Chapter 9

1. Chapter 9 Geometry Vocabulary

2. Lesson 9-1 Introduction to Geometry: Points, Lines, and Planes

3. A B M D C P Q R C

4. Intersecting, Parallel, and Skew Lines • Two lines that lie in the same plane and do not intersect are parallel. Use the symbol ║to indicate “is parallel to”. • Two lines intersect if they have exactly one point in common. • Skew lines are lines that do not lie in the same plane. AB ║ PQ EF intersects BF AB and DE are skew

5. Parts of an Angle • An angle is formed by two rays with a common endpoint. • The rays are the sides of the angle. • The common endpoint is the vertex. A Ray ● Angle B ● Endpoint or Vertex C

6. An acute angle is less than 90○. A right angle is 90○. An obtuse angle is greater than 90○and less than 180○. A straight angle is equal to 180○. Classifying Angles Acute Angle Right Angle Obtuse Angle Straight Angle

7. Lesson 9-2 Angle Relationships and Parallel Lines

8. Adjacent angles share a vertex and a side but no points in their interiors. Vertical angles are formed by two intersecting lines and are opposite each other. Adjacent and Vertical Angles 1 Common Side 3 4 2 Angles 1 & 2 are vertical angles. Angles 3 & 4 are vertical angles.

9. Angle Relationships • If the sum of the measures of two angles is 90○, the angles are complementary. • If the sum of the measures of two angles is 180○, the angles are supplementary. Complementary Angles Supplementary Angles

10. Relating Angles and Parallel Lines A line that intersects two other lines in different points is a transversal. Alternate interior angles are in the interior of a pair of lines and on opposite sides of the transversal. d and e are alternate interior angles. When a transversal intersects two parellel lines, corresponding and alternate interior angles are congruent. Corresponding angles lie on the same side of the transversal and in corresponding positions. d and h are corresponding angles.

11. Lesson 9-3 Classifying Polygons

12. Classifying Triangles • A triangle is a polygon with three sides. Acute triangle three acute sides Right triangle one right angle Obtuse triangle one obtuse angle Equilateral triangle three congruent sides Isosceles triangle at least two congruent sides Scalene triangle no congruent sides

13. Classifying Quadrilaterals Trapezoid exactly one pair of parallel sides Quadrilateral four sides Parallelogram both pairs of opposite sides parallel Rhombus four congruent sides Rectangle four 90○ angles Square four 90○ angles and four congruent sides

14. Classifying Quadrilaterals Cont. • All parallelograms have opposite sides parallel. • Parallelograms include rectangles, rhombuses, and squares. • Quadrilaterals that have four right angles include the rectangles and squares.

15. Regular Polygons • A regular polygon has all sides congruent and all angles congruent. • The formula for the perimeter of a regular polygon is P = number of sides  the length of the sides. Triangle Pentagon Square Hexagon

16. Quadrilaterals and Their Properties Quadrilateral Quest: Do You Know Their Properties?

17. Lesson 9-5 Congruence

18. Congruent Triangles • Congruent figures have the same size and shape, and their corresponding parts have equal measures. • Triangles are congruent when all corresponding sides and interior angles are congruent. • You use corresponding parts of triangles to identify congruent triangles.

19. Congruent Triangles Angle-Side-Angle (ASA) If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. Side-Side-Side (SSS) If three sides of one triangle are congruent to three sides of a second triangle, the two triangles are congruent. Congruent figures have the same size and shape, and their corresponding parts have equal measures. Side-Angle-Side (SAS) If two sides and the included angle are congruent to two sides and the included angle of a second triangle, the two triangles are congruent.

20. Lesson 9-6 Circles

21. Circle Circumference is the distance around the circle. Radius is a segment that has one endpoint at the center and the other point on the circle. Diameter is a chord that passes through the center of a circle. Chord is a segment whose endpoints are on the circle.

22. Circumference of a Circle • The circumference of a circle is π times the diameter. C = π d C = 2 π r C = π d Write the formula C ≈ (3.14)6 Replace π with 3.14 and d with 6 = 18.84 Simplify 6 ft

23. Making a Circle Graph • To make a circle graph, you find the measure of each central angle. • A central angle is an angle whose vertex is the center of a circle. • There are 360○ in a circle. • Use proportions to find the measures of the central angles. 20 = _r 25 = _r_ 100 360 100 360 r = 72 ○ r = 90 ○ • Use a compass to draw a circle. • Draw the central angles with a protractor. • Label each section. • Add a title and necessary information.

24. Lesson 9-7 Constructions

25. Construction Vocabulary Perpendicular lines, segments, or rays intersect to form right angles. A perpendicular bisector is a line, segment, or ray that is perpendicular to the segment it bisects. A segment bisector is a line, segment, or ray that divides a segment into two congruent segments.

26. Construction Vocabulary Cont. An angle bisector is a ray that divides an angle into two congruent angles.

27. Steps for Constructing a(n) . . . Congruent Segment Pearson Prentice Hall Mathematics Video Congruent Angle Pearson Prentice Hall Mathematics Video Perpendicular Bisector Pearson Prentice Hall Mathematics Video Angle Bisector Pearson Prentice Hall Mathematics Video

28. Lesson 9-8 Translations

29. Translation Vocabulary • You perform a translation by sliding, flipping, or turning an object. • A transformation is a change of position or size of a figure. • A translation is a transformation that moves points the same distance and in the same direction. • The figure you get after a transformation is called the image. • Use prime notation (A1) to name the image of a point.

30. Examples of Translations Example of a Flip Example of a Slide Pearson Prentice Hall Mathematics Video Translating a Figure Example of a Turn

31. Writing a Rule to Describe a Translation Pearson Prentice Hall Mathematics Video

32. Lesson 9-9 Symmetry and Reflections

33. Symmetry • A figure has reflectional symmetry when one half is a mirror image of the other half. • A line of symmetry divides a figure with reflectional symmetry into two congruent halves.

34. Reflections • A reflection is a transformation that flips a figure over a line of reflection. • The reflected figure, or image, is congruent to the original figure. • Together, an image and its reflection have line symmetry, the line of reflection being the line of symmetry.

35. Graphing Reflections of a Shape Pearson Prentice Hall Mathematics Video

36. Lesson 9-10 Rotations

37. Rotations • A rotation is a transformation that turns a figure about a fixed point called the center of rotation. • The angle measure of the rotation is the angle of rotation.

38. Rotational Symmetry • A figure has rotational symmetry if you can rotate it 360○, or less, so that its image matches the original figure. • The angle (or its measure) through which the figure rotates is the angle of rotation.