[b]What kind of figure is formed?[br][/b] The midpoint quadrilateral is a parallelogram.[br][b][br]Complete the following sentence: [br][/b] "If the midpoints of an arbitrary quadrilateral are connected in order, the resulting midpoint quadrilateral will be [u]a parallelogram[/u]."

[b]What do you observe about these angles?[/b][br][list][*]Opposite angles are congruent (i.e., [math]\angle HEF\cong\angle FGH[/math] and [math]\angle EFG\cong\angle GHE[/math])[/*][*]Consecutive angles are supplementary. (i.e. [math]m\angle HEF+m\angle EFG=180^\circ[/math], [math]m\angle EFG+m\angle FGH=180^\circ[/math], and [math]m\angle FGH+m\angle GHE=180^\circ[/math]).[/*][*]The sum of all four angles is [math]360^\circ[/math].[/*][/list]

[b]What do you observe about these areas?[br][/b] The area of the midpoint quadrilateral is half the area of the original quadrilateral.[br][br][b]Express this observation as a conjecture by completing the following sentence[/b]:[br] "If the midpoints of an arbitrary quadrilateral are connected in clockwise order, the area of the resulting midpoint quadrilateral will be [u]half the area of the original quadrilateral[/u]."[br][br][b]Does the conjecture continue to hold?[br][/b] Yes.

[b]What do you observe about the shape of the quadrilateral?[br][/b] When the diagonals of the quadrilateral don't intersect, one of the interior angles exceeds [math]180^\circ[/math]. This means the quadrilateral is concave.[br] Additionally, one of the diagonals is exterior to the quadrilateral.

[b]Write some conjectures about your observations to summarize what you notice by completing the following sentences[/b]:[br][br] "If the diagonals of the quadrilateral intersect each other, then the quadrilateral [u]is convex[/u]."[br] [br] "If the diagonals of the quadrilateral don't intersect each other, then the quadrilateral [u]has an interior angle greater than 180[math]^\circ[/math][/u].[br][br] "If the diagonals of the quadrilateral don't intersect each other, then the quadrilateral [u]is concave[/u]."

[b]What if the diagonals bisect each other? write a conjecture to summarize what you learn in this exploration by completing the statement:[br][/b] "If the diagonals of a quadrilateral bisect each other then [u]the quadrilateral is a parallelogram[/u]."[br] "If the diagonals of a quadrilateral bisect each other, then [u]the quadrilateral is convex[/u]."

[b]Write the converse of one or more of the conjectures you formulated in this activity. Does the converse mean the same thing as the original statement?[br][/b] If the quadrilateral is a parallelogram, then its diagonals bisect each other.[br] [br] The converse is true.[br]

[b]Write the converse of one or more of the conjectures you formulated in this activity. Does the converse mean the same thing as the original statement?[br][/b] If the quadrilateral is a parallelogram, then its diagonals bisect each other.[br] [br] The converse is true.[br]

[b]Complete the following sentence[/b]:[br] "The sum of the measures of the interior angles of an arbitrary triangle is [math]180^\circ[/math].

[b]Complete the following sentence[/b]:[br] "The sum of the measures of the interior angles of an arbitrary triangle is [math]180^\circ[/math].

[b]Write a conjecture about the angle sum of a quadrilateral.[br][/b] The sum of the measures of the interior angles of a quadrilateral is [math]360^\circ[/math].

[b]Write a conjecture about the angle sum of a quadrilateral.[br][/b] The sum of the measures of the interior angles of a quadrilateral is [math]360^\circ[/math].

[b]Are your conjectures about the angle sum for triangles and the angle sum for quadrilaterals related to each other? Why or why not?[br][/b] The conjectures are related to each other for the following reasons:[br] 1) The quadrilateral has one more interior angle than the triangle, and the sum of its angles is [math]180^\circ[/math]greater.[br] 2) Consider [math]\overline{BD}[/math], a diagonal of the quadrilateral ABCD, which cuts the quadrilateral into two triangles.

[b]Are your conjectures about the angle sum for triangles and the angle sum for quadrilaterals related to each other? Why or why not?[br][/b] The conjectures are related to each other for the following reasons:[br] 1) The quadrilateral has one more interior angle than the triangle, and the sum of its angles is [math]180^\circ[/math]greater.[br] 2) Consider [math]\overline{BD}[/math], a diagonal of the quadrilateral ABCD, which cuts the quadrilateral into two triangles.