Geometry Flashcards: Always, Sometimes, Never

Vertical Angles and Definitions

• Vertical angles are always congruent, meaning that they are equal in measure.
• Definitions are always reversible, allowing for flexibility in understanding concepts.

Theorems and Rays

• Theorems are sometimes reversible, depending on the specific theorem.
• Opposite rays are never collinear rays that extend in the same direction; they diverge from their common endpoint.

Angles and Triangles

• Supplementary angles are sometimes congruent, such as in special cases.
• If two angles are complementary, they are always acute, meaning they measure less than 90 degrees.
• The supplement of an acute angle is never acute; it is always obtuse or equal to 90 degrees.

Additional Properties of Angles

• The bisectors of vertical angles are always opposite rays, forming straight angles.
• Vertical angles can sometimes be complementary, depending on their measures.

Triangle Characteristics

• An obtuse triangle never has three obtuse angles; it can only have one obtuse angle.
• A right triangle is sometimes isosceles, but it's not a requirement.
• The base of an isosceles triangle is sometimes shorter than either leg, depending on its configuration.

Median and Triangle Properties

• A median of a triangle always bisects the side to which it is drawn, creating two equal segments.
• If a median of a triangle is also an altitude, it is always an angle bisector, ensuring balanced triangles.

Basic Triangle Facts

• A triangle always has three sides, a fundamental characteristic of its definition.
• Any angle is always congruent to itself, highlighting the reflexive property of equality.

Congruence and Angles

• If all pairs of corresponding angles of two triangles are congruent, then the triangles are sometimes congruent under specific conditions.
• A median sometimes forms a right angle with the opposite side of the triangle, depending on the triangle's type.

Points, Lines, and Segments

• One point never determines a line; at least two points are needed for a line to exist.
• A line segment is always the shortest path between two points, adhering to geometric principles.

Perpendicular Bisectors and Parallel Lines

• If two points are equidistant from the endpoints of a segment, those points always determine the perpendicular bisector of the segment.
• Two lines that do not intersect are sometimes parallel; they can be skew lines in three-dimensional space.

Angles and Slopes

• Alternate interior angles never share a common vertex, as they lie on opposite sides of a transversal.
• The slopes of perpendicular lines are sometimes opposites, indicating a right angle between them.

Properties of Isosceles Triangles

• If the base of an isosceles triangle is positioned on the x-axis, the slopes of its legs are always opposite of each other.
• Alternate exterior angles are sometimes acute, varying with the specific angles involved.

Polygon and Quadrilateral Properties

• A kite is sometimes a parallelogram, depending on its specific properties.
• A convex polygon never has an interior angle greater than 180 degrees, defining its convexity.
• A rectangle is sometimes a square, only when all sides are of equal length.

Parallelograms and Equiangular Properties

• If a parallelogram is equiangular, it is always a rectangle, ensuring right angles at all corners.
• If a parallelogram is equilateral, it is always a rhombus, as all sides are equal in length.

Quadrilateral and Polygon Definitions

• A quadrilateral never has five sides; it is defined specifically as having four sides.
• The diagonals of a trapezoid are sometimes congruent, depending on the nature of the trapezoid.

Polygon Characteristics

• A polygon always has the same number of sides as vertices, which is a fundamental characteristic of polygonal shapes.
• If two lines are cut by a transversal, alternate interior angles are sometimes congruent, based on their positions relative to the transversal.

Slope of Parallelograms

• Only one side of a parallelogram can never have a slope of three, as the slopes of opposite sides are equal in a parallelogram.