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Chapter 9
Theorem: Opposite sides of a parallelogram are congruent.
Theorem: Opposite angles of a parallelogram are congruent.
Theorem: The diagonals of a parallelogram bisect each other.
Theorem: If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off
congruent segments on every transversal.
Theorem: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Theorem: If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is
a parallelogram.
Theorem: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral in a
parallelogram.
Theorem: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is
parallelogram.
Theorem: Each diagonal of a rhombus bisects two angles of a rhombus.
Theorem: The diagonals of a rhombus are perpendicular.
Theorem: The area of a rhombus is equal to half the product of the lengths of its diagonals.
Theorem: The diagonals of a rectangle are congruent.
Theorem: If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a
rhombus.
Theorem: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
Theorem: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
Theorem: Base angles of an isosceles trapezoid are congruent.
Theorem: The diagonals of an isosceles trapezoid are congruent.
Theorem: The diagonals of a kite are perpendicular
Theorem: The Trapezoid Midsegment Theorem – The midsegment of a trapezoid is parallel to the bases and
half as long as the sum of the lengths of the bases.
Chapter 10
Angle-Angle Similarity Postulate (AA~): If two angles of one triangle are congruent to two angles of another
triangle, then the triangles are similar.
Side-Angle-Side Similarity Theorem (SAS~): If an angle of one triangle is congruent to an angle of another
triangle, and the sides including the angles are proportional, then the triangles are similar.
Side-Side-Side Similarity Theorem (SSS~): If the corresponding sides of two triangles are proportional, then the
triangles are similar.
Theorem: The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar
to the original triangle and to each other.
Corollary 1: The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths
of the segments of the hypotenuse.
Corollary 2: The altitude to the hypotenuse of a right triangle intersects it so that the length of each leg is the
geometric mean of the length of its adjacent segment of the hypotenuse and the length of the entire hypotenuse.
Side-Splitter Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, then it
divides those side proportionally.
Triangle-Angle-Bisector Theorem: If a ray bisects an angle of a triangle, then it divides the opposite side into
two segments that are proportional to the other two sides of the triangle.
Perimeters and Areas of Similar Figures: If the similarity ratio of two similar figures is a : b, then the ratio of
their perimeters is a : b, and the ratio of their areas is ∶ Areas and Volumes of Similar Solids: If the similarity ratio of two similar solids is a : b, then the ratio of their
areas is ∶ , and the ratio of their volumes is ∶ .