Homework 1

Author:
kjs36

Activity 1

1a.

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

1b.

What do you observe about these angles?
  • Opposite angles are congruent (i.e., and )
  • Consecutive angles are supplementary. (i.e. , , and ).
  • The sum of all four angles is .

1c.

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

Activity 2

2a.

What do you observe about the shape of the quadrilateral? When the diagonals of the quadrilateral don't intersect, one of the interior angles exceeds . This means the quadrilateral is concave. Additionally, one of the diagonals is exterior to the quadrilateral.

2b.

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

2c.

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

2c: Construction of a quadrilateral with bisecting diagonals.

2c: Construction of a quadrilateral with bisecting diagonals.

2d.

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? If the quadrilateral is a parallelogram, then its diagonals bisect each other. The converse is true.

2d.

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? If the quadrilateral is a parallelogram, then its diagonals bisect each other. The converse is true.

Activity 3

Activity 3

3a

Complete the following sentence: "The sum of the measures of the interior angles of an arbitrary triangle is .

3a

Complete the following sentence: "The sum of the measures of the interior angles of an arbitrary triangle is .

3b.

3b.

3b

Write a conjecture about the angle sum of a quadrilateral. The sum of the measures of the interior angles of a quadrilateral is .

3b

Write a conjecture about the angle sum of a quadrilateral. The sum of the measures of the interior angles of a quadrilateral is .

3c.

Are your conjectures about the angle sum for triangles and the angle sum for quadrilaterals related to each other? Why or why not? The conjectures are related to each other for the following reasons: 1) The quadrilateral has one more interior angle than the triangle, and the sum of its angles is greater. 2) Consider , a diagonal of the quadrilateral ABCD, which cuts the quadrilateral into two triangles.

3c.

Are your conjectures about the angle sum for triangles and the angle sum for quadrilaterals related to each other? Why or why not? The conjectures are related to each other for the following reasons: 1) The quadrilateral has one more interior angle than the triangle, and the sum of its angles is greater. 2) Consider , a diagonal of the quadrilateral ABCD, which cuts the quadrilateral into two triangles.