Download Theorems Involving Triangles and Quadrilaterals

Theorems Involving Triangles and
Quadrilaterals
If a segment joins the midpoints of two sides of a triangle, then it forms two similar
triangles with a 1:2 (2:1) ratio.
If a segment joins the midpoints of two sides of a triangle, then it is parallel to and 1/2 the
length of the third side.
If a segment joining two sides of a triangle is parallel to the third side, then it divides the two
sides proportionally.
If the altitude, angle bisector, or median is drawn from the corresponding vertices of two
similar triangles, then the ratio of these segments is equal to the ratios of the corresponding
sides of the two similar triangles.
The bulleted items below are options for the third proof packet.
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The diagonals of a parallelogram divide the parallelogram into two congruent
triangles.
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Opposite sides of a parallelogram are congruent. (Converse: If both pairs of opposite
sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.)
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If one pair of sides of a quadrilateral are both parallel and congruent, the quadrilateral
is a parallelogram.
The diagonals of a parallelogram bisect each other. (Converse: If the diagonals of a
quadrilateral bisect each other, then the quadrilateral is a parallelogram.)
Opposite angles of a parallelogram are congruent. (Converse: If both pairs of opposite
angles in a quadrilateral are congruent, then the quadrilateral is a parallelogram.)
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The consecutive angles of a parallelogram are supplementary. (Converse: If all of the
consecutive angles of a quadrilateral are supplementary, then the quadrilateral is a
parallelogram.)
The diagonals of a rhombus (or a kite) are perpendicular. (Converse – only true with a
kite: If the diagonals of a quadrilateral are perpendicular, then the quadrilateral is a
kite.)
If the midpoints of the sides of a quadrilateral are connected (in order), the resulting
quadrilateral is a parallelogram.
If a point lies on the perpendicular bisector of a segment, then it is equidistant from
the endpoints of the segment.
****Other good proofs you could include are found in the “Looking Back” section of Unit 3.
Quadrilateral Definitions
Quadrilateral
Kite
Trapezoid
Parallelogram
Rhombus
Rectangle
Square
Quadrilateral Relationships: List all that apply.
A kite is always a …
A trapezoid is always a …
A parallelogram is always a …
A rhombus is always a …
A rectangle is always a …
A square is always a …