# Homework 1

##### 1a.
[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]."
##### 1b.
[b]What do you observe about these angles?[/b][br][list][*]Opposite angles are congruent (i.e., $\angle HEF\cong\angle FGH$ and $\angle EFG\cong\angle GHE$)[/*][*]Consecutive angles are supplementary. (i.e. $m\angle HEF+m\angle EFG=180^\circ$, $m\angle EFG+m\angle FGH=180^\circ$, and $m\angle FGH+m\angle GHE=180^\circ$).[/*][*]The sum of all four angles is $360^\circ$.[/*][/list]
##### 1c.
[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.
##### 2a.
[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 $180^\circ$. This means the quadrilateral is concave.[br] Additionally, one of the diagonals is exterior to the quadrilateral.
##### 2b.
[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$^\circ$[/u].[br][br] ﻿"If the diagonals of the quadrilateral don't intersect each other, then the quadrilateral [u]is concave[/u]."
##### 2c.
[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]."
##### 2d.
[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]
##### 2d.
[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]
##### 3a
[b]Complete the following sentence[/b]:[br] ﻿"The sum of the measures of the interior angles of an arbitrary triangle is $180^\circ$.
##### 3a
[b]Complete the following sentence[/b]:[br] ﻿"The sum of the measures of the interior angles of an arbitrary triangle is $180^\circ$.
##### 3b
[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 $360^\circ$.
##### 3b
[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 $360^\circ$.
##### 3c.
[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 $180^\circ$greater.[br] ﻿2) Consider $\overline{BD}$, a diagonal of the quadrilateral ABCD, which cuts the quadrilateral into two triangles.
##### 3c.
[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 $180^\circ$greater.[br] ﻿2) Consider $\overline{BD}$, a diagonal of the quadrilateral ABCD, which cuts the quadrilateral into two triangles.