Proof Geometry

 All quadrilaterals have four sides.  They also have four angles.  The sum of the four angles totals 360°.  These properties are what make quadrilaterals alike, but what makes them different?

 Two sets of parallel sides. (Definition)  Two sets of congruent sides. (Theorem)  The angles that are opposite each other are congruent. (Theorem)  The diagonals bisect each other. (Theorem)  Consecutive angles are supplementary. (Theorem)

 Has all properties of quadrilateral and parallelogram (Definition)  A rectangle also has four right angles. (Theorem)  The diagonals are congruent. (Theorem)

 Prove it is a parallelogram with 1 right angle Theorem: If a parallelogram has one right angle then it is a rectangle.  Prove that the diagonals are congruent to each other and bisect each other. Theorem: If the diagonals of a quadrilateral are congruent and bisect each other, then it is a rectangle.

 All properties of parallelogram (Definition)  All sides are congruent (Theorem)  The diagonals bisect the angles. (Theorem)  The diagonals are perpendicular bisectors of each other (Theorem)

 Prove it is a parallelogram with all 4 sides congruent.  Prove it is a parallelogram with each diagonal bisecting a pair of opposite angles.  Prove it is a quadrilateral with diagonals that bisect each other and are perpendicular to each other.

 All the properties of a parallelogram apply (Definition)  All the properties of a rectangle apply (Definition)  All the properties of a rhombus apply (Definition)  Prove it is a rectangle and a rhombus

 Trapezoid has one and only one set of parallel sides.  Prove a set of sides is parallel

 Never assume a trapezoid is isosceles unless it is given  An isosceles trapezoid has two equal sides. These equal sides are called the legs of the trapezoid, which are the non- parallel sides of the trapezoid.  Both pair of base angles in an isosceles trapezoid are also congruent.  The diagonals are congruent

 Prove that is a trapezoid with the non parallel set of sides congruent  Prove that it is a trapezoid with congruent diagonals  Prove that it is a trapezoid with base angles congruent.

 Two adjacent sets of consecutive congruent sides  Diagonals are perpendicular to each other.  Opposite angles are congruent

 Prove that it is a quadrilateral with two sets of consecutive congruent sides.  Prove that one diagonal is the perpendicular bisector of the other.

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