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Transcript
Triangle Congruence, SAS, and Isosceles Triangles Recall the definition of a triangle: A triangle is the union of three segments (called its sides), whose endpoints (called its vertices) are taken, in pairs, from a set of three noncollinear points. Thus, if the vertices of a triangle are A, B and C, then its sides are , , and , and the triangle is then the set defined by , denoted by ªABC. The angles of ªABC are pA = pBAC, pB = pABC, and pC = pACB. There are other associated terms that we could formally define. These include: • A side opposite an angle: In ªABC, side and so on. • A vertex opposite a side: In ªABC, vertex B is opposite side is opposite pC, , and so on. • An angle included between two sides: In ªABC, pA is included between • and , etc. A side included between two angles: In ªABC, side included between pB and pC, etc. is Recall also the definition of congruence for segments and angles: Two segments and are said to be congruent (we write ) if and only if AB = XY. Two angles pABC and pXYZ are said to be congruent (and we write pABC pXYZ) if and only if µpABC = µpXYZ. Definition: A correspondence between vertices of two triangles is a one-to-one, onto mapping from the set of vertices of the first triangle to the second. Intuitively, a correspondence is a matching of vertices between two triangles. This matching also establishes a correspondence between sides and angles. Shorthand: ABC : XYZ iff A : X, B : Y, and C : Z. This also establishes : , : , : pA : pX, pB : pY, pC : pZ. , as well as Definition: If, under a certain correspondence between the vertices of two triangles, corresponding sides and corresponding angles are congruent, the triangles are said to be congruent. Recall that a definition is really an “if and only if” statement. We could reword this as: Two triangles are said to be congruent if and only if there is a correspondence between the vertices such that corresponding sides and corresponding angles are congruent. We abbreviate this definition by saying, “Corresponding parts of congruent figures are congruent,” and further abbreviate this to CPCF. We use this expression all the time as a shorthand way to express the definition of congruence. Notation: If ªABC is congruent to ªXYZ, we write ªABC ªXYZ. Note: This notation not only says that the two triangles ªABC and ªXYZ are congruent, but also establishes the correspondence under which they are congruent. (The correspondence is ABC : XYZ). Thus, ªABC ªXYZ if, and only if, , , pA pX, pB pY, pC pZ Properties of Congruence for Triangles: 1. 2. 3. ªABC ªABC (Symmetric) If ªABC ªXYZ, then ªXYZ ªABC (Transitive) If ªABC ªXYZ, and ªXYZ ªUVW, then ªABC ªUVW (Reflexive) These are not in your textbook, but they follow easily from the definition, since congruence is based on equality of the real numbers that are the measures of the segments and angles, and equality of real numbers is reflexive, symmetric, and transitive. Note: Because the statement ªABC ªXYZ is a statement about a particular correspondence, the statements ªABC ªCBA says something very different from ªABC ªABC. Among other things, ªABC ªCBA implies that pApC, which is not implied by ªABC ªABC The SAS Postulate Under the correspondence ABC : XYZ, let two sides and the included angle of ªABC be congruent, respectively, to the corresponding two sides and the included angle of ªXYZ (that is, for example, , , and pA pX). Then ªABC ªXYZ. Before we fully accept this as an axiom we will show that it is independent of our current axiom set, by showing a model in which it is true and one in which it is false. The model in which it is true is regular Euclidean space as developed in the Cartesian coordinate system. Since trigonometry holds in this system, we can show that the third side of any triangle can be uniquely determined if we know the opposite angle and the two other sides (by using the Law of Cosines) and the other two angles will be uniquely determined from the Law of Sines. In the Taxicab geometry, in which angles are measured in the same way as in the regular Euclidean model, SAS does not hold, as can be demonstrated by these two triangles: Definition: A triangle is said to be isosceles if it has a pair of congruent sides. The angles opposite the congruent sides are the base angles and the third angle is the apex of the triangle. The side between the base angles is the base. Theorem (The Isosceles Triangle Theorem): If a triangle is isosceles, its base angles are congruent. Two proofs: ~ Let ÎABC be isosceles with we can find a point D interior to s of so . By a previous theorem uch that . By crossbar, this ray intersects is the bisector at point E. Now (given), and pACE pBCE (bisector) and and by CPCF, . ~ Let ÎABC be isosceles with have by SAS. Then so the base angles are congruent. . Since , , we by CPCF, Yes, I know this seems like cheating, but it demonstrates the power of the correspondence that is part of the definition of congruence. This theorem is called the pos asinorum or “bridge of asses” because of yet a third proof, the one used by Euclid. He extended the segments to points D and E at equal distances from B and A, respectively, then formed . The result looks a little like a bridge. It is also said that this proof was the bridge over which students of mathematics had to pass if they were to be successful in their study of geometry. It may be worth thinking about how you would use this diagram in a proof.