One figure is congruent to another if it can be moved with translations, rotations, and reflections to fit exactly over the other.

In this figure, Triangle A is congruent to Triangles B, C, and D.

• A translation takes Triangle A to Triangle B.
• A rotation takes Triangle B to Triangle C.
• A reflection takes Triangle C to Triangle D.

The coordinate plane is one way to represent pairs of numbers. The plane is made of a horizontal number line and a vertical number line that cross at 0.

Pairs of numbers can be used to describe the location of a point in the coordinate plane.

Point \(R\) is located at \((3,\text-2)\). This means \(R\) is 3 units to the right and 2 units down from \((0,0)\).

Corresponding parts are the parts that match up between a figure and its scaled copy. They have the same relative position. Points, segments, angles, or distances can be corresponding.

Point \(B\) in the first triangle corresponds to point \(E\) in the second triangle. Segment \(AC\) corresponds to segment \(DF\).

An image is the result of translations, rotations, and reflections on an object. Every part of the original object moves in the same way to match up with a part of the image.

Triangle \(ABC\) has been translated up and to the right to make triangle \(DEF\). Triangle \(DEF\) is the image of the original triangle \(ABC\).

A reflection is a transformation that “flips” a figure over a line. Every point on the figure moves to a point directly on the opposite side of the line. The new points are the same distance from the line as they are in the original figure.

This diagram shows a reflection of A over line \(\ell\) that makes the mirror image B.

A right angle is half of a straight angle. It measures 90 degrees.

A rigid transformation is a move that does not change any measurements of a figure. Translations, rotations, and reflections are rigid transformations. So is any sequence of these.

A rotation is a transformation that “turns” a figure. Every point on the figure moves around a center by a given angle in a specific direction.

This diagram shows Triangle A rotated around center \(O\) by 55 degrees clockwise to get Triangle B.

A sequence of transformations is a set of translations, rotations, reflections, and dilations on a figure. The transformations are performed in a given order.

This diagram shows a sequence of transformations to move Figure A to Figure C. 

First, A is translated to the right to make B. Next, B is reflected across line \(\ell\) to make C.

A straight angle is an angle that forms a straight line. It measures 180 degrees.

A tessellation is a repeating pattern of 1 or more shapes. The sides of the shapes fit together with no gaps or overlaps. The pattern goes on forever in all directions.

This diagram shows part of a tessellation.

A transformation is a translation, rotation, reflection, or dilation, or a combination of these.

A translation is a transformation that “slides” a figure along a straight line. Every point on the figure moves a given distance in a given direction.

This diagram shows a translation of Figure A to Figure B using the direction and distance given by the arrow.

A transversal is a line that crosses parallel lines.

This diagram shows a transversal line \(k\) intersecting parallel lines \(m\) and \(\ell\).

Vertical angles are opposite angles that share the same vertex. They are formed when two lines cross each other. Their angle measures are equal.

Angles \(AEC\) and \(DEB\) are vertical angles. If angle \(AEC\) measures \(120^\circ\), then angle \(DEB\) must also measure \(120^\circ\).

Angles \(AED\) and \(BEC\) are another pair of vertical angles.