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Q: Identify the theorem you can use to show that the quadrilateral is a parallelogram?

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A quadrilateral, in general, is not a parallelogram. If it is a parallelogram then you will have some additional information about its sides and angles. If you do not have such information it is not possible to prove that it is a parallelogram. Draw a diagonal which will divide the quadrilateral into two triangles and use the additional information that you have to show that the triangles are congruent. This can then be used to show equality of sides or of angles: the latter can then be used to show that sides are parallel. Note that the choice of which diagonal may influence how (if at all) you proceed.

This cannot be proven, because it is not generally true. If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. And conversely, the diagonals of any parallelogram bisect each other. However not every parallelogram is a rhombus.However, if the diagonals are perpendicular bisectors, then we have a rhombus.Consider quadrilateral ABCD, with diagonals intersecting at X, whereAC and BD are perpendicular;AX=XC;BX=XD.Then angles AXB, BXC, CXD, DXA are all right angles and are congruent.By the ASA theorem, triangles AXB, BXC, CXD and DXA are all congruent.This means that AB=BC=CD=DA.Since the sides of the quadrilateral ABCD are congruent, it is a rhombus.

Yes.Although the definition of a parallelogram is "a quadrilateral with both pairs of opposite sides parallel", the only way for a quadrilateral to include opposite sides of equal length is if the included angles are the same, and hence the sides are parallel.(Hint : draw a diagonal to a parallelogram. You can show that one of the two triangles formed is the mirror image of the other, which immmediately proves that each pair of opposite sides is equal.)

The diagonals divide the quadrilateral into four sections. You can then use the bisection to prove that opposite triangles are congruent (SAS). That can then enable you to show that the alternate angles at the ends of the diagonal are equal and that shows one pair of sides is parallel. Repeat the process with the other pair of triangles to show that the second pair of sides is parallel. A quadrilateral with two pairs of parallel lines is a parallelogram.

draw a diagonal through opposite corners of the quadrilateral. This makes two triangles. Prove the triangles are congruent using SSA (side side angle) congruence. Then show that the other two sides of the quadrilater must be congruent to each other, so it is a parallelogram.

It is a quadrilateral but not a parallelogram since it does not have two pair of parallel sides. no, but all parallelograms are quadrilaterals. Quadrilateral means four sides, paralleograms means the sides are parallel. The only quadrilateral parallelograms are Square, rectangle, rhombus. * * * * * ... and parallelograms themselves: 2 pairs of parallel sides which are also equal.

a trapazoid

The statement cannot be proved because it is not true.As a counter example, consider a parallelogram which is not a rhombus. Its diagonals do bisect each other but, by definition, it is not a rhombus.

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Converse: If the diagonals of a quadrilateral are congruent and bisect each other, then the quadrilateral is a rectangle. Given: Quadrilateral ABCD with diagonals , . and _ bisect each other Show: ABCD is a rectangle Because the diagonals are congruent and bisect each other, . Using the Vertical Angles Theorem, AEB CED and BEC DEA. So ∆AEB ∆CED and ∆AED ∆CEB by SAS. Using the Isosceles Triangle Theorem and CPCTC, 1 2 5 6, and 3 4 7 8. By the Angle Addition Postulate each angle of the quadrilateral is the sum of two angles, one from each set. For example, mDAB = m1 + m8. By the addition property of equality, m1 m8 m2 m3 m5 m4 m6 m7. So by substitution mDAB mABC mBCD mCDA. Therefore the quadrilateral is equiangular. Using 1 5 and the Converse of AIA, . Using 3 7 and the Converse of AIA, . Therefore ABCD is an equiangular parallelogram, so it is a rectangle by definition of rectangle.

just have 4 lines that join up... a quadrilateral is a 4 sided shape

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