Mathematics
Isosceles Triangles
Isosceles triangles are a type of triangle that has two sides of equal length. This means that two of its angles are also equal. The third side and angle of an isosceles triangle are typically unequal to the other two sides and angles.
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11 Key excerpts on "Isosceles Triangles"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 1 Introduction to Triangle Triangle A triangle Edges and vertices 3 Schläfli symbol {3} (for equilateral) Area various methods; see below Internal angle (degrees) 60° (for equilateral) A triangle is one of the basic shapes of geometry: a polygon with three corners or ver-tices and three sides or edges which are line segments. A triangle with vertices A , B , and C is denoted ABC . In Euclidean geometry any three non-collinear points determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space). ________________________ WORLD TECHNOLOGIES ________________________ Types of triangles Euler diagram of types of triangles, using the definition that Isosceles Triangles have at least 2 equal sides, i.e. equilateral triangles are isosceles. By relative lengths of sides Triangles can be classified according to the relative lengths of their sides: • In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°. • In an isosceles triangle , two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length; this fact is the content of the Isosceles triangle theorem. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles Isosceles Triangles. The 45-45-90 Right Triangle, which appears in the Tetrakis square tiling, is isosceles. • In a scalene triangle , all sides are unequal. The three angles are also all different in measure. Some (but not all) scalene triangles are also right triangles.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 1 Introduction to Triangle Triangle A triangle Edges and vertices 3 Schläfli symbol {3} (for equilateral) Area various methods; see below Internal angle (degrees) 60° (for equilateral) A triangle is one of the basic shapes of geometry: a polygon with three corners or ver-tices and three sides or edges which are line segments. A triangle with vertices A , B , and C is denoted ABC . In Euclidean geometry any three non-collinear points determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space). ________________________ WORLD TECHNOLOGIES ________________________ Types of triangles Euler diagram of types of triangles, using the definition that Isosceles Triangles have at least 2 equal sides, i.e. equilateral triangles are isosceles. By relative lengths of sides Triangles can be classified according to the relative lengths of their sides: • In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°. • In an isosceles triangle , two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length; this fact is the content of the Isosceles triangle theorem. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles Isosceles Triangles. The 45-45-90 Right Triangle, which appears in the Tetrakis square tiling, is isosceles. • In a scalene triangle , all sides are unequal. The three angles are also all different in measure. Some (but not all) scalene triangles are also right triangles.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 1 Introduction to Triangle Triangle A triangle Edges and vertices 3 Schläfli symbol {3} (for equilateral) Area various methods; see below Internal angle (degrees) 60° (for equilateral) A triangle is one of the basic shapes of geometry: a polygon with three corners or verti-ces and three sides or edges which are line segments. A triangle with vertices A , B , and C is denoted ABC . In Euclidean geometry any three non-collinear points determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space). ________________________ WORLD TECHNOLOGIES ________________________ Types of triangles Euler diagram of types of triangles, using the definition that Isosceles Triangles have at least 2 equal sides, i.e. equilateral triangles are isosceles. By relative lengths of sides Triangles can be classified according to the relative lengths of their sides: • In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°. • In an isosceles triangle , two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length; this fact is the content of the Isosceles triangle theorem. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles Isosceles Triangles. The 45-45-90 Right Triangle, which appears in the Tetrakis square tiling, is isosceles. • In a scalene triangle , all sides are unequal. The three angles are also all different in measure. Some (but not all) scalene triangles are also right triangles.
- Allen Ma, Amber Kuang(Authors)
- 2022(Publication Date)
- For Dummies(Publisher)
165. 14 An isosceles triangle is a triangle that has two congruent sides. In this isosceles triangle, because E is the vertex, DF FE . Set the two sides equal to each other to determine the value of x: 2 10 3 4 10 4 14 x x x x 166. 46 An equilateral triangle is a triangle that has all three sides congruent. Set any two sides of the triangle equal to each other to determine the value of x: 2 5 14 22 1 5 14 22 1 5 36 24 . . . x x x x x To find the length of LI , plug 24 in for x in either of the expressions: LN 24 22 46 167. 9 Using the given relationships, let HP x PY x HY x 3 2 3 A right triangle is a triangle whose sides satisfy the Pythagorean theorem. Set up the Pythagorean theorem and solve for x: a b c x x x x x x x x x x 2 2 2 2 2 2 2 2 2 2 3 2 3 6 9 4 1 2 9 6 9 2 12 + = + + = -+ + + = -+ + = -( ) ( ) ( ) x x x x x x x x x x + = -= -= -= = 9 6 2 12 0 2 18 0 2 9 0 9 2 2 ( ) or Therefore, HP x 9 . 188 PART 2 The Answers ANSWERS 101–200 168. 4 An isosceles triangle is a triangle that has two congruent sides. In this isosceles triangle, because H is the vertex, SH HE . Set the two sides equal to each other to determine the value of x: x x x x x x x x 2 2 12 12 0 3 4 0 3 4 ( )( ) or 169. 4 An equilateral triangle is a triangle that has all three sides congruent. Set any two sides of the triangle equal to each other to determine the value of x: x x x 3 3 3 3 64 64 4 170. Right The sum of the three angles in a triangle is 180°: 4 0 50 180 90 180 90 x x x Because the largest angle in the triangle is 90°, the triangle is a right triangle. 171. Equiangular The sum of the three angles in a triangle is 180°: 6 0 60 180 120 180 60 x x x Because all three angles are equal, the triangle is equiangular. 172. Acute The sum of the three angles in a triangle is 180°: 50 70 180 120 180 60 x x x Because the largest angle in the triangle is 90 and no angles are equal, the triangle is acute.
eBook - PDFMathematics for Elementary Teachers
A Contemporary Approach
- Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
We will discuss other triangles that are classified according to angles after we have clearly defined the size of an angle. TABLE 12.2 MODEL NAME ABSTRACTION DESCRIPTION Bird beak Scalene triangle Triangle with three sides of different lengths. Pennant Isosceles triangle Triangle with at least two sides the same length. Yield sign Equilateral triangle Triangle with three sides the same length. Staircase Right triangle Triangle with one right angle. 570 Chapter 12 Geometric Shapes We will analyze some triangles and identify properties of each one shown in Example 12.5. Identify the characteristics of and name the triangles shown on the square lattices in Figure 12.24. (b) (a) Figure 12.24 SOLUTION The distinctive characteristics of each triangle follow: (a) All three sides are different lengths so it is a scalene triangle. (b) Two of the sides are the same length so the triangle is isosceles and there is a right angle at the upper left vertex. ■ Identify the characteristics of and name the triangles shown on the triangular geoboard in Figure 12.25. SOLUTION The distinctive characteristics of each triangle follow: (a) All three sides are the same length so this triangle is equilateral. (b) All three sides are of different lengths so the triangle is scalene. The triangle also has a right angle at the lower left vertex. (b) (a) Figure 12.25 ■ Quadrilaterals This discussion of quadrilaterals begins by describing in more detail quadrilaterals like those that were covered in level 0. A quadrilateral literally means four (quadri) sides (lateral). The sides, which are line segments, are called sides of the quadrilateral. Also, a quadrilateral has four angles—these are called angles of the quadrilateral. Finally, the vertex of each of the four angles is called a vertex of the quadrilateral. It is interesting to note that triangles are composed of three line segments and yet they are named according to the number of angles.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 8 Triangle and Circle Triangle Triangle A triangle Edges and vertices 3 Schläfli symbol {3} A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A , B , and C is denoted ABC . ________________________ WORLD TECHNOLOGIES ________________________ In Euclidean geometry any three non-collinear points determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space). Types of triangles By relative lengths of sides Triangles can be classified according to the relative lengths of their sides: • In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°. • In an isosceles triangle , two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length; this fact is the content of the Isosceles triangle theorem. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles Isosceles Triangles. • In a scalene triangle , all sides are unequal. The three angles are also all different in measure. Some (but not all) scalene triangles are also right triangles. Equilateral Isosceles Scalene By internal angles Triangles can also be classified according to their internal angles, measured here in degrees. • A right triangle (or right-angled triangle , formerly called a rectangled triangle ) has one of its interior angles measuring 90° (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side of the right triangle. The other two sides are called the legs or catheti (singular: cathetus ) of the triangle.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 2 Triangle and Circle Triangle Triangle A triangle Edges and vertices 3 Schläfli symbol {3} A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A , B , and C is denoted ABC . ________________________ WORLD TECHNOLOGIES ________________________ In Euclidean geometry any three non-collinear points determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space). Types of triangles By relative lengths of sides Triangles can be classified according to the relative lengths of their sides: • In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°. • In an isosceles triangle , two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length; this fact is the content of the Isosceles triangle theorem. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles Isosceles Triangles. • In a scalene triangle , all sides are unequal. The three angles are also all different in measure. Some (but not all) scalene triangles are also right triangles. Equilateral Isosceles Scalene By internal angles Triangles can also be classified according to their internal angles, measured here in degrees. • A right triangle (or right-angled triangle , formerly called a rectangled triangle ) has one of its interior angles measuring 90° (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side of the right triangle. The other two sides are called the legs or catheti (singular: cathetus ) of the triangle.
eBook - PDF- Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
V T R U S W Exercise 33 D C B E A Run Rise A B C D E Exercises 41, 42 S T R V (a) W X Y Z V (b) Exercises 43, 44 B A C F D E 3.3 ■ Isosceles Triangles 153 In an isosceles triangle, the two sides of equal length are its legs, and the third side is the base. See Figure 3.22. The point at which the two legs meet is the vertex of the isosceles triangle, and the angle formed by the legs (and opposite the base) is the vertex angle. The two remaining angles are the base angles of the isosceles triangle. If AC _ BC in Figure 3.23, then nABC is isosceles with legs AC and BC , base AB , vertex C, vertex angle C, and base angles at A and B. With AC _ BC , we see that the base AB of this isosceles triangle is not necessarily the “bottom” side. Helping lines known as auxiliary lines are needed to prove many theorems. To this end, we consider some of the lines (line segments) that may prove helpful. Each angle of a triangle has a unique angle bisector; this may be indicated by a ray or segment from the vertex of the bisected angle. See Figure 3.24(a). Just as an angle bisector begins at the vertex of an angle, the median also joins a vertex to the midpoint of the opposite side. See Figure 3.24(b). Generally, the median from a vertex of a triangle is not the same as the angle bisector from that vertex. An altitude is a line segment drawn from a vertex to the opposite side so that it is perpendicular to the opposite side. See Figure 3.24(c). Finally, the perpendicular bisector of a side of a triangle is shown as a line in Figure 3.24(d). A segment or ray could also perpendicularly bisect a side of the triangle.
eBook - PDF- J Daniels, N Solomon, J Daniels, N Solomon(Authors)
- 2014(Publication Date)
- Future Managers(Publisher)
X U T R S O 25° a c b Determine the numerical values of a , b and c , with reasons. 206 Mathematics: Hands-On Training Geometry of triangles A triangle is defined as a polygon with three angles and therefore three sides or edges, which are line segments. c b a B C A Triangles are classified according to their sides or interior angles. Type of triangle Description Sketch Scalene triangle Sides are all different lengths and all three angles different Isosceles triangle Two equal sides and the angles opposite the sides are also equal x x Equilateral triangle All three sides are equal and each angle measures 60° 60° 60° 60° Acute-angled triangle All interior angles are less than 90° x z y Obtuse-angled triangle One interior angle is more than 90° x Right-angled triangle One interior angle is a right angle, that is, 90° Two main properties of the interior angles of a triangle Description Sketch The sum of the interior angles of a triangle is equal to 180° x y z ˆ ˆ ˆ x y z + + = ° 180 The exterior angle of a triangle is equal to the sum of the two opposite interior angles x y z ˆ ˆ ˆ z x y = + Note Isosceles is a Greek word, meaning “equal legs”. 207 Chapter 3 Space, shape and measurement Additional to the main properties of the interior angles of a triangle, the Theorem of Pythagoras states that in a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. c b a a 2 + b 2 = c 2 Example 5 3 BCD is an isosceles triangle with BC = BD. A B C D 110° z y x Compute the numerical values x , y and z , with reasons. Solution If 3 BCD is an isosceles triangle with BC = BD, BC ˆ D = BDC ˆ , therefore ˆ x = ˆ z . BC ˆ D + BDC ˆ = ABD ˆ • Exterior angle of a triangle is equal to the sum of the two opposite interior angles. ˆ x + ˆ z = 110° • Substitute ˆ z = ˆ x , as 3 BCD is an isosceles triangle. ˆ x + ˆ x = 110° 2 ˆ x = 110° ˆ x = 55° CBD ˆ + ABD ˆ = 180° • Straight angle, ABC ˆ = 180°.
eBook - PDFThe Story of Proof
Logic and the History of Mathematics
- John Stillwell(Author)
- 2022(Publication Date)
- Princeton University Press(Publisher)
Equality of parallelogram sides implies, by subtraction and addition again, that the 28 CHAPTER 2 EUCLID Figure 2.11 : Equal rectangle and parallelogram triangles have equal width, and then equality of angles implies they are congruent, by SAS. Next, one notices that any triangle, added to a copy of itself, makes a parallelogram (figure 2.12). α α β γ β γ Figure 2.12 : Triangle and parallelogram It follows that the area of a triangle is half that of a parallelogram with the same base and height, a result Euclid uses in his proof of the Pythagorean theorem. (This makes for a somewhat longer path to the theorem than the one described in the previous section.) In general, Euclid considers regions “equal” if one can be converted to the other by addition and subtraction of finitely many equal figures. Remarkably, this definition coincides with the modern concept of “equal area” for polygons. However, the “product” ab has very limited algebraic properties. One has the commutative law ab = ba, because the rectangle with adjacent sides a and b is the same as the rectangle with adjacent sides b and a. And one has the distributive law a(b + c)= ab + ac, 2.6 NUMBER THEORY AND INDUCTION 29 which is Euclid’s proposition 1 of book 2. However, there is very little else. A product ab of two lengths is not a length, so if c is a length then ab + c does not make sense. Also, while abc is considered meaningful (it is a box with adjacent edges a, b, and c), abcd is not, because the Greeks did not believe there could be mutually perpendicular lengths a, b, c, and d. Another limitation is that finitely many additions and subtractions do not generally work for curved regions; for example, one would not expect to find a square equal in area to a circle by this method. More disappointing, the method does not generally work for polyhedra either.
eBook - PDFDr. Math Introduces Geometry
Learning Geometry is Easy! Just ask Dr. Math!
- (Author)
- 2004(Publication Date)
- Jossey-Bass(Publisher)
Dear Lorraine, A useful trick in trying to remember these names and many others is to think about the pieces of words that they’re made from. For example, “lateral” always has to do with sides. The fins on the side of a fish are “lateral fins” (as opposed to “dorsal fins,” which are on the back). Trade between two countries is “bilateral trade.” In football, a “lateral” is when the quarterback tosses the ball to the side instead of throwing it forward, as in a regular pass. And so on. So “equilateral” means “equal sides,” and in fact, all the sides of an equilateral triangle are equal. (That means its angles are also the same, and figures with sides and angles all the same are called regular. The prefix “iso-” means “same.” An “isometric” exercise is one in which the position of the muscles stays the same (as when you press your two hands together). Two things that have the same shape are “isomorphic.” And so on. “Sceles” comes from the Greek “skelos,” which means “leg.” So an isosceles triangle is one that has the “same legs” as opposed to “equal sides.” In an equilateral triangle, all the sides are the same; but in an isosceles triangle, only two of the sides, called the legs, must have the same measure. The other side is called the base. 32 Dr. Math Introduces Geometry Introduction to Two-Dimensional (2-D) Geometric Figures 33 “Scalene” comes from the Greek word for “uneven,” and a sca- lene triangle is uneven: no side is the same length as any other. But to be honest, usually I just remember that “scalene” means “not equilateral or isosceles.” So what can be learned from this? One lesson is that when you’re having trouble remembering a word, it’s often a good idea to consult a dictionary to find out the history of the word, because understand- ing how a word was created can help it seem less random. Another lesson is that many of the words that we find in math and science were made up by people who were familiar with Latin and Greek.
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