Section 12.2:
Analyzing Shapes
• Notation:
AB denotes line segment connecting A and B.
−−→
AB denotes a ray with endpoint A passing through B.
←→
AB denotes a line passing through A and B.
∠ABC denotes angle with vertex B and sides BA and BC.
4ABC denotes triangle with vertices A, B, C.
• Reflection symmetry: A figure has reflection symmetry if there is a line that the
figure can be “folded over” so that one-half of the figure matches the other half. This
line is called the line (axis) of symmetry.
Example 1: Give examples of geometric figures that have reflection symmetry and identify
the axis of symmetry.
Example 2: Suppose that ABCD is an isosceles trapezoid with side AB congruent to side
CD. How can reflection symmetry be used to illustrate that ∠BAD is congruent to ∠CDA?
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SECTION 12.2: ANALYZING SHAPES
• Rotation Symmetry: A figure has rotation symmetry if there is a point around
which the figure can be rotated (less than a full term) so that the image matches the
original figure perfectly.
◦ To calculate the angle of rotation divide 360◦ by the number of rotations. (Note
the number of rotations are also called the order of rotation.)
◦ Note: figures for which only a full turn produces an identical image do not have
rotation symmetry.
Example 3: Give examples of geometric figures that have rotation symmetry and identify
how many rotations.
Example 4: Suppose that ABCD is a parallelogram. How could you use rotation symmetry
to illustrate that the diagonals of a parallelogram bisect each other? (A diagonal is a line
segment formed by connecting nonadjacent vertices.)
SECTION 12.2: ANALYZING SHAPES
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• Perpendicular Line Segment Test: Let P be the point of intersection of l and m.
Fold l at point P so that l folds across P onto itself. Then l and m are perpendicular
if and only if m lies along the fold line.
Example 5: Let ABCD be a kite. Show that the diagonals of a kite are perpendicular.
• Parallel Line Segment Test: Fold the figure so that ` folds onto itself. Then ` and
m are parallel if and only if m folds onto itself or an extension of m.
Example 6: Suppose ABCD is a rhombus. Show that the opposite sides of a rhombus are
parallel.
• Simple closed curves: can be traced with the same starting and stopping points and
without crossing or retracing any part of the curve.
• Polygon: is a simple closed curve made up of line segments.
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SECTION 12.2: ANALYZING SHAPES
• Regular Polygon (Regular n-gon): is a polygon with all n sides congruent and all
n vertex angles congruent.
◦ A regular n-gon has n rotations and n lines of symmetry.
◦ A vertex angle also called an interior angle is formed by two consecutive sides.
◦ A central angle is formed by the segments connecting consecutive vertices to the
center of the regular n-gon.
◦ An exterior angle is formed by one side and the extension of an adjacent side. A
regular n-gon has 2n exterior angles.
SECTION 12.2: ANALYZING SHAPES
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• Convex: A figure is convex if any line segment joining two points inside the figure lies
completely inside the figure.
• Concave: A figure is concave if it is not convex.
• Circle: is the set of all points in the plane that are a fixed distance from a given point
called the center.
• Diameter of a circle: is the length of the line segment whose endpoints are on the
circle and which contain the center of the circle.
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SECTION 12.2: ANALYZING SHAPES
Example 7: Draw an example of the following or tell why it is not possible.
(a) A simple closed curve with exactly two lines of symmetry that is not a polygon.
(b) A convex polygon with no reflection symmetry and no rotation symmetry.
(c) A regular polygon with reflection symmetry, but no rotation symmetry.
(d) A concave quadrilateral with reflection symmetry.