Mathematics
HL ASA and AAS
HL ASA and AAS are two methods used to prove that two triangles are congruent. ASA stands for Angle-Side-Angle, which means that if two triangles have two angles and the included side equal, then they are congruent. AAS stands for Angle-Angle-Side, which means that if two triangles have two angles and a non-included side equal, then they are congruent.
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No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
The ASA Postulate was contributed by Thales of Miletus (Greek). In most systems of axioms, the three criteria— SAS , SSS and ASA —are established as theorems. In the School Mathematics Study Group system SAS is taken as one (#15) of 22 postulates. • AAS (Angle-Angle-Side): If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent. ________________________ WORLD TECHNOLOGIES ________________________ • RHS (Right-angle-Hypotenuse-Side): If two right-angled triangles have their hypotenuses equal in length, and a pair of shorter sides are equal in length, then the triangles are congruent. Side-Side-Angle The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS, or Angle-Side-Side) does not by itself prove congruence. In order to show congruence, additional information is required such as the measure of the corresponding angles and in some cases the lengths of the two pairs of corresponding sides. There are four possible cases: If two triangles satisfy the SSA condition and the corresponding angles are either obtuse or right, then the two triangles are congruent. In this situation, the length of the side opposite the angle will be greater than the length of the adjacent side. Where the angle is a right angle, also known as the Hypotenuse-Leg (HL) postulate or the Right-angle-Hypotenuse-Side (RHL) postulate, the third side can be calculated using the Pythagorean Theorem thus allowing the SSS postulate to be applied. If two triangles satisfy the SSA condition and the corresponding angles are acute and length of the side opposite the angle is greater than or equal to the length of the adjacent side, then the two triangles are congruent.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
• AAS (Angle-Angle-Side): If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent. • RHS (Right-angle-Hypotenuse-Side): If two right-angled triangles have their hypotenuses equal in length, and a pair of shorter sides are equal in length, then the triangles are congruent. Side-Side-Angle The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS, or Angle-Side-Side) does not by itself prove congruence. In order to show congruence, additional information is required such as the measure of the corresponding angles and in some cases the lengths of the two pairs of corresponding sides. There are four possible cases: If two triangles satisfy the SSA condition and the corresponding angles are either obtuse or right, then the two triangles are congruent. In this situation, the length of the side opposite the angle will be greater than the length of the adjacent side. Where the angle is a right angle, also known as the Hypotenuse-Leg (HL) postulate or the Right-angle-Hypotenuse-Side (RHL) postulate, the third side can be calculated using the Pythagorean Theorem thus allowing the SSS postulate to be applied. If two triangles satisfy the SSA condition and the corresponding angles are acute and length of the side opposite the angle is greater than or equal to the length of the adjacent side, then the two triangles are congruent. If two triangles satisfy the SSA condition and the corresponding angles are acute and length of the side opposite the angle is equal to the length of the adjacent side multiplied by the sine of the angle, then the two triangles are congruent.
eBook - PDF- Michael Hvidsten(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
A B C X Y Z This proposition is one of the axioms in Hilbert’s axiomatic basis for Euclidean geometry. Hilbert chose to make this result an axiom rather than a theorem to avoid the trap that Euclid fell into in his proof of the SAS result. In Euclid’s proof, he moves points and segments so as to overlay one triangle on top of the other and thus prove the result. How-ever, there is no axiomatic basis for such transformations in Euclid’s original set of five postulates. Most modern treatments of Euclidean ge-ometry assume SAS congruence as an axiom. Birkhoff chooses a slightly different triangle comparison result, the SAS condition for triangles to be similar , as an axiom in his development of Euclidean geometry. Theorem 2.11. ( ASA: Angle-Side-Angle , Prop. 26 of Book I) If in two triangles there is a correspondence in which two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. A B C X Y Z 64 � Exploring Geometry Theorem 2.12. ( AAS: Angle-Angle-Side , Prop. 26 of Book I) If in two triangles there is a correspondence in which two angles and the side subtending one of the angles are congruent to two angles and the side sub-tending the corresponding an-gle of another triangle, then the triangles are congruent. A B C X Y Z Theorem 2.13. ( SSS: Side-Side-Side , Prop. 8 of Book I) If in two triangles there is a correspondence in which the three sides of one triangle are congruent to the three sides of the other triangle, then the tri-angles are congruent. A B C X Y Z We note here for future reference that the four fundamental triangle congruence results are independent of the parallel postulate; that is, their proofs do not make reference to any result based on the parallel postulate. Let’s see how triangle congruence can be used to analyze isosceles triangles. Definition 2.15. An isosceles triangle is a triangle that has two sides congruent.
eBook - ePub- Marshall Whittlesey(Author)
- 2019(Publication Date)
- CRC Press(Publisher)
π, and this sum is not generally the same from one triangle to another.However, the angle sum theorem in the plane depends on the parallel postulate, and in fact the SAA congruence theorem in the plane does not depend on the parallel postulate. The reader will find a proof of SAA congruence in the plane in [Mo1963] (or [MD1982]) which makes use of the exterior angle theorem in the plane (which states that an exterior angle of a triangle has measure greater than the measures of either of the opposite interior angles). The spherical analogue of this theorem also turns out to be false.In plane geometry the SSA correspondence does not guarantee congruence of triangles, although we might say that it almost does. Knowledge of the measures of two sides and an angle that is not included leads to the so-called “ambiguous case” in determining the other sides of the triangle. The other sides are uniquely determined if the angle is a right angle, but otherwise there are either two possibilities for the other sides and angles, or the triangle cannot be constructed at all.We can make an immediate observation for spherical geometry: either SSA and SAA both guarantee congruence of triangles or neither does. The reason is that if one guarantees congruence, then we could prove that the other does as well by employing the same argument with polar triangles used in the proofs of ASA and AAA congruence. For example, suppose there were an SAA congruence theorem. To prove that an SSA correspondence guarantees congruence, we would consider two triangles which have an SSA correspondence. Their polar triangles would have an SAA correspondence by Theorem 11.19, and hence would be congruent. The congruence of the pair of polar triangles would in turn guarantee (by Theorem 11.19 again) that the original triangles be congruent. A similar line of reasoning would allow us to use an SSA congruence theorem to prove an SAA congruence theorem.
eBook - PDFMathematics for Elementary Teachers
A Contemporary Approach
- Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
Example 14.2 gives an illustration. From the result of Example 14.2, we can conclude that ∠ ≅ ∠ ABP ACP in Figure 14.6, since these angles correspond in the congruent triangles. Thus the base angles of an isosceles triangle are congruent. In Section 14.3, Set A, Problem 21, we investigate the converse result, namely, that if two angles of a triangle are congruent, the triangle is isosceles. Figure 14.6 ASA Triangle Congruence A second congruence property for triangles involves two angles and their common side. Suppose that we are given two angles, the sum of whose measures is less than 180°. Also suppose that they share a common side. Figure 14.7 shows an example. The extensions of the noncommon sides of the angles intersect in a unique point C . That is, a unique triangle, nABC, is formed. This observation can be generalized as the angle–side–angle congruence property. Figure 14.7 722 Chapter 14 Geometry Using Triangle Congruence and Similarity Angle–Side–Angle (ASA) Congruence If two angles and the included side of a triangle are congruent, respectively, to two angles and the included side of another triangle, then the two triangles are congru- ent. Here, n n ABC DEF ≅ . P R O P E R T Y NOTE: Although we are assuming ASA congruence as a property, it actually can be shown to be a theorem that follows from the SAS congruence property. Example 14.3 illustrates an application of the ASA congruence property. NCTM Standard All students should create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship. Show that the diagonal in Figure 14.8 divides a parallelogram into two congruent triangles. Figure 14.8 SOLUTION Line DB is a transversal for lines AB and DC. Since AB DC || , we know (i) ∠ ≅ ∠ ABD CDB by the alternate interior angle property. Similarly, ∠ADB and ∠CBD are alternate interior angles formed by the transversal BD and parallel lines AD and BC.
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