ASA Theorem
The ASA (Angle-Side-Angle) Theorem states that if two triangles have two corresponding angles and the included side congruent, then the triangles are congruent. This theorem is a fundamental concept in geometry and is used to prove the congruence of triangles in various geometric proofs.
6 Key excerpts on "ASA Theorem"
eBook - ePubThe Metaphysics of the Pythagorean Theorem
Thales, Pythagoras, Engineering, Diagrams, and the Construction of the Cosmos out of Right Triangles
- Robert Hahn(Author)
- 2017(Publication Date)
- SUNY Press(Publisher)
...What these propositions show is thatif certain things are equal in two triangles, other things will be equal as well.And the overall, general strategy of Book I shows that the common notions are axioms forequality and include a fundamental test, namely, that things that can be superimposed are equal; I.4 begins an exploration of equal figures that are equal via superimposition, that is, they are equal and identical in shape; I.35 shows that figures can be equal that cannot be superimposed, that is, figures can be equal but not identical in shape; I.47 shows that it is possible to have two figures of a given shape, a square, that can be equal to a third figure of the same shape.I.4: (SAS—Side-Angle-Side) Two triangles are equal if they share two side lengths in common and the angle between them:Figure 1.12.I.8: (SSS—Side-Side-Side) Two triangles are equal (i.e., congruent) if they share the lengths of all three sides:Figure 1.13.I.26: (ASA—Angle-Side-Angle) Two triangles are equal (i.e., congruent) if they share two angles in common and the side length between them:Figure 1.14.The third theorem of equality, I.26, is explicitly credited to Thales by Proclus on the authority of Eudemus.3Two triangles are shown to be equal—congruent—if two sides and the angle contained by them are equal (SAS), if all three sides are equal in length (SSS), and iftwo angles and the side shared by them either adjoining or subtending are equal (ASA). The last of these is credited to Thales because Eudemusinferredit was needed for the measurement of the distance of a ship at sea. We shall investigate such a measurement inchapter 2. An understanding of it was needed to grasp the measurement, by one approach, while an understanding of similarity was needed by all the other approaches. But let us get clear about the general matter; one does not come to grasp angle-side-angle equality without having recognized side-side-side and side-angle-side equality...
- Rebecca Dayton(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...The included angle for both triangles is ∠ A, so ∠ A ≅ ∠ A. Therefore, Δ ABC ∼ Δ ADE by SAS. Measurements in Similar Triangles As previously stated, the corresponding sides of similar triangles are in the same ratio and their corresponding angles are congruent. If the ratio of the corresponding side lengths is, the ratio of the areas is. Example: Given two similar triangles, the area of the first is 27 in 2. If the lengths of the sides in the first triangle are triple those of the second, what is the area of the second triangle? The ratio of the side lengths is because the first triangle is triple (three times) the second. Therefore, the ratio of the areas is. To find the area of the second triangle set up the following proportion: The area of the second triangle is 3 in 2. Exercise 2 1. Determine if Δ LMN ≅ Δ WXY. 2. Melissa stated that Δ ABC ≅ Δ CDA by the AAS theorem. Is she correct? 3. Given two similar triangles, the area of the first triangle is 20 ft 2 and the area of the second triangle is 45 ft 2...
eBook - ePub- Thomas Heath(Author)
- 2015(Publication Date)
- Routledge(Publisher)
...5.) F IG. 7 (2) The theorem that the angles of a triangle are together equal to two right angles depends upon the Euclidean theory of parallels. There are two traditional proofs the Pythagorean, handed down by Proclus on the authority of Eudemus, and the Euclidean (I. 32). F IG. 8 Given a triangle ABC, in the Pythagorean proof a straight line DA E is drawn through A parallel to BC ; Euclid produces BC to D and draws through C, in the angle ACD, a straight line parallel to BA. In the former case it is shown that the angles B, C are equal to the angles DAB, EAC, the alternate angles, respectively; hence the three angles together are equal to the sum of the three angles about A, which sum is equal to two right angles (I. 13). In the second case the angle A is proved eaual to the alternate angle ACE, and the angle B to the exterior angle ECD, and hence the sum of the three angles A, B, C is equal to the sum of the three angles about C, which sum again is equal to two right angles (I. 13). The proof known to Aristotle was that of Eucl. I. 32, as we gather from the wording of another passage (Metaph. Θ. 9. 1051 a 24). F IG. 9 If now, in Euclid’s figure, the three angles of the triangle are together greater than the sum of the three angles at C, either the angle A must be greater than the alternate angle ACE, or the angle B must be greater than the exterior angle ECD, or both. Suppose the angle A to be greater than the angle ACE. Then, if we produce BA to F, the angles FAC, ACE are together less than the sum of the angles BAC, FAC, and therefore less than two right angles. Hence, as before, the supposed parallels BF, CE will meet beyond F, E. The hypothesis that the angle ABC is greater than the angle ECD has already been dealt with. (g) Incommensurability of diagonal and Zeno’s bisection. Ckrystal’s proof An. Pr. II. 17...
eBook - ePubDyslexia, Dyscalculia and Mathematics
A practical guide
- Anne Henderson(Author)
- 2013(Publication Date)
- Routledge(Publisher)
...The theorem is used to find the length of a side in a right-angled triangle when the lengths of the other two sides are known. The theorem states: In a right-angled triangle the area of the square on the hypotenuse is equal to the sum ofthe areas of the squares on the other two sides. How to help ● Find a picture or drawing depicting Pythagoras and talk about the man. Often students cannot remember the word Pythagoras on its own but they remember the picture and with a little prompting remember his theorem. ● Make a memory card (number 24, see page 144) to show that the longest side, the hypotenuse, is always opposite to the right angle. ● Present the triangle in different ways. Figure 9.21 A right-angled triangle presented in different way to show the hypotenuse A good multi-sensory exercise is to draw a triangle with the two smaller sides with lengths of 3 cm and 4 cm so that the hypotenuse will be 5 cm. Draw squares on the three sides. Cut the 9 cm into 1 cm squares and fit these around the 16 cm square to make a 25 cm square. Figure 9.22 A multi-sensory exercise to explain the Pythagorean theorem To find the length of the hypotenuse of a triangle when the other two sides are known. ● Square the numbers (multiply them by themselves). ● Add the numbers. ● Take the square root to get the answer. Section I: Trigonometry Abbreviations O = opposite side A = adjacent side H = hypotenuse S = sine (angle) C = cosine (angle) T = tangent (angle) Figure 9.23 Identify the sides of the triangle in relation to the angle XZY How to help ● Identify the sides of the triangle, discussing which sides are adjacent and which are opposite. This seemingly simple item can take a great deal of time as many students cannot connect up angles and lines easily. ● Indicate with an arrow the hypotenuse. ● The following mnemonic may help:...
eBook - ePub- Mel Friedman(Author)
- 2012(Publication Date)
- Research & Education Association(Publisher)
...By the Pythagorean Theorem, the lengths of the three sides of a right triangle are related by the formula c 2 = a 2 + b 2 where c is the hypotenuse and a and b are the other two sides (the legs). The Pythagorean Theorem is discussed in more detail in the next section. An altitude, or height, of a triangle is a line segment from a vertex of the triangle perpendicular to the opposite side. For an obtuse triangle, the altitude sometimes is drawn as a perpendicular line to an extension of the opposite side. The area of a triangle is given by where h is the altitude and b is the base to which the altitude is drawn. A line segment connecting a vertex of a triangle and the midpoint of the opposite side is called a median of the triangle. A line that bisects and is perpendicular to a side of a triangle is called a perpendicular bisector of that side. An angle bisector of a triangle is a line that bisects an angle and extends to the opposite side of the triangle. The line segment that joins the midpoints of two sides of a triangle is called a midline of the triangle. An exterior angle of a triangle is an angle formed outside a triangle by one side of the triangle and the extension of an adjacent side. PROBLEM The measure of the vertex angle of an isosceles triangle exceeds the measure of each base angle by 30°. Find the value of each angle of the triangle. SOLUTION In an isosceles triangle, the angles opposite the congruent sides (the base angles) are, themselves, congruent and of equal value. Therefore, 1. Let x = the measure of each base angle 2. Then x + 30 = the measure of the vertex angle We can solve for x algebraically by keeping in mind that the sum of all the measures of the angles of a triangle is 180°. Therefore, the base angles each measure 50°, and the vertex angle measures 80°. THE PYTHAGOREAN THEOREM The Pythagorean Theorem pertains to a right triangle, which, as we saw, is a triangle that has one 90° angle...
- Sandra Rush(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...If you know one side and two angles in a triangle, the lengths of the other two sides are predetermined. Can you see above (right) that if we extend the two partial sides, they will meet at a point (let’s call it T) that depends on R and S ? Thus, there is one and only one triangle that can be drawn if you know the two angles and the distance between them. Again, finding the lengths of any of the two remaining sides requires a knowledge of trigonometry. Note that if all you know are the lengths of two sides of a triangle and nothing about the angles, you don’t automatically know the third side of the triangle—it actually could be anything, depending on what the angle between the two sides is. In the figures in this chapter, if sides are equal, they will be marked with the same tick marks (either one or two). Likewise, if angles are equal, they will be marked with the same angle mark. Right angles are marked with a small box. Types of Triangles Triangles can be classified by their side measurements. Scalene : A triangle in which the lengths of all of the sides are different. Isosceles: A triangle with two equal side lengths. The two equal sides are called the legs, and the third side is called the base. Equilateral : All three side lengths are equal. The name comes from the two parts of the word: equi (“equal”) and lateral (“sides”). An equilateral triangle is also called an equiangular triangle because, if all three sides are equal, so are all three angles. The two parts of the word equiangular are: equi (“equal”) and angular (“angle”). The angles in an equilateral triangle are each 60° because there are three of them and they have to add up to 180°. The sum of the lengths of any two sides of a triangle must be greater than the length of the longest side, or we don’t have a triangle (the two sides won’t meet)...
