Congruent Squares

These squares are congruent.  You can tell because they have parallel lines which are cut by a transversal. It follows that the interior angles are congruent.  It also shows that the same-side interior angles are supplementary.  This is obvious to see because a square has four 90 degree angles.  The squares also share a side and the same parallel lines. In conclusion, these two squares are congruent. (~KM5)

Carpeted Rectangles

Earlier today, I was looking at the carpet on my living room floor and I saw what appeared to be three similar quadrilaterals, more specifically rectangles.

I then took a picture of it and uploaded it to sketchpad to prove Theorem 5-12, which states that the diagonals of a rectangle are congruent. I measured the lengths and proved that it was a rectangle and then proved that the diagonals were congruent.

(~WM5)

Parallel Bricks

These bricks demonstrate rectangles and, by definition, parallel lines. In fact, they demonstrate many of the postulates and theorems about parallel lines.

A rectangle is a special type of quadrilateral with opposite sides parallel and consecutive sides perpendicular. Because the consecutive sides are perpendicular, each consecutive angle must be a right angle and the opposite sides must be parallel because the same-side interior angles are supplementary. Each bricks is also parallel to the two next it, as shown above. (~JC1)

Tennis Court

I was playing tennis this weekend and I realized that geometry included both the wall in the background and the net in the image below.

Each individual square is a parallelogram. With drawings on the Geometer's Sketchpad, I was able to prove theorems 5-1 to 5-8.

The measure of angle C is 106.66 degrees, angle D is 73.34 degrees, angle B is 106.66 degrees, and angle A is 73.34 degrees. Same side interior angles measured 180 degrees, and the total angle measure was 360 degrees. (~WM5)

Trapezoids in Candy

This candy piece demonstrates the definition of a trapezoid (specifically an isosceles trapezoid), and Theorem 5-18, which states that the base angles of an isosceles trapezoid are congruent.

Lines AB and DC are parallel while AD and BC are not, which fits the qualification: "A quadrilateral with exactly one pair of parallel sides is called trapezoid." It also fits the qualification of an isosceles trapezoid by having congruent legs. Finally, the candy also fulfills Theorem 5-18, because, as you can see, its base angles are congruent. -IR1