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# 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

## 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

## 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