Definition: A line extends infinitely in both directions, represented with arrows on both ends.
Naming: Can be named in multiple ways (e.g., line AB, line AC, line BC).
Ray
Definition: A ray has a starting point and extends infinitely in one direction.
Naming: Must start with the initial point (e.g., ray AB).
Segment
Definition: A segment has defined endpoints (beginning and end).
Naming: Can be referred to as segment AB or segment BA.
Angles
Types of Angles
Acute Angle: Measures between 0 and 90 degrees (less than 90).
Right Angle: Measures exactly 90 degrees.
Obtuse Angle: Measures greater than 90 degrees but less than 180.
Straight Angle: Measures exactly 180 degrees (looks like a line).
Angle Formation
Angles are formed by the union of two rays at a vertex.
Midpoint and Bisectors
Midpoint
Definition: The midpoint divides a segment into two equal parts.
Notation: If B is the midpoint of segment AC, then segment AB is congruent to segment BC.
Segment Bisector
Definition: A line or ray that intersects a segment at its midpoint, creating two equal segments.
Angle Bisector
Definition: A ray that divides an angle into two equal angles.
Notation: For angle ABC, ray BD is its angle bisector if it splits angle ABC evenly.
Parallel and Perpendicular Lines
Parallel Lines
Definition: Lines that never intersect and have the same slope.
Notation: Line A is parallel to line B (symbol: ||).
Perpendicular Lines
Definition: Lines that intersect at right angles (90 degrees).
Slope: The slopes of perpendicular lines are negative reciprocals of each other.
Angle Relationships
Complementary Angles
Definition: Two angles that add up to 90 degrees.
Example: In a right triangle, if one angle is 40 degrees, the other must be 50 degrees.
Supplementary Angles
Definition: Two angles that add up to 180 degrees.
Example: If angle ABD is 110 degrees, angle DBC must be 70 degrees.
Properties of Congruence
Transitive Property
Definition: If two angles are congruent to the same angle, then they are congruent to each other.
Example: If angle 1 = angle 2 and angle 3 = angle 2, then angle 1 = angle 3.
Vertical Angles
Definition: Angles opposite each other when two lines intersect are congruent.
Example: Angle 1 is congruent to angle 3; angle 2 is congruent to angle 4.
Medians and Altitudes
Median
Definition: A line segment from a vertex to the midpoint of the opposite side of a triangle.
Example: In triangle ABC, segment BD is the median to side AC.
Altitude
Definition: A line segment from a vertex perpendicular to the opposite side of a triangle.
Example: In triangle ABC, segment BD is an altitude if it forms a right angle with AC.
Perpendicular Bisectors
Definition: A line that is perpendicular to a segment at its midpoint.
Properties: Divides the segment into two equal parts and forms right angles.
Triangle Congruence Postulates
SSS Postulate: If all three sides of one triangle are congruent to all three sides of another, the triangles are congruent.
SAS Postulate: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another, the triangles are congruent.
ASA Postulate: If two angles and the included side of one triangle are congruent to two angles and the included side of another, the triangles are congruent.
AAS Postulate: If two angles and a non-included side of one triangle are congruent to two angles and a corresponding non-included side of another triangle, the triangles are congruent.
Additional Resources
For practice problems and further study, check the description section for links to more videos and resources.