Exercise Write the opposite of 7. – 7

Exercise Write the opposite of – 3. 3

Exercise Calculate the measure of each angle in an equilateral triangle. 60°

Exercise Calculate the measure of the angles formed by intersecting diagonals of a square. 90°

Exercise Calculate the measure of the angles formed by intersecting diagonals of a regular hexagon. 60°

Transformation A transformation is the movement of an original geometric shape to another according to a predefined rule.

preimage—figure’s position before the transformation image—figure’s position after the transformation

A A' read A prime

Perpendicular Bisector A perpendicular bisector is a line perpendicular to a segment that intersects the segment at its midpoint.

Reflection A reflection is a transformation in which each point of the figure’s image is the same distance from the line of reflection as the corresponding point of the preimage.

Reflection A D D ' B ' C ' A ' B C

Example 1 Reflect point A through line j. j A

Example 1 Reflect BCD through line j. D j B C

Translation A reflection through a pair of parallel lines is a translation.

Example 2 Translate point E by reflecting it first through line k and then through line l. E k l E ' E ''

Example 2 Translate FGH. G k l H H ' G' F ' F F '' H '' G''

Rotation Reflecting a figure through each of two intersecting lines in succession produces a rotation.

Example 3 Rotate point M around X by reflecting it first through line k and then through line l. k M M ' l M ''

Example 3 Rotate NOP. k P O N ' O' P ' N l O'' N '' P ''

Line Symmetry A figure has line symmetry if and only if each half of the figure is the image of the other half under a reflection in some line.

Rotational Symmetry A figure has rotational symmetry if and only if there is a rotation of less than 360° around a center point such that the image of the figure coincides with the original figure.

Point Symmetry A figure has point symmetry if and only if it has 180° rotational symmetry.

Example Use this figure for the following problems.

yes; two—vertical and horizontal lines through the center Example Does the figure have any lines of symmetry? If so, how many? yes; two—vertical and horizontal lines through the center

Example Does the figure have rotational symmetry? If so, state all the degrees of rotational symmetry less than 360°. yes; 180°

Example Does the figure have point symmetry? yes

Example An equilateral triangle, a square, and a regular pentagon are illustrated below.

The triangle has 3, the square has 4, and the pentagon has 5. Example How many lines of symmetry does each have? The triangle has 3, the square has 4, and the pentagon has 5.

Example In general, how many lines of symmetry does a regular n-gon have? n

Example How many rotational symmetries does each figure have? Give their degrees. triangle: 2; 120° and 240°; square: 3; 90°, 180°, and 270°; pentagon: 4; 72°, 144°, 216°, and 288°

Example In general, how many rotational symmetries does a regular n-gon have? n – 1

The square; it is the only one that has 180° rotational symmetry. Example Which of the figures have point symmetry? The square; it is the only one that has 180° rotational symmetry.

Exercise If n is even, how many lines of symmetry can be drawn through a regular n-gon? Describe where they pass.

Exercise If n is odd, how many lines of symmetry can be drawn through a regular n-gon? Describe where they pass.