Quadrilateral Theorems
Quadrilaterals:Parallelograms
About Sides
* If a quadrilateral is a parallelogram, the opposite
sides are parallel.
* If a quadrilateral is a parallelogram, the opposite
sides are congruent.
About Angles=
* If a quadrilateral is a parallelogram, the opposite
angles are congruent.
* If a quadrilateral is a parallelogram, the
consecutive angles are supplementary.
About Diagonals=
* If a quadrilateral is a parallelogram, the diagonals
bisect each other.
* If a quadrilateral is a parallelogram, the diagonals
form two congruent triangles.
Parallelogram Converses
About Sides=
* If both pairs of opposite sides of a quadrilateral
are parallel, the quadrilateral is a parallelogram.
* If both pairs of opposite sides of a quadrilateral
are congruent, the quadrilateral is a
parallelogram.
About Angles=
* If both pairs of opposite angles of a quadrilateral
are congruent, the quadrilateral is a
parallelogram.
* If the consecutive angles of a quadrilateral are
supplementary, the quadrilateral is a parallelogram.
About Diagonals=
* If the diagonals of a quadrilateral bisect each
other, the quadrilateral is a
parallelogram.
* If the diagonals of a quadrilateral form two
congruent triangles, the quadrilateral is a
parallelogram.
Parallelogram= If one pair of sides of a quadrilateral is BOTH parallel and congruent, the quadrilateral is a parallelogram.
Rectangle= If a parallelogram has one right angle it is a rectangle
A parallelogram is a rectangle if and only if its diagonals are congruent.
A rectangle is a parallelogram with four right angles.
Rhombus= A rhombus is a parallelogram with four congruent sides.
If a parallelogram has two consecutive sides congruent, it is a rhombus.
A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.
A parallelogram is a rhombus if and only if the diagonals are perpendicular.
Square= A square is a parallelogram with four congruent sides and four right angles.
A quadrilateral is a square if and only if it is a rhombus and a rectangle.
Trapezoid= A trapezoid is a quadrilateral with exactly one pair of parallel sides.
Isosceles Trapezoid= An isosceles trapezoid is a trapezoid with congruent legs.
A trapezoid is isosceles if and only if the base angles are congruent
A trapezoid is isosceles if and only if the diagonals are congruent
If a trapezoid is isosceles, the opposite angles are supplementary.
Circles:
Radius= In a circle, a radius perpendicular to a chord bisects the chord and the arc.
In a circle, a radius that bisects a chord is perpendicular to the chord.
In a circle, the perpendicular bisector of a chord passes through the center of the circle.
If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency.
Chords=
In a circle, or congruent circles, congruent chords are equidistant from the center. (and converse)
In a circle, or congruent circles, congruent chords have congruent arcs. (and converse)
In a circle, parallel chords intercept congruent arcs
In the same circle, or congruent circles, congruent central angles have congruent chords (and converse)
Tangents= Tangent segments to a circle from the same external point are congruent
Arcs In the same circle, or congruent circles, congruent central angles have congruent arcs. (and converse)
Angles= An angle inscribed in a semi-circle is a right angle.
In a circle, inscribed angles that intercept the same arc are congruent.
The opposite angles in a cyclic quadrilateral are supplementary
In a circle, or congruent circles, congruent central angles have congruent arcs.
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