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Definition: Rectangle A rectangle is a parallelogram in which all four angles are congruent. A B D C A parallelogram in which all four angles are congruent is a rectangle. (reverse of definition) A B D C Each pair of angles must be congruent and supplementary, so they must all be right angles. A B D C This quadrilateral is given as having four right angles. A B D C The four right angles allow us to prove that the opposite sides are parallel, since congruent and supplementary angles on the same side of a transversal prove lines parallel. A B D C A quadrilateral with four right angles must be a rectangle. A B D C This parallelogram is given with one known right angle. A B D C Since opposite angles of a parallelogram are congruent, we can mark angle B as a right angle. A B D C Since angles on the same side of a transversal of two parallel lines are supplementary, we can mark the other two angles as right angles. A B D C A parallelogram with a right angle is a rectangle. A B D C This parallelogram is given with congruent diagonals, . A B D C Opposite sides of a parallelogram are congruent. A B D C Triangle ADC is congruent to triangle BCD by SSS. A B D C Angle ADC is congruent to angle BCD by CPCTC. A B D C The parallel sides of the parallelogram mean that the two congurent angles are also supplementary. The both must be right angles. A B D C Opposite angles of a parallelogram are congruent, so all four of the angles are right angles, and the parallelogram is a rectangle. A B D C If a parallelogram has congruent diagonals, it is a rectangle. Definition: Rhombus A rhombus is a parallelogram in which all four sides are congruent. A D B C A parallelogram in which all four sides are congruent is a rhombus. (reverse of definition) A D B C This parallelogram is given with two consecutive sides congruent. A D B C Opposite sides of a parallelogram are congruent, so all four sides are congruent. A D B C A parallelogram with two consecutive sides congruent is a rhombus. A D B C This parallelogram has a diagonal that bisects two of the angles. A D B C Since the opposite angles of a parallelogram are congruent, we know that all four of the angles marked in red are congruent. A D B C Triangles ABC and CDA are isosceles because the base angles are congruent. The triangles are congruent by ASA. A D B C All four sides are congruent, so the parallelogram is a rhombus. A D B C If either diagonal of a parallelogram bisects two angles of the parallelogram, then it is a rhombus. A D B C Here is a quadrilateral with diagonals that are perpendicular bisectors of each other. A B M D C Four congruent right triangles, DMC, DMA, BMA, and BMC are formed. The are congruent by SAS. A B M D C The four sides of the quadrilateral are congruent by CPCTC. A B M D C The quadrilateral is a parallelogram, because both pairs of opposite sides are congruent. It is a rhombus because all four sides are congruent. A B M D C A quadrilateral in which the two diagonals are perpendicular bisectors of each other is a rhombus. Definition: Square A square is a parallelogram in which all four sides and all four angles are congruent. If a quadrilateral is both a rectangle and a rhombus, then it is a square. Definition: Kite A kite is a quadrilateral with two distinct pairs of congruent adjacent sides. Definition: Kite A kite is a quadrilateral with two pairs of congruent adjacent sides. A quadrilateral with two distinct pairs of congruent adjacent sides is a kite. (reverse of definition) A quadrilateral with two pairs of congruent adjacent sides is a kite. (reverse of definition) B A M D C Here is a quadrilateral in which one of the diagonals is the perpendicular bisector of the other. B A M D C Triangle AMB is congruent to triangle AMC by SAS. Triangle DMC is congruent to triangle DMB by SAS. B A M D C AB is congruent to AC, and DB is congruent to DC by CPCTC. The quadrilateral is a kite, because two distinct pairs of adjacent sides are congruent. B A M D C If exactly one diagonal of a quadrilateral is the perpendicular bisector of the other, then the quadrilateral is a kite. If one or both of the diagonals of a quadrilateral is the perpendicular bisector of the other, then the quadrilateral is a kite. Definition: Isosceles trapezoid An isosceles trapezoid is a trapezoid in which the two sides that are not parallel are congruent. Those two sides are called the legs. A trapezoid in which the two sides that are not parallel are congruent is an isosceles trapezoid. A D B C Here is a trapezoid in which the lower two base angles are congruent. E A D B C Since DA and CB are not parallel, the lines containing the segments must intersect. We can extend the segments until they intersect at point E. E A D B C Angle EAB is congruent to angle EDC, because they are corresponding angles. Angle EBA is congruent to angle ECD, because they are corresponding angles. E A D B C EA is congruent to EB, because "if angles, then sides." ED is congruent to EC for the same reason. E A D B C AD is congruent to BC by subtraction. A D B C If either the upper or lower two base angles of a trapezoid are congruent, then the trapezoid is isosceles. Ways to prove a quadrilateral is a rectangle. A parallelogram in which all four angles are congruent is a rectangle. Ways to prove a quadrilateral is a rectangle. A parallelogram in which all four angles are congruent is a rectangle. A quadrilateral with four right angles must be a rectangle. Ways to prove a quadrilateral is a rectangle. A parallelogram in which all four angles are congruent is a rectangle. A quadrilateral with four right angles must be a rectangle. A parallelogram with a right angle is a rectangle. Ways to prove a quadrilateral is a rectangle. A parallelogram in which all four angles are congruent is a rectangle. A quadrilateral with four right angles must be a rectangle. A parallelogram with a right angle is a rectangle. If a parallelogram has congruent diagonals, it is a rectangle. Ways to prove a quadrilateral is a rhombus. A parallelogram in which all four sides are congruent is a rhombus. Ways to prove a quadrilateral is a rhombus. A parallelogram in which all four sides are congruent is a rhombus. A parallelogram with two consecutive sides congruent is a rhombus. Ways to prove a quadrilateral is a rhombus. A parallelogram in which all four sides are congruent is a rhombus. A parallelogram with two consecutive sides congruent is a rhombus. If either diagonal of a parallelogram bisects two angles of the parallelogram, then it is a rhombus. Ways to prove a quadrilateral is a rhombus. A parallelogram in which all four sides are congruent is a rhombus. A parallelogram with two consecutive sides congruent is a rhombus. If either diagonal of a parallelogram bisects two angles of the parallelogram, then it is a rhombus. A quadrilateral in which the two diagonals are perpendicular bisectors of each other is a rhombus. Ways to prove a quadrilateral is a square. If a quadrilateral is both a rectangle and a rhombus, then it is a square. Ways to prove a quadrilateral is a kite. A quadrilateral with two distinct pairs of congruent adjacent sides is a kite. A quadrilateral with two pairs of congruent adjacent sides is a kite. Ways to prove a quadrilateral is a kite. A quadrilateral with two distinct pairs of congruent adjacent sides is a kite. A quadrilateral with two pairs of congruent adjacent sides is a kite. If exactly one diagonal of a quadrilateral is the perpendicular bisector of the other, then the quadrilateral is a kite. If one or both of the diagonals of a quadrilateral is the perpendicular bisector of the other, then the quadrilateral is a kite. Ways to prove a quadrilateral is an isosceles trapezoid. A trapezoid in which the two sides that are not parallel are congruent is an isosceles trapezoid. Ways to prove a quadrilateral is an isosceles trapezoid. A trapezoid in which the two sides that are not parallel are congruent is an isosceles trapezoid. If either the upper or lower two base angles of a trapezoid are congruent, then the trapezoid is isosceles.