8.1 Building Blocks of Geometry

• Point: an exact location [notation]• Line: a straight path with no thickness, extending forever in opposite

directions [notation]• Plane: flat surface with no thickness that extends forever along two

dimensions [notation]• Ray: part of a line. It has an end point, and extends forever in one direction

[notation]• Line segment: part of a line or a ray. Has two end points. [notation]• Vertex: the point where two lines, rays or line segments meet to form an

angle.• Congruent: figures that have the same shape and size. Congruent line

segments have the same length. [notation]• Complementary angle pairs are two angles that form a _______________

(think Corner) …• Supplementary angle pairs are two angles that form a

______________________ (think Straight)…

8.2 Classifying Angles

• Angle: two rays with a common end point• Vertex: the common endpoint of two rays• Right: an angle that measures exactly 90°.• The symbol ∟ on the inside of an angle means that it is a right

angle.• Acute: an angle that measures more than 0° and less than 90°.• Obtuse: an angle that measures more than 90° and less than 180°.• Straight: an angle that measures exactly 180°.• Reflex: an angle that measures more than 180° and less than 360°.• Complementary: two angles where the sum of their measurements

is 90°.• Supplementary: two angles where the sum of their measurements is

180°.

8.3 Line and Angle Relationships• Perpendicular: two rays, lines, or line segments that intersect, forming a

90° angle.• Parallel lines: lines on the same plane that never intersect.• Skew lines: lines on different planes do not intersect and that are not

parallel.• Adjacent angles: are supplementary angles with a common vertex and a

common side but no common interior points. (1,2), ( , )• Vertical angles: opposite angles formed by intersecting lines, they have

the same measure (are congruent). (2, 3), ( , )• Transversal: a line that intersects two or more lines in the same plane.

1 2

3 4

8.3 Line and Angle RelationshipsTransversals with Parallel Lines

• Corresponding Angles: are congruent (1,5), (3,7), ( , ), ( , )

• Alternate Interior Angles: are congruent (4, 5), ( , )

• Alternate Exterior Angles: are congruent (1, 8), ( , )

1 2

3 4

5 6

7 8

8.4 Properties of Circles

Circle: set of points in plain that are all the same distance from a given point (the center of the circle).

Arc: part of a circle named by its endpoints.Diameter: line segment that passes through the center of a circle, and

whose endpoints lie on the circle.Radius: line segment whose endpoints are the center of the circle and

any point on the circle.Chord: line segment whose endpoints are any two points on a circle.Central angle: angle formed by two radii (plural of radius). The sum of

all non-overlapping central angles is 360°.Sector: the area of a circle formed by two radii and an arc connecting

the radii (think of a slice of pie or pizza).

8.5 Classifying Polygons

Polygon: Closed plane; made by three or more line segments that meet at their end points, but do not intersect.

Vertex: points where line segments meetRegular: all sides and angles are congruent

Name Number of SidesTriangle 3Quadrilateral 4Pentagon 5Hexagon 6Heptagon 7Octagon 8Nonagon 9Decagon 10

8.6 Classifying Triangles

Scalene: no congruent sides of anglesIsosceles: at least two congruent sides and anglesEquilateral: all the sides and angles are congruent

Acute: all of the angles are acuteObtuse: one of the angles is obtuseRight: one angle is a right angle

Angles opposite of congruent sides are congruent.Sides opposite of congruent angles are congruent.

8.7 Classifying Quadrilaterals

Parallelogram: opposite sides are parallel and congruent. Opposite angles are congruent.

Rectangle: Parallelogram with four right angles

Rhombus: Parallelogram with four congruent sides, opposite angles are congruent

Square: Parallelogram with four congruent sides and four right angles.

Trapezoid: Only one pair of opposite sides is parallel.

8.8 Angles in Polygons

Triangle Sum Rule: the sum of the measures of the angles in a triangle is 180°.

Quadrilateral: If you draw a diagonal from one vertex of a quadrilateral to the opposite vertex, you form two triangles. 180° + 180° = 360°.

1. Draw a pentagon

2. Draw a diagonal from one vertex to a non-adjacent vertex.

3. Start at the original vertex and draw a diagonal to another non-

adjacent vertex. How many triangles are there?

Formula for degrees in a polygon is:

180 × (n – 2), where n is the number of sides.

8.9 Congruent Figures

• Congruent: figures with the same size and shape.

• Triangles are congruent if they pass the:– Side-Side-Side rule – Side-Angle-Side rule– Angle-Side-Angle rule

• Figures may not look congruent at first– you may have to flip them.

• For polygons with more than three sides, you need to compare sides and angles.

• If figures are congruent, you can find missing measurements.

8.9 Congruent Figures

Similar:

Congruent

What about these?

8.10 Translations, Reflections, and Rotations

Preimage is the original and is changed to the image

Translation (slide): each point of a figure is move the same distance in the same direction.

Notation for translation: (x,y) → (x + a, y + b)

Moving a figure with coordinate (5,3) to the right 5 units and down 6 units:(5,3) → (5 + 5, 3 + (-6)) → (10, -3)

Reflection (flip): the figure is flipped or reflected on a line called the line of reflection to make a mirror image.

Rotation (turn or pivot): the figure is rotated around a fixed point (center of rotation). Only one point should stay the same.

Point Notation for rotation where a triangle rotates on A: ABC → AB’C’

8.11 Symmetry

Line Symmetry: the figure can be divided by a line that creates two sides that are mirror images of each other. The line of reflection or line of symmetry is the location where that mirror image is created. You can fold one part over the other and they are identical. (Flags, faces, rugs,…)

Rotational Symmetry: the figure can be turned less than 360 ° to produce an image that fits exactly over the original figure. (sand dollar, snowflake, …)

Figures without symmetry are said to have asymmetry or be asymmetrical.