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Theorem 17 : Equal corresponding angles mean that lines are parallel.
Corollary 1 : Equal alternate interior angles mean that lines are parallel.
Corollary 2 : Supplementary interior angles on the same side of a transversal
mean that lines are parallel.
Corollary 3 : In a plane, two lines perpendicular to a third line are parallel.
The Parallel Postulate – Through a point not on a line, there is exactly
one line parallel to the given line.
Theorem 18 : In a plane, two lines parallel to a third line are parallel to each other.
Theorem 19 : Parallel lines form equal corresponding angles.
Corollary 1: Parallel lines form equal alternate interior angles.
Corollary 2: Parallel lines form supplementary interior angles on the same side of a transversal.
Corollary 3: In a plane, a line perpendicular to one of two parallel lines is also perpendicular to the other.
Theorem 20 : The Triangle Sum Theorem – The sum of the angles of a triangle is 180°.
Corollary 1: If two angles of one triangle are equal to two angles of another
triangle, the third angles are equal.
Corollary 2: The acute angles of a right triangle are complementary.
Corollary 3: Each angle of an equilateral triangle is 60 °.
Theorem 21 :An exterior angle of a triangle is equal to the sum of the remote
interior angles.
Theorem 22 : The AAS Theorem– If two angles and the side opposite one of them in one triangle are equal
to the corresponding parts of another triangle, the triangles are congruent.
Theorem 23 : The HL Theorem– If the hypotenuse and a leg of one right triangle are equal to the
corresponding parts of another right triangle, the triangles are congruent.
Def: Adiagonal of a polygon is a line segment that connects any two
nonconsecutive vertices.
Theorem 24:The sum of the angles of a quadrilateral is 360°.
Def: Arectangle is a quadrilateral each of whose angles is a right angle.
Corollary to Theorem 24:A quadrilateral is equiangular iff it is a rectangle.
In general, if a polygon has n sides, in terms of n,
n3 diagonals can be drawn from one vertex
these diagonals form n2 triangles
the sum of the angles of an ngon is (n2)*180°
If the ngon is equiangular, each angle measures (n2)*180°/n
Def: Aparallelogramis a quadrilateral whose opposite sides are parallel.
A figure has point symmetry if it looks exactly the same when it is rotated about apoint.
Def: Two points aresymmetric with respect to a point iff it is the midpoint of the line segment joining them.
Parallelograms have point symmetry about the point in which their diagonals intersect.
Theorem 25 : The opposite sides and angles of a parallelogram are equal.
Theorem 26 : The diagonals of a parallelogram bisect each other.
Theorem 27 :A quadrilateral is a parallelogram, if its opposite sides are equal.
Theorem 28 :A quadrilateral is a parallelogram if its opposite angles are equal.
Theorem 29 :A quadrilateral is a parallelogram if two opposite sides are both parallel and equal.
Regular dodecagon
- How many sides does a dodecagon have?
Aregular polygon is one that is equilateral and equiangular.
- How many regular quadrilaterals do there seem to be in the figure?
- What is a regular quadrilateral called?
- How many rectangles do there seem to be in the figure?
- How many rhombuses are in the figure?
- How many different shapes of rhombuses does the figure seem to contain?
8.1 – Transformations
Def: Atransformation is a onetoone
correspondence between two sets of points.
Atranslation slides an object a certain
distance without turning it.
Areflection flips an object over a mirror line.
Arotation turns an object a certain number
of degrees about a fixed point.
Adilation enlarges or reduces the
size of an object.
Def: an isometry is a transformation that
preserves distance and angle measure.
Translations, reflections, and rotations
are all examples of isometries,
but dilations are not.
M.C. Escher. Fish (No. 20)
What type of translation seems to relate
- Two fish of the same color?
- A pair of red and white fish?
- A pair of blue and white fish?
Are there any pairs of fish in the figure for which one fish of the pair seems to be
- A dilation of the other?
- A reflection of the other?
Complete the figures by including the reflection image of the object through the mirror line.
N
A E
R Z
Which figures look the same as their mirror images?
17. What is it about these figures that causes them and their mirror images to
look the same?
Def: A translation is the composite of two successive reflections through parallel lines.
The distance between a point of the original figure and its translation image is called the magnitude of the
translation.
Def: A rotation is the composite of two successive reflections through intersecting lines.
The point in which the lines intersect is the center of rotation, and the measure of the angle through
which a point of the original figure turns to coincide with its rotation image is called the magnitude of the
rotation.
8.3 ‐ Isometries and Congruence
Def: Two figures are congruent if there is an isometry such that one figure is the image of the other.
Def: A glide reflection is the composite of a translation and a reflection in a line parallel to the direction of
the translation.
M.C. Escher
Koloman Moser
8.4 ‐ Transformations and Symmetry
Def: A figure has rotation symmetry with respect to a point iff it coincides with its rotation image through
less that 360° about the point.
A figure is said to have n‐fold rotation symmetry iff the smallest angle through which it can be turned to
look exactly the same is 360°/n.
