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8.1 Building Blocks of Geometry
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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
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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
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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.
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8.3 Line and Angle Relationships
Transversals 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), ( , )
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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 meet
Regular: all sides and angles are congruent
Name
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Number of Sides
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8.6 Classifying Triangles
Scalene: no congruent sides of angles
Isosceles: at least two congruent sides and angles
Equilateral: all the sides and angles are congruent
Acute: all of the angles are acute
Obtuse: one of the angles is obtuse
Right: 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°.
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Draw a pentagon
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Draw a diagonal from one vertex to a non-adjacent vertex.
3.
Start at the original vertex and draw a diagonal to another nonadjacent 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.