# Chapter 1 Geometry: Project 1

Topic:
Geometry

## Explanation: 2 Opposite Pairs of Parallel Lines

I created a line segment AB and a separate point C. I connected points A and C, creating line segment AC. I used the “parallel lines” tool through point B and in reference to line segment AC, to create a line passing through B and parallel to AC. I used the “parallel lines” tool through point C and in reference to line segment AB, to create a line passing through C and parallel to AB. I created a point of intersection D at the overlap of the line parallel to AB and the line parallel to AC. Thus I created a parallelogram with two opposite pairs of parallel lines.

## Explanation: 1 Pair of Parallel and Congruent Lines

I created a line AB and a separate point C. I connected point B to point C, and then made a circle centered at B with radius BC. I then made a second circle with radius BC, centered at A.  I also used the “parallel lines” tool through point A and in reference to line segment BC, to create a line through A and parallel to BC. I created a point of intersection D at the overlap of the parallel line through A and the circle centered at A. This line segment (radius length BC) falls on the line parallel to BC. Therefore, BC and AD are both congruent and parallel. Thus I created a parallelogram with one pair of parallel and congruent sides.

## Explanation: 2 Opposite Pairs of Congruent Lines

I created a line segment AB, as well as a used the “parallel lines” tool to create a line through point C, parallel to AB. I then used the “perpendicular lines” tool to create two lines perpendicular to AB, passing through points A and B. I made points of intersection D and E at the overlap between the two lines perpendicular to AB and the line parallel to AB. The encased, resulting parallelogram has two opposite pairs of congruent lines (as proven in the algebra view by the side lengths).

## Explanation: 2 Opposite Pairs of Congruent Angles

I created a line segment AB, and a separate point C. I made a line parallel to AB, through point C, and I put an additional point D on line segment AB. I made a line passing through points C and D, creating line segment CD. I then made a line parallel to CD, through point F on line segment AB.I created a point of intersection where lines b and d meet, and connected parallelogram DCGF. Adjacent angles along line bisections are supplementary and opposite angles on pairs of bisected lines are congruent, as confirmed by the angle measurement tool. Thus the created parallelogram has two opposite pairs of congruent angles.

## Explanation: 2 Pairs of Diagonals that Bisect

I created line segment AB and separate point C. I created a line parallel to AB, through point C. I then connected points A and C, creating line segment AC and made a line parallel to AC, through point B. I created point D at the intersection of lines b and d. created parallelogram ABDC. I then connected point B and point C to create line segment BC, and used the midpoint tool to find the line segment’s midpoint E. I then connected point A and point D to create line segment AD, which passes through midpoint E. To confirm this, I used the midpoint tool to find line segment AD’s midpoint, which of course overlapped with point E. Thus I created a parallelogram with two pairs of diagonals that bisect one another.