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1

What is it?

The figures on the following page are inaccurate pictures of 

quadrilaterals. They are marked so you know how they shouldhave been drawn:

•Angles that have the same number of marks are definitelysupposed to be congruent (that is, their measures are equal).If they have differing numbers of marks, they might not becongruent (but still could be).

•The same goes for markings on line segments. If they havethe same number of marks, they are congruent (they havethe 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 pointingin the same direction, they are supposed to be parallel. Infigure 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   ex-actly what the figure is. For example, saying that a figure is aparallelogram is more specific than saying it is a quadrilateral.A rhombus is more specific than a parallelogram. A square ismore specific than a rectangle, rhombus, or parallelogram.

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

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

Problems with a Point: August 22, 2001   c EDC 2001

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What is it?: Problem 2

Problems with a Point: August 22, 2001   c EDC 2001

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What is it?: Hints 1

Hints

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

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

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

Problems with a Point: August 22, 2001   c EDC 2001

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What is it?: Answers 1

Answers

See solutions for justification.

1. Parallelogram

2. Rectangle

3. Kite (or simply quadrilateral)   Some people consider thedefinition of a kite—aquadrilateral with exactly twodistinct pairs of congruentadjacent sides—to benonstandard.

4. Parallelogram

5. Rhombus

6. Rectangle

7. Rectangle

8. Rhombus

9. Impossible figure   Teacher’s Note:  For anextension activity, students mayenjoy creating their own problemsand finding out what kind of figure they create (if, in fact, thefigure is even possible.) Somegood discussion may emerge fromtheir work.

10. Parallelogram

11. Trapezoid

12. Rhombus

13. Isosceles trapezoid

14. Parallelogram

15. A 60◦-120◦ rhombus

16. Square

Problems with a Point: August 22, 2001   c EDC 2001

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What is it?: Solutions 1

Solutions

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 markedangles), so the two unmarked sides are also parallel. Thusthe quadrilateral has opposite sides parallel and is a par-allelogram.

2. A pair of alternate interior angles (1 and   2) are congru-ent, making a pair of opposite sides parallel. Because op-posite angles of a parallelogram are congruent, two supple-mentary adjacent angles are congruent (2 and   3). Thusthe angles are right angles and the figure is a rectangle.

3. The quadrilateral can’t be a parallelogram because alter-

nate interior angles aren’t congruent. The congruent baseangles in the triangles make adjacent sides congruent. Thefigure is a quadrilateral with two pairs of adjacent anglescongruent (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 aresupplementary, and the opposite sides of the quadrilateralare 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 paral-lel. If opposite sides of a quadrilateral are congruent andparallel, it is a parallelogram. Because two adjacent sidesare also congruent, this parallelogram is a rhombus.

6. The four congruent angles of the quadrilateral sum to 360◦,so the measure of each angle is 90◦.

7. The quadrilateral is a parallelogram because its diagonalsbisect each other. The parallelogram is a rectangle becauseits diagonals are congruent.

8. The marked sides are parallel because alternate interiorangles are congruent. The figure is a parallelogram becausetwo sides are congruent and parallel. The measures of thedouble marked angles must sum to 180◦, so the diagonalsare perpendicular. That means it is a rhombus.

Problems with a Point: August 22, 2001   c EDC 2001

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What is it?: Solutions 2

9. The double marked angles are corresponding angles, so thelowest side must be parallel to the diagonal going from thelower left corner to the upper right. This is impossible.

10. Extend one of the parallel sides to form an exterior an-

gle (1). This is alternate interior to one of the markedinterior angles (2), and so it’s congruent to that angle.However, it is also a corresponding angle to the remainingmarked interior angle. Since these angles are congruent,the remaining two sides are parallel.

11. Two of the sides are parallel because two alternate interiorangles are congruent. If the two marked sides were parallel,the figure would be a parallelogram and hence the sideswould be congruent. They aren’t congruent, and so thesides 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 anglescongruent. This means the remaining angles are congru-ent, so the triangles are similar. They share a correspond-ing side (the diagonal), so the triangles are also congruent.That means all four sides are the same length, so this is arhombus.

13. The two isosceles triangles have congruent vertex angles,and so their base angles are congruent. These form con-gruent alternate interior angles with opposite sides, and sothese two sides are parallel. The two remaining two trian-gles are congruent, but not in a way that causes alternateinterior angles to be congruent. The remaining two sidesare, therefore, congruent but not parallel, making the fig-ure an isosceles trapezoid.

14. The diagonals bisect each other, so this is a parallelogram.

15. The two isosceles triangles are equiangular, hence equi-lateral. Congruent alternate interior angles make oppositesides parallel. The fact that adjacent sides are equal makes

the parallelogram a 60◦

-120◦

rhombus.16. Congruent corresponding angles (1 and   2) make two

sides parallel; congruent alternate interior angles (1 and3) make the other sides parallel. The marked supplemen-tary angles (2 and   4) are congruent, so each measures90◦. Therefore the figure is a rectangle with adjacent sidesequal—a square.

Problems with a Point: August 22, 2001   c EDC 2001