1. Transformations and Congruence
Geometry
Topic 1
Transformations and Congruence

2. Vocabulary

3. Ray – a ray has one endpoint and extends without end in one direction.
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.

4. K, L and M are collinear points.
Coplanar – points that lie in the same plane.
Collinear – points that lie on the same line.
K, L and M are collinear points.
Midpoint - divides a segment into two congruent segments.
Vertex – the common endpoint of two or more rays or line segments.
Segment bisector – a line, ray or segment that divides a segment into two congruent segments.
Angle – a figure forms by two rays with a common endpoint.

5. Side of an angle – one of the two rays that form an angle.
Linear pair – a pair of adjacent angles whose non-common sides are opposite rays or share one common side.
∠1 and ∠2 form a linear pair
Side of an angle – one of the two rays that form an angle.
𝐷𝐸 and 𝐸𝐹 are sides of ∠𝐷𝐸𝐹
Angle bisector – a ray that divides an angle into two congruent angles.
Perpendicular bisector - a segment, ray, line, or plane that is perpendicular to a segment at its midpoint.
Perpendicular lines – lines that intersect at 90° angles.
Parallel lines – lines in the same plane that do not intersect.

6. Transformation – a change in the position, size, or shape of a figure
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.
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.

7. Center of rotation – the point around which a figure is rotated.
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.
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’.
Symmetry – the transformation of a figure such that the image coincides with the preimage, the image and preimage have symmetry.
Line of Symmetry – a line that divides a place figure into two congruent reflected halves.

8. 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.
Complementary angles – two angles whose measures have a sum of 90°.
Supplementary angles – two angles whose measures have a sum of 180°.

9. Lines, Angles, and Triangles
Topic 2
Lines, Angles, and Triangles

10. Vocabulary

11. Transversal – a line that intersects at least two other lines.
Vertical angles – the non adjacent angles formed by two intersecting lines.
∠1 and ∠3 are vertical angles. ∠2 and ∠4 are vertical angles.
Transversal – a line that intersects at least two other lines.
Line t is a transversal.
Hypotenuse – the side opposite the right angle in a right triangle.
Exterior angle – an angle formed by one side of a polygon and the extension of an adjacent side.
Interior angle – an angle formed by two sides of a polygon with a common vertex.
Auxiliary line – a line drawn in a figure to aid in a proof.

12. Parallel Lines

13. Isosceles triangle – a triangle with at least two congruent sides.
Equilateral triangle – a triangle with three congruent sides.
Equiangular triangle – a triangle with three congruent angles.
Triangle Inequality Theorem - The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Example:
AB + BC > AC
AC + BC > AB
AB + AC > BC

14. Congruent Triangles
Three possible congruence statements:
∠ABC ≅ ∠FED
∠BCA ≅ ∠EDF
∠BAC ≅ ∠EFD
SSS Triangle Congruence Postulate
SAS Triangle Congruence Postulate

15. ASA Triangle Congruence Postulate
AAS Triangle Congruence Theorem
HL Right Triangle Congruence

16. Lines, Angles, and Triangles – Part B
Topic 3
Lines, Angles, and Triangles – Part B

17. Vocabulary

18. Equidistant – the same distance from two or more objects.
Distance from a point to a line – the length of the perpendicular segment from the point to the line.
Midsegment of a triangle – a segment that joins the midpoints of two sides of the triangle.
The midsegment is always parallel to the third side of the triangle.
The midsegment is always half the length of the third side.
A triangle has three possible midsegments, depending on which pair of sides is initially joined.
Median of a triangle – a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.

19. Circumscribed circle – every vertex of the polygon lies on the circle.
Inscribed circle – a circle in which each side of the polygon is tangent to the circle.
Altitude of a triangle – a segment from a vertex perpendicular to the opposite side.
Vocabulary for Geometry Honors
Point of concurrency – a point where three or more lines coincide. P

20. Circumcenter of a triangle – the point of concurrency of the three perpendicular bisectors of a triangle.
The circumcenter is also the center of the triangle's circumcircle - the circle that passes through all three of the triangle's vertices.
The circumcenter is equidistant from all three vertices of the triangle.
In the special case of a right triangle, the circumcenter lies exactly at the midpoint of the hypotenuse.
Centroid of triangle – the point of concurrency of the three medians of a triangle. Also known as the center of gravity.
The centroid is always inside the triangle
Each median divides the triangle into two smaller triangles of equal area.
The centroid is exactly two-thirds the way along each median.
The centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex.
Incenter of a triangle – the point of concurrency of the three angle bisectors of a triangle.
The incenter is the center of the triangle's incircle, the largest circle that will fit inside the triangle and touch all three sides.
The triangle's incenter is always inside the triangle.