Can Other Figures Other Than Triangles Be Proven To Be Congruent?

You have learned how triangles can be proven congruent time and time again. Is it possible to prove the congruency of quadrilaterals? Before we answer that, the definition of a quadrilateral must be clearly defined. A quadrilateral, according to Kay, is this: if A, B, C, and D are any four points lying in a plane such that no three of them are collinear, and if the points are so situated that no pair of open segments (segment AB, segment BC, segment CD, and segment DA) have any points in common, then the set

quadrilateral ABCD = segment AB U segment BC U segment CD U segment DA

is a quadrilateral, with vertices A, B, C, D; sides AB, BC, CD, DA; diagonals AC, BD, and angles DAB, ABC, BCD, and CDA.

Before we can continue with answering our question, we must learn a few more terms that come up frequently. Adjacent sides, also called consecutive sides, are those sides that have a common endpoint. An example of adjacent sides is BC and CD, in which they share point C. Adjacent angles, also called consecutive angles, are those angles that contain a common side. An example of adjacent angles is angles BCD and CDA, in which side CD is common.

What are the criteria for proving quadrilaterals congruent? For our purposes, we will be looking at convex quadrilaterals. A convex quadrilateral is a quadrilateral in which the diagonals intersect and lie between the vertices of the quadrilateral. Convex quadrilaterals have these useful properties according to Kay:

2. If quadrilateral ABCD is convex, then D lies on the interior of angle

3. If A, B, C, and D are consecutive vertices of a convex quadrilateral,

Let's look at these quadrilaterals: quadrilaterals ABCD and quadrilateral EFGH. In order for these quadrilaterals to be congruent, they must be congruent under the following correspondence: side AB is congruent to side EF, side BC is congruent to side FG, side CD is congruent to side GH, side DA is congruent to side HE; angle A is congruent to angle E, angle B is congruent to side F, angle C is congruent to angle G, angle D is congruent to angle H. This is shown in figure 1.



Quadrilateral ABCD = Quadrilateral EFGH means AB = EF, angle A = angle E

BC = FG, angle B = angle F

CD = GH, angle C = angle G

DA = HE, angle D = angle H

Now we can start looking at the different types of congruencies for quadrilaterals. The first one we will look at is the SASAS congruence hypothesis. This states that if you are given two quadrilaterals, quadrilateral ABCD and quadrilateral EFGH, three consecutive sides and the included two angles of quadrilateral ABCD is congruent to the corresponding three consecutive sides and the included two angles of quadrilateral EFGH. This is shown in Figure 2.



The following is the proof of this hypothesis:

1. By segment construction, construct segments AC and segment EG

2. Segment AC is inside quadrilateral ABCD by definition of convex polygon

3. Segment EG is inside quadrilateral EFGH by definition of convex polygon

4. Assumptions: Segment AB is congruent to Segment EF

Angle B is congruent to Angle F

Segment BC is congruent to Segment FG

5. Triangle ABC is congruent to Triangle EFG by SAS

6. Segment AC is congruent to Segment EG by CPCF

7. Angle BAC is congruent to Angle FEG by CPCF

8. Angle BCA is congruent to Angle FGE by CPCF

9. Implication: Angle ACD is congruent to Angle EGH since Angle BCA is congruent to Angle FGE

10. Assumption: Segment CD is congruent to Segment GH

11. Triangle ACD is congruent to Triangle EGH by SAS

12. Measure of Angle BCD = measure of Angle BCA + measure of Angle ACD by angle addition

13. Measure of Angle FGH = measure of Angle FGE + measure of Angle EGH by angle addition

14. Angle BCD is congruent to Angle FGH by algebra and substitution

15. Therefore SASAS congruence

Another theorem we will look at is the ASASA Theorem, which states that the three angles and the two corresponding sides of one quadrilateral is congruent to the corresponding three angles and the two corresponding sides of one quadrilateral. This is proven below with the use of figure 3.



The proof of the ASASA theorem:

1. By segment construction, construct segments AC and segment EG

2. Segment AC is inside quadrilateral ABCD by definition of convex

polygon

3. Segment EG is inside quadrilateral EFGH by definition of convex polygon

4. Assumptions: Segment AB is congruent to Segment EF

Angle B is congruent to Angle F

Angle BAC is congruent to Angle FEG 5. Triangle ABC is congruent to Triangle EFG by ASA

6. Segment BC is congruent to Segment FG by CPCF

7. Segment AC is congruent to Segment EG by CPCF

8. Angle BCA is congruent to Angle FGE by CPCF

9. The measure of angle BCD = measure of Angle BCA + measure of Angle ACD by angle addition

10. The measure of Angle FGH = measure of Angle FGE + measure of Angle EGH by angle addition

11. Angle ACD is congruent to Angle EGH by substitution and algebra

12. The measure of Angle BAD = measure of angle BAC + measure of Angle CAD by angle addition

13. The measure of Angle FEH = measure of Angle FEG + measure of Angle GEH by angle addition

14. Angle CAD is congruent to Angle GEH by substitution and algebra

15. Triangle ACD is congruent to Triangle EGH by ASA

16. Segment CD is congruent to Segment GH by CPCF

17. Quadrilateral ABCD is congruent to Quadrilateral EFGH by SASAS

18. Angle A is congruent to Angle E by CPCF

19. Therefore ASASA congruence

Another popular congruence theorem for quadrilaterals is the SASSS Theorem. This theorem states that all four sides and an included angle of one quadrilateral is congruent to the corresponding four sides and the included angle of another quadrilateral. This is proven with the use of figure 4.



The proof of SASSS goes as follows:

1. By segment construction, construct segments AC and segment EG

2. Segment AC is inside quadrilateral ABCD by definition of convex

polygon

3. Segment EG is inside quadrilateral EFGH by definition of convex

polygon

4. Assumptions: Segment AB is congruent to Segment EF

Angle B is congruent to Angle F

Segment BC is congruent to Segment FG

5. Triangle ABC is congruent to Triangle EFG by SAS

6. Segment AC is congruent to Segment EG by CPCF

7. Angle BAC is congruent to Angle FEG by CPCF

8. Angle BCA is congruent to Angle FGE by CPCF

9. Assumption: Segment CD is congruent to Segment GH

10. Implication: Angle ACD is congruent to Angle EGH since Angle BCA is

congruent to Angle FGE because by angle addition, Angle

BCD = Angle BCA + Angle ACD and Angle FGH = Angle FGE =

11. Triangle ACD is congruent to Triangle EGH by SAS

12. Segment AD is congruent to Segment EH by CPCF

13. Therefore SASSS congruence

Another popular congruence theorem for quadrilaterals is the SASAA Theorem, which states that two sides and the three included angles of one quadrilateral is congruent to the corresponding two sides and three angles of another quadrilateral. This will be proven with the use of Figure 5



Proof of the SASAA Theorem:

1. By segment construction, construct segments AC and segment EG

3. Segment EG is inside quadrilateral EFGH by definition of convex

polygon

4. Assumptions: Segment AB is congruent to Segment EF

Angle B is congruent to Angle F

Segment BC is congruent to Segment FG

5. Triangle ABC is congruent to Triangle EFG by SAS

6. Segment AC is congruent to Segment EG by CPCF

7. Angle BAC is congruent to Angle FEG by CPCF

8. Angle BCA is congruent to Angle FGE by CPCF

9. Assumption: Segment CD is congruent to Segment GH

10. Implication: Angle ACD is congruent to Angle EGH since Angle BCA is

Congruent to Angle FGE and by angle addition, Angle BCD = Angle BCA

+ Angle ACD and Angle FGH = Angle FGE + Angle EGH

11. Triangle ACD is congruent to Triangle EGH by SAS

12. Angle CDA is congruent to Triangle EGH by CPCF

13. Therefore SASAA congruence

These are the tools that can be used to prove convex quadrilaterals to be congruent. Here are some questions that can be posed to students:

1. Can ASAA be used as a valid congruence criterion?

References

Kay, David C. College Geometry A Discovery Approach. HarperCollins, New York,1994, 182-191.