The figures on the following page are inaccurate pictures of quadrilaterals. They are marked so you know how they should have been drawn:

• Angles that have the same number of marks are definitely supposed to be congruent (that is, their measures are equal). If they have differing numbers of marks, they might not be congruent (but still could be).
• The same goes for markings on line segments. If they have the same number of marks, they are congruent (they have the same length). If they have different numbers of marks, they might not be congruent (but still could be).
• If line segments have the same number of arrows pointing in the same direction, they are supposed to be parallel. In figure 1, two sides are parallel and congruent.
• All lines that look straight really are supposed to be straight.

For each quadrilateral, decide as specifically as possible exactly what the figure is. For example, saying that a figure is a parallelogram is more specific than saying it is a quadrilateral. A rhombus is more specific than a parallelogram. A square is more specific than a rectangle, rhombus, or parallelogram.

A quadrilateral may be marked so that it is impossible to construct accurately. In that case it is no real figure at all!

Be prepared to explain why you know your answer is correct for each case.

Some people find it helpful to redraw the figures, using the markings as a guide.

Hint to problem 1. Draw a diagonal. What can you say about the triangles formed?

Hint to problem 2. What do the two angle measures on the right side of the figure tell you about two of the sides of the quadrilateral?

See solutions for justification.

1. Parallelogram
2. Rectangle
3. Kite (or simply quadrilateral)
   Some people consider the definition of a kite--a quadrilateral with exactly two distinct pairs of congruent adjacent sides--to be nonstandard.
   
4. Parallelogram
5. Rhombus
6. Rectangle
7. Rectangle
8. Rhombus
9. Impossible figure
10. Parallelogram
11. Trapezoid
12. Rhombus
13. Isosceles trapezoid
14. Parallelogram
15. A 60^o-120^o rhombus
16. Square

1. Draw a diagonal and prove the two triangles congruent (SAS, using the single marked angle shown here). Then two alternate interior angles are congruent (double marked angles), so the two unmarked sides are also parallel. Thus the quadrilateral has opposite sides parallel and is a parallelogram.
   
   
2. A pair of alternate interior angles (1 and 2) are congruent, making a pair of opposite sides parallel. Because opposite angles of a parallelogram are congruent, two supplementary adjacent angles are congruent (2 and 3). Thus the angles are right angles and the figure is a rectangle.
   
   
3. The quadrilateral can’t be a parallelogram because alternate interior angles aren’t congruent. The congruent base angles in the triangles make adjacent sides congruent. The figure is a quadrilateral with two pairs of adjacent angles congruent (sometimes called a kite).
   
   
4. The sum of the angles of a quadrilateral is 360 degrees. Let each of a pair of congruent angles be x and each of the second pair of congruent angles be y. Then 2x + 2y = 360 and so x + y = 180. So pairs of interior angles are supplementary, and the opposite sides of the quadrilateral are therefore parallel.
5. The base angles of the isosceles triangle are congruent. This means that two alternate interior angles (1 and 2) are congruent, and so the marked opposite sides are parallel. If opposite sides of a quadrilateral are congruent and parallel, it is a parallelogram. Because two adjacent sides are also congruent, this parallelogram is a rhombus.
   
   
6. The four congruent angles of the quadrilateral sum to 360^o, so the measure of each angle is 90^o.
7. The quadrilateral is a parallelogram because its diagonals bisect each other. The parallelogram is a rectangle because its diagonals are congruent.
8. The marked sides are parallel because alternate interior angles are congruent. The figure is a parallelogram because two sides are congruent and parallel. The measures of the double marked angles must sum to 180^o, so the diagonals are perpendicular. That means it is a rhombus.
9. The double marked angles are corresponding angles, so the lowest side must be parallel to the diagonal going from the lower left corner to the upper right. This is impossible.
10. Extend one of the parallel sides to form an exterior angle (1). This is alternate interior to one of the marked interior angles (2), and so it’s congruent to that angle. However, it is also a corresponding angle to the remaining marked interior angle. Since these angles are congruent, the remaining two sides are parallel.
    
    
11. Two of the sides are parallel because two alternate interior angles are congruent. If the two marked sides were parallel, the figure would be a parallelogram and hence the sides would be congruent. They aren’t congruent, and so the sides can’t be parallel. A quadrilateral with exactly one pair of sides parallel is a trapezoid.
    
    
12. The two triangles are isosceles triangles with vertex angles congruent. This means the remaining angles are congruent, so the triangles are similar. They share a corresponding side (the diagonal), so the triangles are also congruent. That means all four sides are the same length, so this is a rhombus.
13. The two isosceles triangles have congruent vertex angles, and so their base angles are congruent. These form congruent alternate interior angles with opposite sides, and so these two sides are parallel. The two remaining two triangles are congruent, but not in a way that causes alternate interior angles to be congruent. The remaining two sides are, therefore, congruent but not parallel, making the figure an isosceles trapezoid.
    
    
14. The diagonals bisect each other, so this is a parallelogram.
15. The two isosceles triangles are equiangular, hence equilateral. Congruent alternate interior angles make opposite sides parallel. The fact that adjacent sides are equal makes the parallelogram a 60^o-120^o rhombus.
16. Congruent corresponding angles (1 and 2) make two sides parallel; congruent alternate interior angles (1 and 3) make the other sides parallel. The marked supplementary angles (2 and 4) are congruent, so each measures 90^o. Therefore the figure is a rectangle with adjacent sides equal--a square.