1.6 Motion in Geometry Objective Identify and draw the three basic rigid transformations: translation, rotation, and reflection.

Glossary Terms image preimage reflection rigid transformation rotation 1.6 Motion in Geometry Glossary Terms image preimage reflection rigid transformation rotation translation

Key Skills Identify translations, rotations, and reflections. 1.6 Motion in Geometry Key Skills Identify translations, rotations, and reflections.

Key Skills Translate a figure along a line. B A C B’ A’ C’ 1.6 Motion in Geometry Key Skills Translate a figure along a line. B A C B’ A’ C’

1.6 Motion in Geometry Key Skills Rotate a figure about a point.

1.6 Motion in Geometry Key Skills Reflect a figure across a line. TOC

1.7 Motion in the Coordinate Plane Objectives Review the algebraic concepts of coordinate plane, origin, x- and y-coordinates, and ordered pair. Construct translations, reflections across axes, and rotations about the origin on a coordinate plane.

Theorems, Postulates, & Definitions 1.7 Motion in the Coordinate Plane Theorems, Postulates, & Definitions Horizontal and Vertical Coordinate Translations Horizontal translation of h units: H(x, y) = (x + h, y) Vertical translation of v units: H(x, y) = (x, y + v)

Theorems, Postulates, & Definitions 1.7 Motion in the Coordinate Plane Theorems, Postulates, & Definitions Reflections Across the x- or y-Axes Reflection across the x-axis: M(x, y) = (x, –y) Reflection across the y-axis: N(x, y) = (–x, y) 180° Rotation About the Origin R(x, y) = (–x, –y)

Key Skills Use the coordinate plane to transform geometric figures. 1.7 Motions in the Coordinate Plane Key Skills Use the coordinate plane to transform geometric figures. x +6 To translate the triangle 6 units right, use the rule H(x, y) = (x + 6, y). x +6 x +6

Key Skills Use the coordinate plane to transform geometric figures. 1.7 Motions in the Coordinate Plane Key Skills Use the coordinate plane to transform geometric figures. y To reflect the triangle across the x-axis, use the rule M(x, y) = (x, –y). y –y –y TOC

Key Skills Write a definition of an object. These objects are 2.3 Definitions Key Skills Write a definition of an object. These objects are rhombuses. These objects are not rhombuses.

Key Skills a. b. c. a d d. Which of these objects are rhombuses? 2.3 Definitions Key Skills a. b. c. a d d. Which of these objects are rhombuses? Write a definition of a rhombus. Objects a and d are rhombuses. Definition: A rhombus is a four-sided figure in which all four sides have the same length.

Objectives Define polygon. 3.1 Symmetry in Polygons Objectives Define polygon. Define and use reflectional symmetry and rotational symmetry. Define regular polygon, center of a regular polygon, central angle of a regular polygon, and axis of symmetry.

Glossary Terms axis of symmetry center of a regular polygon 3.1 Symmetry in Polygons Glossary Terms axis of symmetry center of a regular polygon central angle of a regular polygon equiangular polygon equilateral polygon polygon reflectional symmetry regular polygon rotational symmetry

Theorems, Postulates, & Definitions 3.1 Symmetry in Polygons Theorems, Postulates, & Definitions Triangles Classified by Number of Congruent Sides Three congruent sides equilateral At least two congruent sides isosceles No congruent sides scalene

Theorems, Postulates, & Definitions 3.1 Symmetry in Polygons Theorems, Postulates, & Definitions Reflectional Symmetry: A figure has reflectional symmetry if and only if its reflected image across a line coincides exactly with the preimage. The line is called an axis of symmetry. Rotational Symmetry: A figure has rotational symmetry if and only if it has at least one rotation image, not counting rotation images of 0° or multiples of 360°, that coincides with the original image.

Key Skills Identify reflectional symmetry of figures. 3.1 Symmetry in Polygons Key Skills Identify reflectional symmetry of figures. Draw all the axes of symmetry of this figure.

Key Skills Identify rotational symmetry of figures. 3.1 Symmetry in Polygons Key Skills Identify rotational symmetry of figures. The figure has 4-fold rotational symmetry. The image will coincide with the original figure after rotations of 90°, 180°, 270° and 360°. At 360°, the figure returns to its original position. TOC

3.3 Parallel Lines and Transversals Objectives Define transversal, alternate interior angles, alternate exterior angles, same-side interior angles, and corresponding angles. Make conjectures and prove theorems by using postulates and properties of parallel lines and transversals.

Glossary Terms alternate exterior angles alternate interior angles 3.3 Parallel Lines and Transversals Glossary Terms alternate exterior angles alternate interior angles corresponding angles same-side interior angles transversal

Theorems, Postulates, & Definitions 3.3 Parallel Lines and Transversals Theorems, Postulates, & Definitions Corresponding Angles Postulate 3.3.2: If two lines cut by a transversal are parallel, then corresponding angles are congruent. corresponding angles 2  3

Theorems, Postulates, & Definitions 3.3 Parallel Lines and Transversals Theorems, Postulates, & Definitions Alternate Interior Angles Theorem 3.3.3: If two lines cut by a transversal are parallel, then alternate interior angles are congruent. alternate interior angles 1  3

Theorems, Postulates, & Definitions 3.3 Parallel Lines and Transversals Theorems, Postulates, & Definitions Alternate Exterior Angles Theorem 3.3.4: If two lines cut by a transversal are parallel, then alternate exterior angles are congruent. alternate exterior angles 2  5

Theorems, Postulates, & Definitions 3.3 Parallel Lines and Transversals Theorems, Postulates, & Definitions Same-Side Interior Angles Theorem 3.3.5: If two lines cut by a transversal are parallel, then same-side interior angles are supplementary. same-side interior angles 1 + 4 = 180

Key Skills Identify special pairs of angles. Corresponding angles 3.3 Parallel Lines and Transversals Key Skills Identify special pairs of angles. Corresponding angles 1 and 5 1 and 3 Alternate interior angles Same-side interior angles 1 and 4 Alternate exterior angles 2 and 5

Key Skills Find angle measures formed by parallel lines 3.3 Parallel Lines and Transversals Key Skills Find angle measures formed by parallel lines and transversals. m || n and m1 = 135°. Then m2 = m3 = m5 = 135° and m4 = 45°. TOC

3.6 Angles in Polygons Objective Develop and use formulas for the sums of the measures of interior and exterior angles of a polygon.

3.6 Angles in Polygons Glossary Terms concave polygon convex polygon

Theorems, Postulates, & Definitions 3.6 Angles in Polygons Theorems, Postulates, & Definitions Sum of the Interior Angles of a Polygon 3.6.1: The sum, s, of the measures of the interior angles of a polygon with n sides is given by s = (n – 2)180. The Measure of an Interior Angle of a Regular Polygon 3.6.2: The measure, m, of an interior angle of a regular polygon with n sides is given by m = 180  360 n .

Theorems, Postulates, & Definitions 3.6 Angles in Polygons Theorems, Postulates, & Definitions Sum of the Exterior Angles of a Polygon 3.6.3: The sum of the measures of the exterior angles of a polygon is 360.

Key Skills Find interior and exterior angle measures of polygons. 3.6 Angles in Polygons Key Skills Find interior and exterior angle measures of polygons. Find the sum of the measures of the interior angles and of the exterior angles of a polygon with 15 sides. Sum of the measures of the interior angles: (15 – 2)180 = 2340 Sum of the measures of the exterior angles: 360

Key Skills Find interior and exterior angle measures of polygons. 3.6 Angles in Polygons Key Skills Find interior and exterior angle measures of polygons. Find the measure, m, of each interior angle of a regular polygon with 15 sides. m = 180  360 n 360 15 = 180  = 156 TOC

3.7 Midsegments of Triangles and Trapezoids Objectives Define midsegment of a triangle and midsegment of a trapezoid. Develop and use formulas based on the properties of triangle and trapezoid midsegments.

Glossary Terms midsegment of a trapezoid midsegment of a triangle 3.7 Midsegments of Triangles and Trapezoids Glossary Terms midsegment of a trapezoid midsegment of a triangle

Theorems, Postulates, & Definitions 3.7 Midsegments of Triangles and Trapezoids Theorems, Postulates, & Definitions Midsegment of a Triangle: A midsegment of a triangle is a segment whose endpoints are the midpoints of two sides. Midsegment of a Trapezoid: A midsegment of a trapezoid is a segment whose endpoints are the midpoints of the nonparallel sides.

Key Skills Solve problems by using triangle and trapezoid midsegments. 3.7 Midsegments of Triangles and Trapezoids Key Skills Solve problems by using triangle and trapezoid midsegments. Find DE and FG. DE is a midsegment of ABC, so DE = 80 2 = 40 meters FG is a midsegment of trapezoid ABED, so FG = 80 + 40 2 = 60 meters TOC

Theorems, Postulates, & Definitions 3.8 Analyzing Polygons With Coordinates Theorems, Postulates, & Definitions Definition of Slope: The slope of a nonvertical line that contains the points (x1, y1) and (x2, y2) is equal to the ratio: y2  y1 x2  x1 . Parallel Lines Theorem 3.8.2: Two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel.

Theorems, Postulates, & Definitions 3.8 Analyzing Polygons With Coordinates Theorems, Postulates, & Definitions Perpendicular Lines Theorem 3.8.3: Two non- vertical lines are perpendicular if and only if the product of their slopes is –1. Any vertical line is perpendicular to any horizontal line. Midpoint Formula: The midpoint of a segment with endpoints (x1, y1) and (x2, y2) has the following coordinates: x1 + x2 2 , y1 + y2 .

Key Skills Use slope to determine whether lines and 3.8 Analyzing Polygons With Coordinates Key Skills Use slope to determine whether lines and segments are parallel or perpendicular. A quadrilateral has vertices A(2, 3), B(6, 4), C(8, 7), and D(4, 6). Is ABCD a parallelogram? and are parallel (same slope: ). 1 4 CD AB and are parallel (same slope: ). BC DA 3 2 ABCD is a parallelogram (opposite sides of quadrilateral ABCD are parallel). TOC