Geometry Topic 1 Transformations and Congruence Point a

Geometry Topic 1 Transformations and Congruence

Point – a point has no dimension. It is a location on a plane. It is represented by a dot.

Line – a line has one dimension. It is an infinite set of points represented by a line with two arrowheads that extends without end.

Plane – a plane has two dimensions extending without end. It is often represented by a parallelogram.

Line segment – a line segment consists of two endpoints and all the points between them.

Ray – a ray has one endpoint and extends without end in one direction. Note: Name the endpoint first. BC and CB are different rays.

Collinear – points that lie on the same line. K, L and M are collinear points.

Coplanar – points that lie in the same plane.

Midpoint - divides a segment into two congruent segments.

Angle – a figure formed by two rays with a common endpoint.

Vertex – the common endpoint of two or more rays or line segments.

Side of an angle – one of the two rays that form an angle.

Complementary angles – two angles whose measures have a sum of 90°.

Supplementary angles – two angles whose measures have a sum of 180°.

Transformation – a change in the position, size, or shape of a figure. A transformation maps the preimage to the image.

Rigid motion– a transformation of the plane or space, which preserves distance and angles.

Translation - a transformation in which all the points of a figure move the same distance in the same direction; the figure is moved along a vector so that all of the segments joining a point and its image are congruent and parallel. Preimage A(1, 2) B(3, 2) C(4, 3) D(3, 4) E(1, 4) Image A (-2, -3) B (0, -3) C (1, -2) D (0, -1) E (-2, -1)

Reflection – a transformation across a line, called the line of reflection. The line of reflection is the perpendicular bisector of each segment joining a point and its image. Preimage D (1, -2) E (3, -2) F (3, 2) Image D (-1, -2) E (-3, -2) F (-3, 2)

Rotation – a transformation about a point P, also known as the center of rotation, such that each point and its image are the same distance from P. All of the angles with vertex P formed by a point and its image are congruent. Pre-image has been transformed by a 90° clockwise rotation about the origin. Preimage A (-3, 0) B (-3, 3) C (-1, 3) D (-1, 0) Image A (0, 3) B (3, 3) C (3, 1) D (0, 1)

Symmetry – the transformation of a figure such that the image coincides with the preimage, the image and preimage have symmetry.

Center of rotation – the point around which a figure is rotated. Pre-image A has been transformed by a 90° clockwise rotation about the point (2, 0) to form image A’.

Postulate – a statement that is accepted as true without proof.

Conjecture – a statement that is believed to be true. Theorem – a statement that has been proven.

Deductive reasoning - method using logic to draw conclusions based upon definitions, postulates, and theorems. Inductive reasoning – method of drawing conclusions from a limited set of observations

Counterexample - specific case for which a conjecture is false. One counterexample proves a conjecture false.

Conditional Statement - a logical argument consisting of a set of premises, hypothesis (p), and conclusion (q).

Segment bisector – a line, ray or segment that divides a segment into two congruent segments.

Angle bisector – a ray that divides an angle into two congruent angles.

Linear pair – a pair of adjacent angles whose non-common sides are opposite rays.

Parallel lines – lines in the same plane that do not intersect.

Perpendicular lines – lines that intersect at 90° angles.

Perpendicular bisector - a segment, ray, line, or plane that is perpendicular to a segment at its midpoint.

Line of Symmetry – a line that divides a figure into two congruent reflected halves.

Rotational symmetry – a figure that can be rotated about a point by an angle less than 360° so that the image coincides with the preimage has a rotational symmetry.