior angles

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

Third Angle Theorem

If two angles of two triangles are congruent, then the third angles are also congruent.

CPCTC

An abbreviation for "Corresponding Parts of Congruent Triangles are Congruent," used after proving triangles are congruent.

Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Circumcenter

The point of concurrency of the perpendicular bisectors of a triangle; it is equidistant from the vertices.

Incenter

The point of concurrency of the angle bisectors of a triangle; it is equidistant from the sides.

Centroid

The point of concurrency of the medians of a triangle, located two-thirds of the distance from the vertex to the midpoint of the opposite side.

Orthocenter

The point of concurrency of the altitudes of a triangle.

Midsegment of a triangle

A segment connecting the midpoints of two sides; it is half the length of the third side and parallel to it.

Triangle Inequality

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Geometric mean

The positive square root of the product of two positive numbers.

Pythagorean Theorem

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs: $a^2 + b^2 = c^2$.

Sine ($\sin$)

Defined as $\frac{\text{opposite}}{\text{hypotenuse}}$ (SOH).

Cosine ($\cos$)

Defined as $\frac{\text{adjacent}}{\text{hypotenuse}}$ (CAH).

Tangent ($\tan$)

Defined as $\frac{\text{opposite}}{\text{adjacent}}$ (TOA).

Unit rate

Describes how many units of one quantity correspond to exactly one unit of a second quantity.

Slope formula

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

Distance Formula

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Midpoint Formula

$$m = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$