Geometry: Lines, Segments, Rays, Planes, and Angles

• The following are notes on lines, segments, rays, planes, angles, angle pairs, polygons, circles, and triangles

Line Segment

• A part of a straight line
• Bounded by two distinct endpoints
• It contains two distinct endpoints and every point in-between
• Named by two capital letters representing the endpoints (e.g., AB or BA)
• The distance between two points is the absolute value of the difference of their coordinates.
• The number of segments given 'n' points is calculated by: n(n-1)/2
• A bisector of a line segment is a point, line, ray, or plane passing through the segment's midpoint

Ray

• A subset of a line
• Starts at a single endpoint, also known as the point of origin
• Extends infinitely in one direction
• Rays are named after their endpoint, followed by any other point on the ray, also called the direction point
• For example, ray RS where R is the endpoint
• The order of letters cannot be interchanged
• A symbol for a ray is a small arrow placed above the two-lettered name of the ray
• Rays have one endpoint (finite) and extend infinitely on the other side

Opposite Rays

• A pair of rays with the same endpoint
• They extend in opposite directions, forming a straight line
• When naming opposite rays, the first letter (endpoint) must be the same for both rays
• Example: EF and EG share the endpoint E
• The number of rays for collinear points is calculated by: 2(n-1)
• For non-collinear points, the formula is: n(n-1)

Kinds of Planes

• Parallel planes never meet or intersect
• Intersecting planes are not parallel and intersect at a line
• Two planes cannot intersect at more than one line

Kinds of Lines

• Intersecting lines share exactly one common point, called the point of intersection
• Transversal lines intersect two or more lines at distinct points
• Concurrent lines are three or more lines sharing exactly one common point
• The common point of intersection is the concurrent point
• Parallel lines maintain the same distance apart and never intersect, even when extended infinitely
• They are always equidistant and run in the same direction
• Perpendicular lines are two intersecting lines crossing at a right angle (90 degrees)
• Skew lines continue indefinitely in two directions, are in different planes, and never appear parallel or intersect

Angles

• Angles are a combination of two rays with a common endpoint
• The primary parts of an angle are the vertex and sides
• Vertex being the point of intersection of two rays
• Sides being the two intersecting rays
• The secondary parts of an angle
• Interior of the angle is the space inside the opening
• Exterior is the space outside of the opening
• An angle can be named by its vertex, by three points on the angle or by a letter/number inside the opening

Classification of Angles

• Acute angles measure between 0° and 90°
• Right angles measure exactly 90°
• All right angles are congruent
• Obtuse angles measure between 90° and 180°
• Straight angles measure exactly 180°
• All straight angles are congruent
• Reflex angles measure between 180° and 360°
• Full rotation is an angle that measures 360°

Angle Bisectors and Congruent Angles

• An angle bisector is a ray dividing an angle into two equal parts
• Every angle has a bisector
• Congruent angles have the same measure
• The length or direction of the angles has no effect on their congruency

Pair of Angles

• Adjacent angles are coplanar, share a common side and vertex, but have no common interior points
• Vertical angles are formed by two intersecting lines and their sides form two pairs of opposite rays
• Vertical angles are always congruent
• Dihedral angles are formed by the union of two intersecting half-planes, not coplanar, which share a common edge
• Each half plane, along with the edge, is called the face of the dihedral angle
• The intersection of the half planes is called the edge of the dihedral angle
• Each angle is called the complement of the other
• Complementary angles add up to 90°
• Supplementary angles add up to 180°
• Adjacent supplementary angles with uncommon sides forming opposite rays are a linear pair
• A supplement of an angle is the number/quantity that makes the whole complete
• All linear pairs are supplementary; however, not all supplementary pairs are linear pairs

Polygons

• A plane figure made of line segments intersecting at their endpoints to form a closed polygonal chain
• Segments (sides) intersect at endpoints (vertices)
• No two segments with a common endpoint are collinear
• All sides should be connected

Parts of a Polygon

• Sides are the segments connected at endpoints
• Consecutive sides are two adjacent segments with a common endpoint
• Non-consecutive sides are two non-adjacent segments with no common endpoint
• Vertices/corners are the points where adjacent sides meet
• Consecutive vertices are two consecutive endpoints of a side
• Non-consecutive vertices are non-adjacent endpoints
• Angles are formed by the intersection of two consecutive sides
• Consecutive angles are the adjacent angles of the polygon
• Non-consecutive angles are two non-adjacent angles of the polygon
• A diagonal of a polygon is a line segment joining two non-consecutive vertices

Classification of Polygons

• Convex polygons have all interior angles less than 180° and all diagonals contained within the shape
• Concave polygons have at least one interior angle greater than 180° and must have at least four sides
• Regular polygons are convex polygons that are both equiangular and equilateral
• Irregular polygons have unequal angles and unequal sides

Types of Polygons

• Equiangular polygons are convex polygons with all angles congruent
• Equilateral polygons are convex polygons with all sides congruent
• The number of diagonals from each vertex of a polygon is: Dv = n - 3
• The total number of diagonals of a polygon is calculated by: Dt = n(n-3)/2

Triangles

• A mediah is a line segment joining a vertex (corner) to the midpoint of the opposite side bisecting that line
• Every triangle has three medians that meet at the centroid
• An altitude is a perpendicular line segment from a vertex to the opposite side (or its extension)
• Each triangle has three altitudes and they meet at a point called the orthocenter
• An angle bisector divides the angle into two equal parts
• The incenter is a point of congruency defined by the three-angle bisector of the triangle
• A perpendicular bisector is a line that is perpendicular (90°) to a side and passes through its midpoint
• The three perpendicular bisectors of a triangle meet at the circumcenter
• It is the center of the circumscribed circle (touches all three vertices)

Classification of Triangles by Side

• Scalene triangles have all three sides of different lengths
• Isosceles triangles have at least two equal sides (legs)
• The third side is called the base
• Base angles are opposite legs
• The vertex or apex angle is opposite the base
• The altitude from the apex of an isosceles triangle bisects the base and the apex angle
• Equilateral triangles have all three sides equal

Classification of Triangles by Angle

• In acute triangles, all three angles are less than 90°
• Obtuse triangles have one obtuse angle (greater than 90°)
• The other two angles are acute (less than 90°)
• Right triangles have one right angle (exactly 90°)
• The other two angles are acute
• In right triangles, the two shorter sides forming the 90° angle are called legs
• The longest side opposite the right angle is called the hypotenuse
• The sum of the two acute angles is always 90°

Triangle Angles

• The sum of the measures of the angles of a triangle is 180°

Circles

• A closed plane figure with all points equidistant from a fixed point/center
• The distance from the center to any point on the circle is the radius
• The boundary of the circle is the circumference

Parts of a Circle

• A radius is a line segment from the center to any point on the circumference
• It is half the length of the diameter
• Diameter is a line segment passing through the center with endpoints on the circumference
• It is twice the radius and is the longest chord in a circle
• Circumference is the total distance around the circle, also called the perimeter
• Area is the region covered by its boundary
• A chord is a line segment with both endpoints on the circumference
• The longest chord is the diameter

Lines in Relation to a Circle

• A tangent line touches a circle at exactly one point
• A tangent line is always perpendicular to the radius at the point of tangency
• A secant line intersects the circle at two distinct points
• A secant line is an extended chord

Arcs

• An arc is a part or segment of the circumference
• There are three main types:
• Minor Arc less than half of the circumference (less than 180°)
• Semicircle exactly half of the circumference (180°) is formed when the diameter divides the circle into two equal parts
• Major Arc more than half of the circumference (more than 180°)

Sectors and Segments

• A sector of a circle is a region enclosed by an arc and two radii
• It is a portion of a circle and two radii extending from the center
• A segment of a circle is a region bounded by a chord and the corresponding arc lying inbetween
• A segment does not include the center of the circle

Types of Angles in a Circle

• A central angle has its vertex at the center and sides as radii
• An inscribed angle has its vertex on the circle and sides are chords
• Any two inscribed angles that intercept the same arc are congruent