2.2: The SAS Theorem
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We have said that two triangles are congruent if all their correspond ing sides and angles are equal, However in some cases, it is possible to conclude that two triangles are congruent, with only partial information about their sides and angles.
Suppose we are told that △ABC has ∠A=53∘, AB=5 inches, and AC=3 inches. Let us attenpt to sketch △ABC. We first draw an angle of 53∘ with a protractor and label it ∠A. Using a ruler, we find the point 5 inches from the vertex on one side of the angle and label it B, On the other side of the angle, we find the point 3 inches from the vertex and label it C, See Figure 2.2.1, There is now only one way for us to complete our sketch of △ABC, and that is to connect points B and C with a line segment, We could now measure BC, ∠B, and ∠C to find the remaining parts of the triangle.


Suppose now △DEF were another triangle, with ∠D=53∘, DE=5 inches, and DF=3 inches. We could sketch △DEF just as we did △ABC, and then measure EF, ∠E, and ∠F (Figure 2.2.2). It is clear that we must have BC=EF, ∠B=∠E, and ∠C=∠F because both triangles were drawn in exactly the same way. Therefore △ABC≅△DEF.
- In △ABC, we say that ∠A is the angle included between sides AB and AC.
- In △DEF, we say that ∠D is the angle included between sides DE and DF.
Our discussion suggests the following theorem:
Two triangles are congruent if two sides and the included angle of one are equal respectively to two sides and the included angle of the other,
In Figure 2.2.1 and 2.2.2, △ABC≅△DEF because AB,AC, and ∠A are equal respectively to DE,DF and ∠D.
We sometimes abbreviate Theorem 2.2.1 by simply writing SAS=SAS.
In △PQR name the angle included between sides
- PQ and QR,
- PQ and PR,
- PR and QR,
Solution
Note that the included angle is named by the letter that is common to both sides, For (1), the letter "Q" is common to PQ and QR and so ∠Q is included between sides PQ and QR. Similarly for (2) and (3).
Answer: (1) ∠Q, (2) ∠P, (3) ∠R.
For the two triangles in the diagram
- list two sides and an included angle of each triangle that are respectively equal, using the infonnation given in the diagram,
- write the congruence statement,
and (3) find x by identifying a pair of corresponding sides of the congruent triangles.
Solution
(1) The angles and sides that are marked the same way in the diagram are assumed to be equal, So ∠B in △ABD is equal to ∠D in △BCD. Therefore, "B" corresponds to "D." We also have AB=CD. Therefore "A" must corresponds to "C". Thus, if the triangles are congruent, the correspondence must be
Finally, BD (the same as DB) is a side common to both triangles, Summaryzing,
△ABD_ △CDB_ Side AB=CD (marked = in diagram)Included Angle ∠B=∠D (marked = in diagram)Side BD=DB (common side)
(2) △ABD≅△CDB because of the SAS Theorem (SAS=SAS).
(3) x=AD=CB=10 because AD and CB are corresponding sides (first and third letters in the congruence statement) a.~d corresponding sides of congruent triangles are equal.
Answer:
(1) AB, ∠B, BD of △ABD=CD, ∠D, DB of △CDB.
(2) △ABD≅△CDB.
(3) x=AD=CB=10.
For the two triangles in the diagram
- list two sides and an included angle of each triangle that are respectively equal, using the information given in the diagram.
- write the congruence statement, and
- find x and y by identifying a pair of corresponding sides of the congruent triangles.
Solution
(1) AC=CE and BC=CD because they are mar!,;:ed the same way. We also know that ∠ACB=∠ECD=50∘ because vertical angles are equal. Therefore "C" in △ABC corresponds to "C" in △CDE. Since AC=CE, we must have that "A" in △ABC corresponds to "E" in △CDE. Thus, if the triangles are congruent, the correspondence must be
We summarize:
△ABC_ △EDC_ Side AC=EC (marked = in diagram)Included Angle ∠ACB=∠ECD (vertical angles are =)Side BC=DC (marked = in diagram)
(2) △ABC≅△EDC because of the SAS theorem. (SAS=SAS)
(3) ∠A=∠E and ∠B=∠D because they are corres:9onding angles of the congruent triangles. ∠D=85∘ because the sum of the angles of △EDC must be 180∘. (∠D=180∘−(50∘+45∘)=180∘−95∘=85∘). We obtain a system of two equations in the two unknowns x and y:
Substituting for x in the first original equation,
2x+y=452(20)+y=4540+y=45y=45−40y=5
Check:
Answer:
- AC, ∠ACB, BC of △ABC = EC,∠ECD,DC of △EDC.
- △ABC≅△EDC.
- x=20,y=5.
The following procedure was used to measure the d.istance AB across a pond: From a point C, AC and BC were measured and found to be 80 and 100 feet respectively. Then AC was extended to E so that AC=CE and BC was extended to D so that BC=CD. Finally, DE we found to be 110 feet.
- Write the congruence statement.
- Give a reason for (1).
- Find AB.
Solution
(1) ∠ACB=∠ECD because vertical angles are equal. Therefore the "C's" correspond, AC=EC so A must correspond to E. We have
(2) SAS=SAS. Sides AC, BC, and included angle C of ABC are equal respectively to EC,DC, and included angle C of ∠EDC.
(3) AB=ED ecause they are corresponding sides of congruent triangles, Since ED=110, AB=110.
Answer
(1) △ABC≅△EDC.
(2) SAS=SAS: AC, ∠C, BC of △ABC=EC, ∠C, DC of △EDC.
(3) AB=110feet.
The SAS Theorem is Proposition 4 in Euclid's Elements, Both our discussion and Suclit's proof of the SAS Theoremimplicitly use the following principle: If a geometric construction is repeated in a different location (or what amounts to the same thing is "moved" to a different location) then the size and shape of the figure remain the same, There is evidence that Euclid used this principle reluctantly, and many mathematicians have since questioned its use in formal proofs, They feel that it makes too strong an assumption about the nature of physical space and is an inferior form of geometric reasoning. Bertrand Russell (1872 - 1970), for example, has suggested that we would be better off assuming the SAS Theorem as a postulate, This is in fact done in a system of axioms for Euclidean geometry devised by David Hilbert (1862 - 1943), a system that has gained much favor with modern mathematicians. Hilbert was the leading exponent of the "formalist school," which sought to discover exactly what assumptions underlie each branch of mathematics and to remove all logical ambiguities, Hilbert's system, however, is too formal for an introductory course in geometry,
Problems
1 - 4. For each of the following (1) draw the triangle with the two sides and the included angle and (2) measure the remaining side and angles:
1. AB=2 inches, AC=1 inch, ∠A=60∘.
2. DE=2 inches, DF=1 inch, ∠D=60∘.
3. AB=2 inches, AC=3 inches, ∠A=40∘.
4. DE=2 inches, DF=3 inches, ∠D=40∘.
5 - 8. Name the angle included between sides
5. AB and BC in △ABC.
6. XY and YZ in △XYZ.
7. DE and DF in △DEF.
8. RS and TS in △RST.
9 - 22. For each of the following.
(1) list two sides and an included angle of each triangle that are respectively equal, using the information given in the diagram,
(2) write the congruence statement,
(3) find x, or x and y.
Assume that angles or sides marked in the same way are eQual.
9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
21. 22.
23. 24.
25. 26.