Geometry EOC

Geometry CHEAT Sheet

Basic Geometric Shapes

Point:
- A point is an exact location. It has no size, only position.
- Notation: Represented by a dot and labeled with a capital letter (e.g., $A$ ).
Line:
- A line is a set of points in a straight path that extends infinitely in two directions.
- Notation: Denoted by $\overleftrightarrow{AB}$ , where $A$ and $B$ are points on the line.
Line Segment:
- A line segment is a part of a line with two endpoints.
- Notation: Denoted by $\overline{AB}$ , where $A$ and $B$ are the endpoints.
Ray:
- A ray is a part of a line that starts at one endpoint and extends infinitely in one direction.
- Notation: Denoted by $\overrightarrow{AB}$ , where $A$ is the endpoint and the ray extends through point $B$ .
Angle:
- An angle is formed by two rays that share a common endpoint (vertex).
- Notation: Denoted by $\angle ABC$ , where $B$ is the vertex.

Types of Angles

Acute Angle:
- An angle that measures greater than $0^\circ$ and less than $90^\circ$ . (0^\circ < \theta < 90^\circ)
Right Angle:
- An angle that measures exactly $90^\circ$ . $\theta = 90^\circ$
Obtuse Angle:
- An angle that measures greater than $90^\circ$ and less than $180^\circ$ . (90^\circ < \theta < 180^\circ)
Straight Angle:
- An angle that measures exactly $180^\circ$ . $\theta = 180^\circ$
Reflex Angle:
- An angle that measures greater than $180^\circ$ and less than $360^\circ$ . (180^\circ < \theta < 360^\circ)

Angle Relationships

Complementary Angles:
- Two angles are complementary if their measures add up to $90^\circ$ .
- If $\angle A$ and $\angle B$ are complementary, then $m\angle A + m\angle B = 90^\circ$ .
Supplementary Angles:
- Two angles are supplementary if their measures add up to $180^\circ$ .
- If $\angle A$ and $\angle B$ are supplementary, then $m\angle A + m\angle B = 180^\circ$ .
Vertical Angles:
- Vertical angles are pairs of opposite angles formed by the intersection of two lines. Vertical angles are congruent (equal in measure).
- If lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ intersect at point $E$ , then $\angle AEC \cong \angle DEB$ and $\angle AED \cong \angle CEB$ .
Adjacent Angles:
- Adjacent angles are two angles that share a common vertex and a common side but do not overlap.
- $\angle AXB$ and $\angle BXC$ are adjacent angles because they share vertex $X$ and side $\overrightarrow{XB}$ .
Linear Pair:
- A linear pair is a pair of adjacent angles that are supplementary. They form a straight line.
- If $\angle AXB$ and $\angle BXC$ form a linear pair, then $m\angle AXB + m\angle BXC = 180^\circ$ .

Lines

Parallel Lines:
- Parallel lines are lines in a plane that do not intersect. The distance between them is always the same.
- Notation: $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$
Perpendicular Lines:
- Perpendicular lines are lines that intersect at a right angle ( $90^\circ$ ).
- Notation: $\overleftrightarrow{AB} \perp \overleftrightarrow{CD}$
Transversal:
- A transversal is a line that intersects two or more other lines.
- When a transversal intersects two parallel lines, several angle relationships are formed:
  1. Corresponding Angles:
    - Corresponding angles are angles that are in the same position relative to the transversal and the intersected lines. Corresponding angles are congruent.
  2. Alternate Interior Angles:
    - Alternate interior angles are angles on opposite sides of the transversal and inside the two lines. Alternate interior angles are congruent.
  3. Alternate Exterior Angles:
    - Alternate exterior angles are angles on opposite sides of the transversal and outside the two lines. Alternate exterior angles are congruent.
  4. Same-Side Interior Angles:
    - Same-side interior angles (also called consecutive interior angles) are angles on the same side of the transversal and inside the two lines. Same-side interior angles are supplementary (add up to $180^\circ$ ).

Triangles

Definition:
- A triangle is a polygon with three sides and three angles. The sum of the angles in a triangle is always $180^\circ$ .
Types of Triangles based on Sides:
1. Equilateral Triangle:
  - All three sides are equal in length. All three angles are equal to $60^\circ$ .
2. Isosceles Triangle:
  - Two sides are equal in length. The angles opposite the equal sides are also equal.
3. Scalene Triangle:
  - All three sides have different lengths. All three angles have different measures.
Types of Triangles based on Angles:
1. Acute Triangle:
  - All three angles are acute (less than $90^\circ$ ).
2. Right Triangle:
  - One angle is a right angle ( $90^\circ$ ).
3. Obtuse Triangle:
  - One angle is obtuse (greater than $90^\circ$ ).
Triangle Angle Sum Theorem:
- The sum of the interior angles in any triangle is $180^\circ$ . If the angles are $A$ , $B$ , and $C$ , then $m\angle A + m\angle B + m\angle C = 180^\circ$ .
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. If the sides are $a$ , $b$ , and $c$ , then:
  - a + b > c
  - a + c > b
  - b + c > a

Congruence

Corresponding Parts of Congruent Triangles are Congruent (CPCTC):
- If two triangles are congruent, then their corresponding sides and corresponding angles are congruent.
Triangle Congruence Postulates and Theorems:
1. Side-Side-Side (SSS):
  - If all three sides of one triangle are congruent to the corresponding three sides of another triangle, then the two triangles are congruent.
2. Side-Angle-Side (SAS):
  - If two sides and the включённый angle between them in one triangle are congruent to the corresponding two sides and included angle in another triangle, then the two triangles are congruent.
3. Angle-Side-Angle (ASA):
  - If two angles and the включённый side between them in one triangle are congruent to the corresponding two angles and included side in another triangle, then the two triangles are congruent.
4. Angle-Angle-Side (AAS):
  - If two angles and a non-included side in one triangle are congruent to the corresponding two angles and non-included side in another triangle, then the two triangles are congruent.
5. Hypotenuse-Leg (HL):
  - If the hypotenuse and a leg of one right triangle are congruent to the corresponding hypotenuse and leg of another right triangle, then the two triangles are congruent.

Quadrilaterals

Definition:
- A quadrilateral is a polygon with four sides and four angles. The sum of the angles in a quadrilateral is $360^\circ$ .
Types of Quadrilaterals:
1. Square:
  - A quadrilateral with four equal sides and four right angles.
2. Rectangle:
  - A quadrilateral with four right angles. Opposite sides are equal in length.
3. Parallelogram:
  - A quadrilateral with two pairs of parallel sides. Opposite sides are equal in length, and opposite angles are equal.
4. Rhombus:
  - A quadrilateral with four equal sides. Opposite angles are equal.
5. Trapezoid:
  - A quadrilateral with at least one pair of parallel sides.
  1. Isosceles Trapezoid:
    - A trapezoid in which the non-parallel sides (legs) are congruent.
Angle Sum Theorem for Quadrilaterals:
- The sum of the interior angles in any quadrilateral is $360^\circ$ . If the angles are $A$ , $B$ , $C$ , and $D$ , then $m\angle A + m\angle B + m\angle C + m\angle D = 360^\circ$ .

Circles

Definition:
- A circle is a set of all points in a plane that are at a fixed distance (radius) from a fixed point (center).
Parts of a Circle:
1. Center:
  - The central point from which all points on the circle are equidistant.
2. Radius (r):
  - The distance from the center of the circle to any point on the circle.
3. Diameter (d):
  - The distance across the circle through the center. The diameter is twice the radius ( $d = 2r$ ).
4. Chord:
  - A line segment that connects two points on the circle.
5. Tangent:
  - A line that touches the circle at exactly one point.
6. Secant:
  - A line that intersects the circle at two points.
7. Arc:
  - A portion of the circumference of the circle.
  1. Minor Arc:
    - An arc that is less than 180 degrees.
  2. Major Arc:
    - An arc that is greater than 180 degrees.
  3. Semicircle Arc:
    - An arc that is exactly 180 degrees.
8. Central Angle:
  - An angle whose vertex is at the center of the circle.
9. Inscribed Angle:
  - An angle whose vertex is on the circle and whose sides are chords of the circle.
Circumference:
- The distance around the circle. The formula for the circumference $C$ is $C = 2\pi r$ or $C = \pi d$ , where $r$ is the radius and $d$ is the diameter.
Area:
- The amount of space inside the circle. The formula for the area $A$ is $A = \pi r^2$ , where $r$ is the radius.
Theorems:
1. Tangent Theorem:
- A tangent line is

Basic Geometric Shapes

Point:
- A point is an exact location. It has no size, only position.
- Notation: Represented by a dot and labeled with a capital letter (e.g., $A$ ).
Line:
- A line is a set of points in a straight path that extends infinitely in two directions.
- Notation: Denoted by $\overleftrightarrow{AB}$ , where $A$ and $B$ are points on the line.
Line Segment:
- A line segment is a part of a line with two endpoints.
- Notation: Denoted by $\overline{AB}$ , where $A$ and $B$ are the endpoints.
Ray:
- A ray is a part of a line that starts at one endpoint and extends infinitely in one direction.
- Notation: Denoted by $\overrightarrow{AB}$ , where $A$ is the endpoint and the ray extends through point $B$ .
Angle:
- An angle is formed by two rays that share a common endpoint (vertex).
- Notation: Denoted by $\angle ABC$ , where $B$ is the vertex.

Types of Angles

Acute Angle:
- An angle that measures greater than $0^\circ$ and less than $90^\circ$ . (0^\circ < \theta < 90^\circ)
Right Angle:
- An angle that measures exactly $90^\circ$ . $\theta = 90^\circ$
Obtuse Angle:
- An angle that measures greater than $90^\circ$ and less than $180^\circ$ . (90^\circ < \theta < 180^\circ)
Straight Angle:
- An angle that measures exactly $180^\circ$ . $\theta = 180^\circ$
Reflex Angle:
- An angle that measures greater than $180^\circ$ and less than $360^\circ$ . (180^\circ < \theta < 360^\circ)

Angle Relationships

Complementary Angles:
- Two angles are complementary if their measures add up to $90^\circ$ .
- If $\angle A$ and $\angle B$ are complementary, then $m\angle A + m\angle B = 90^\circ$ .
Supplementary Angles:
- Two angles are supplementary if their measures add up to $180^\circ$ .
- If $\angle A$ and $\angle B$ are supplementary, then $m\angle A + m\angle B = 180^\circ$ .
Vertical Angles:
- Vertical angles are pairs of opposite angles formed by the intersection of two lines. Vertical angles are congruent (equal in measure).
- If lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ intersect at point $E$ , then $\angle AEC \cong \angle DEB$ and $\angle AED \cong \angle CEB$ .
Adjacent Angles:
- Adjacent angles are two angles that share a common vertex and a common side but do not overlap.
- $\angle AXB$ and $\angle BXC$ are adjacent angles because they share vertex $X$ and side $\overrightarrow{XB}$ .
Linear Pair:
- A linear pair is a pair of adjacent angles that are supplementary. They form a straight line.
- If $\angle AXB$ and $\angle BXC$ form a linear pair, then $m\angle AXB + m\angle BXC = 180^\circ$ .

Lines

Parallel Lines:
- Parallel lines are lines in a plane that do not intersect. The distance between them is always the same.
- Notation: $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$
Perpendicular Lines:
- Perpendicular lines are lines that intersect at a right angle ( $90^\circ$ ).
- Notation: $\overleftrightarrow{AB} \perp \overleftrightarrow{CD}$
Transversal:
- A transversal is a line that intersects two or more other lines.
- When a transversal intersects two parallel lines, several angle relationships are formed:
  1. Corresponding Angles:
    - Corresponding angles are angles that are in the same position relative to the transversal and the intersected lines. Corresponding angles are congruent.
  2. Alternate Interior Angles:
    - Alternate interior angles are angles on opposite sides of the transversal and inside the two lines. Alternate interior angles are congruent.
  3. Alternate Exterior Angles:
    - Alternate exterior angles are angles on opposite sides of the transversal and outside the two lines. Alternate exterior angles are congruent.
  4. Same-Side Interior Angles:
    - Same-side interior angles (also called consecutive interior angles) are angles on the same side of the transversal and inside the two lines. Same-side interior angles are supplementary (add up to $180^\circ$ ).

Triangles

Definition:
- A triangle is a polygon with three sides and three angles. The sum of the angles in a triangle is always $180^\circ$ .
Types of Triangles based on Sides:
1. Equilateral Triangle:
  - All three sides are equal in length. All three angles are equal to $60^\circ$ .
2. Isosceles Triangle:
  - Two sides are equal in length. The angles opposite the equal sides are also equal.
3. Scalene Triangle:
  - All three sides have different lengths. All three angles have different measures.
Types of Triangles based on Angles:
1. Acute Triangle:
  - All three angles are acute (less than $90^\circ$ ).
2. Right Triangle:
  - One angle is a right angle ( $90^\circ$ ).
3. Obtuse Triangle:
  - One angle is obtuse (greater than $90^\circ$ ).
Triangle Angle Sum Theorem:
- The sum of the interior angles in any triangle is $180^\circ$ . If the angles are $A$ , $B$ , and $C$ , then $m\angle A + m\angle B + m\angle C = 180^\circ$ .
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. If the sides are $a$ , $b$ , and $c$ , then:
  - a + b > c
  - a + c > b
  - b + c > a

Congruence

Corresponding Parts of Congruent Triangles are Congruent (CPCTC):
- If two triangles are congruent, then their corresponding sides and corresponding angles are congruent.
Triangle Congruence Postulates and Theorems:
1. Side-Side-Side (SSS):
  - If all three sides of one triangle are congruent to the corresponding three sides of another triangle, then the two triangles are congruent.
2. Side-Angle-Side (SAS):
  - If two sides and the включённый angle between them in one triangle are congruent to the corresponding two sides and included angle in another triangle, then the two triangles are congruent.
3. Angle-Side-Angle (ASA):
  - If two angles and the включённый side between them in one triangle are congruent to the corresponding two angles and included side in another triangle, then the two triangles are congruent.
4. Angle-Angle-Side (AAS):
  - If two angles and a non-included side in one triangle are congruent to the corresponding two angles and non-included side in another triangle, then the two triangles are congruent.
5. Hypotenuse-Leg (HL):
  - If the hypotenuse and a leg of one right triangle are congruent to the corresponding hypotenuse and leg of another right triangle, then the two triangles are congruent.

Quadrilaterals

Definition:
- A quadrilateral is a polygon with four sides and four angles. The sum of the angles in a quadrilateral is $360^\circ$ .
Types of Quadrilaterals:
1. Square:
  - A quadrilateral with four equal sides and four right angles.
2. Rectangle:
  - A quadrilateral with four right angles. Opposite sides are equal in length.
3. Parallelogram:
  - A quadrilateral with two pairs of parallel sides. Opposite sides are equal in length, and opposite angles are equal.
4. Rhombus:
  - A quadrilateral with four equal sides. Opposite angles are equal.
5. Trapezoid:
  - A quadrilateral with at least one pair of parallel sides.
  1. Isosceles Trapezoid:
    - A trapezoid in which the non-parallel sides (legs) are congruent.
Angle Sum Theorem for Quadrilaterals:
- The sum of the interior angles in any quadrilateral is $360^\circ$ . If the angles are $A$ , $B$ , $C$ , and $D$ , then $m\angle A + m\angle B + m\angle C + m\angle D = 360^\circ$ .

Circles

Definition:
- A circle is a set of all points in a plane that are at a fixed distance (radius) from a fixed point (center).
Parts of a Circle:
1. Center:
  - The central point from which all points on the circle are equidistant.
2. Radius (r):
  - The distance from the center of the circle to any point on the circle.
3. Diameter (d):
  - The distance across the circle through the center. The diameter is twice the radius ( $d = 2r$ ).
4. Chord:
  - A line segment that connects two points on the circle.
5. Tangent:
  - A line that touches the circle at exactly one point.
6. Secant:
  - A line that intersects the circle at two points.
7. Arc:
  - A portion of the circumference of the circle.
  1. Minor Arc:
    - An arc that is less than 180 degrees.
  2. Major Arc:
    - An arc that is greater than 180 degrees.
  3. Semicircle Arc:
    - An arc that is exactly 180 degrees.
8. Central Angle:
  - An angle whose vertex is at the center of the circle.
9. Inscribed Angle:
  - An angle whose vertex is on the circle and whose sides are chords of the circle.
Circumference:
- The distance around the circle. The formula for the circumference $C$ is $C = 2\pi r$ or $C = \pi d$ , where $r$ is the radius and $d$ is the diameter.
Area:
- The amount of space inside the circle. The formula for the area $A$ is $A = \pi r^2$ , where $r$ is the radius.
Theorems:
1. Tangent Theorem:
- A tangent line is