A triangle that is not 180 degrees is any non-Euclidean triangle. In Euclidean geometry, a triangle has 180 degrees, but in non-Euclidean geometries, such as spherical geometry or hyperbolic geometry, the sum of the angles in a triangle is not always equal to 180 degrees.
In spherical geometry, also known as elliptic geometry or Riemannian geometry, a triangle has more than 180 degrees. This is because spherical geometry takes place on the surface of a sphere, and the three angles of a triangle on a sphere can add up to more than 180 degrees. This is because the surface of a sphere is curved, and lines on the surface of a sphere are curved as well.
In spherical geometry, the sum of the angles in a triangle is proportional to the surface area of the triangle, and the greater the surface area, the more the angles add up to.
In hyperbolic geometry, also known as Lobachevskian geometry or Bolyai-Lobachevskian geometry, a triangle has less than 180 degrees. This is because in hyperbolic geometry, lines diverge and never intersect, which leads to a negative curvature. In hyperbolic geometry, the sum of the angles in a triangle is proportional to the surface area of the triangle, just like in spherical geometry.
However, in hyperbolic geometry, the greater the surface area, the less the angles add up to.
There are many triangles that are not 180 degrees, and these are found in non-Euclidean geometries such as spherical geometry and hyperbolic geometry. These triangles exist because the geometry of the space they exist in is curved or negatively curved, leading to different rules and laws than in traditional Euclidean geometry.
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Does an isosceles triangle 180?
No, an isosceles triangle does not have an angle measure of 180 degrees. An isosceles triangle is a triangle that has two sides of equal length and two angles of equal measure. Since the sum of the angles in a triangle is always 180 degrees, the two equal angles in an isosceles triangle must have a total measure of less than 180 degrees.
To understand why an isosceles triangle cannot have two angles that measure 90 degrees each (which would add up to a total measure of 180 degrees), consider the following: if two angles of a triangle measure 90 degrees each, then the third angle must measure 0 degrees, which means that the third side of the triangle has no length.
In other words, the two sides that meet at the 90-degree angles are actually collinear, which means they lie on the same line. This does not satisfy the definition of a triangle, which requires three sides that do not lie on the same line.
Similarly, if an isosceles triangle had all three angles measuring 60 degrees, then it would be an equilateral triangle rather than an isosceles triangle.
Therefore, an isosceles triangle can have two equal angles that measure less than 90 degrees each, or it can have one angle that measures more than 90 degrees (which would make it an obtuse isosceles triangle), but it cannot have two angles that add up to 180 degrees.
Are all right triangles 180?
No, all right triangles do not equal 180 degrees. A right triangle is a triangle with one angle equaling 90 degrees, which means the other two angles must add up to 90 degrees for the triangle to be classified as a right triangle. Therefore, the sum of angles in a right triangle is 180 degrees, just like any other triangle.
However, the individual angles themselves do not add up to 180 degrees, as one of the angles is already set at 90 degrees. For example, a right triangle with angles measuring 30 and 60 degrees will have a sum of 180 degrees, but it cannot be a right triangle as it does not have an angle measuring 90 degrees.
The only way for a triangle to have a sum of 180 degrees and one angle measuring 90 degrees is for it to be a right triangle.
Is a triangle 90 or 180?
A triangle can be either 90 degrees or 180 degrees depending on the type of triangle you are referring to. If we are talking about a traditional Euclidean triangle, then it has three sides and three angles, and the sum of all the angles in the triangle will always be equal to 180 degrees. This is known as the Triangle Angle Sum Theorem.
It doesn’t matter what the size, shape or orientation of the triangle is, the sum of all the angles within the triangle will always be 180 degrees.
However, if we are referring to a right triangle, which is a triangle that has one angle measuring exactly 90 degrees, then we can say that it has one angle measuring 90 degrees and the sum of the other two angles will always be equal to 90 degrees. This is known as the Pythagorean Theorem, and it is a fundamental theorem in mathematics that deals with the relationship between the sides of a right triangle.
So, to sum it up, a triangle can be either 90 degrees or 180 degrees depending on what kind of triangle you are referring to. If it is a traditional Euclidean triangle, then it will have a sum of angles equal to 180 degrees. If it is a right triangle, then it will have one angle measuring 90 degrees and the sum of the other two angles will always be equal to 90 degrees.
Do obtuse triangles add up to 180?
No, obtuse triangles do not add up to 180 degrees. An obtuse triangle is defined as a triangle that has one angle measuring more than 90 degrees. In fact, the sum of the measures of angles in an obtuse triangle is always greater than 180 degrees, since the other two angles must each measure less than 90 degrees.
To see why, we can use the fact that the sum of the measures of the angles in any triangle is always 180 degrees. If we let A, B, and C represent the measures of the three angles in an obtuse triangle, with angle C being the obtuse angle, we can write the equation:
A + B + C = 180
If we assume that A and B are both less than 90 degrees, then we know that:
A + B < 180 - C
This is because if A + B were equal to or greater than 180 – C, then angle C would not be obtuse. But if A + B is less than 180 – C, then we can substitute this inequality into our original equation and get:
A + B + C < 180
Thus, we see that the sum of the measures of the angles in an obtuse triangle is always less than 180. In fact, the closer the other two angles are to 90 degrees, the larger the obtuse angle must be in order to satisfy the equation A + B + C = 180. For example, if A and B were both 89 degrees, then C would have to be at least 2 degrees greater than 90 in order for the equation to be true, resulting in a sum greater than 180 degrees.
While all triangles must have a sum of 180 degrees for their angles, obtuse triangles do not have angles that add up to 180 degrees specifically.
What is 180 degrees in a triangle?
In a triangle, 180 degrees is the total sum of all the angles within the triangle. A triangle is a closed figure with three sides and three angles. The sum of the internal angles of a triangle can be calculated by using a simple formula.
The formula states that the sum of the interior angles of a triangle is always equal to 180 degrees. Therefore, regardless of the type of triangle (equilateral, isosceles, or scalene), the sum of its angles will always add up to 180 degrees.
It is important to note that the angles within a triangle can have different measures depending on the type of triangle. For example, an equilateral triangle, which has three equal sides, will have three equal angles, each measuring 60 degrees. An isosceles triangle, which has two equal sides, will have two equal angles and one different angle, with the sum of the two equal angles equaling the third angle.
Finally, a scalene triangle, which has no equal sides or angles, will have three different angle measures.
Knowing that the sum of angles in a triangle is 180 degrees is fundamental to solving problems involving triangles. This knowledge helps geometricians and mathematicians to calculate missing angles or to explore the relationships between angles in different types of triangles.
What are the 3 angles of a triangle?
The 3 angles of a triangle are key components that define the shape and properties of the triangle. An angle in geometry is the measure of the degree of the turn between two lines, and in the case of a triangle, the 3 angles are those formed at each vertex where two sides meet.
The sum of the interior angles of a triangle is always equal to 180 degrees, and this is a fundamental property of the triangle that is used in various mathematical applications.
If we are given the measure of two angles of a triangle, we can easily calculate the third using the fact that the sum of all three angles is 180 degrees. For example, if one angle measures 60 degrees and another measures 70 degrees, we can calculate the third angle by subtracting the sum of the two known angles from 180: 180 – (60+70) = 50 degrees.
Therefore, the three angles of this triangle are 60, 70, and 50 degrees.
Another interesting result that can be derived from the three angles of a triangle is the fact that the largest angle is always opposite the longest side of the triangle. This is known as the angle-side relation, and it is a simple but powerful tool that enables us to find missing angles or sides of a triangle.
The 3 angles of a triangle are important components that are used in a variety of mathematical applications, including trigonometry, geometry, and calculus. Knowing these angles helps us understand the shape and properties of a triangle, and enables us to derive many interesting results and formulas related to this fundamental shape.
Why is a triangle 360?
A triangle is 360 because it is a three-sided figure. Triangles are made up of three angles, which add up to create a full circle. This means that all three angles in a triangle must add up to equal 360 degrees, creating a complete circle.
To further illustrate this concept, we can look at a triangle that is divided into two right triangles. A right triangle is composed of two angles that add up to 90 degrees and one angle that equals the remaining degrees in the full triangle.
Therefore, if we add the two 90 degree angles and the remaining angle, we get a total of 180 degrees, doubling this total will give us 360 degrees. This demonstrates that a triangle must equal 360 degrees.
What is the 30 60 90 rule?
The 30 60 90 rule is a standard set of guidelines used by many organizations to measure the progress of new employees during the first three months of their employment. This rule sets out expectations that new employees should have achieved by certain points in time, specifically 30 days, 60 days, and 90 days.
The goal of this rule is to help new employees set realistic goals and to provide them with a framework to focus their efforts during their first three months on the job.
During the first 30 days of the rule, the new employee is expected to get started with their assigned tasks, get to know their team members, understand company culture, and learn more about the job responsibilities. During this period, the employee should focus on adapting and adjusting to the organization’s culture and ethics.
From a performance standpoint, during this period, it’s good to show that the new employee can work well with others, has a positive attitude, asks questions and contributes ideas, and is taking initiative in their role.
During the second 30 days of the rule (60 days), the new employee should have built a deeper understanding of their responsibilities and goals. Their overall performance should show that they are making progress, collaborating well with others who work in the same team and outside of their team too.
It is also important for them to demonstrate an approach to problem-solving, exhibit good communication skills, and be able to take on more projects and responsibilities.
By the end of the first 90 days, the new employee should have established themselves as a productive and valuable member of the team. The employee should have contributed to one or more projects, demonstrated a solid understanding of their role, and have a clear vision of their career goals. They should continue to take personal and professional development opportunities, and remain proactive in identifying new ways they can contribute to the organization.
Overall, the 30 60 90 rule is a valuable tool that helps to set expectations for a new employee’s performance, provide direction, outline clear goals and expectations, and help the employee to integrate into the company culture quickly. It’s a useful framework that ensures new employees have direction and goals while promoting mutual understanding, open communication, and team collaboration between new hires and their managers.
Why are there 360 degrees in a circle and not 400?
The adoption of 360 degrees as the standard for measuring angles in a circle can be traced back to ancient civilizations like the Babylonians and Egyptians. One theory suggests that they chose 360 degrees because it was a useful number for astronomical calculations, which were essential for predicting the seasons and navigating using the stars.
Another theory suggests that 360 may have been chosen because it is a highly composite number, meaning it has a large number of divisors (24 divisors to be exact). This made it easy to divide into smaller segments, such as 30 or 45 degrees, which are commonly used in geometry and navigation.
The idea of dividing a circle into 400 degrees, on the other hand, is a relatively modern concept. It was proposed by French mathematician and engineer Charles René Reynaud in the late 18th century. Reynaud believed that 400 degrees was a more logical and practical measure for angles because it reduces the need for complicated calculations involving fractions and decimals.
Despite its appeal, however, the idea of using 400 degrees never gained widespread acceptance in academic or scientific circles. Today, the use of 360 degrees as the standard for measuring angles in a circle remains the norm, making it an important part of our mathematical and scientific heritage.
The adoption of 360 degrees in a circle dates back to ancient times, and it offers numerous practical advantages in both mathematics and astronomy. While the idea of using 400 degrees offers some benefits, it has not been widely adopted and remains a theoretical concept in the world of mathematics.
Why doesn’t the direction of rotation matter when the angle is 180?
When we talk about the rotation of an object, it can be in a clockwise or counterclockwise direction. The direction of rotation is typically defined as the direction that the object turns or moves from its original position. However, when the angle of rotation is 180 degrees, the direction of rotation does not matter because the object ends up in the same position regardless.
To understand why this is the case, let’s consider a simple example. Imagine a pencil on a desk, and we want to rotate it 180 degrees about its center point. We can do this by picking it up and flipping it over, or by simply rotating it 180 degrees in place. In either case, the end result is the same – the pencil is now upside down with the eraser and tip switched places.
This is because a rotation of 180 degrees essentially means that we are turning the object completely around, so the starting and ending positions are the same.
In mathematical terms, we can represent a 180-degree rotation using a matrix transformation. This matrix has a few key properties, one of which is that it is its own inverse – in other words, if we apply the same matrix to the rotated object, it will return to its original position. This property holds true regardless of whether we use a clockwise or counterclockwise rotation, since both rotations result in the same matrix transformation.
The direction of rotation does not matter when the angle is 180 because a 180-degree rotation means that the object is being turned completely around, so the starting and ending positions are the same. Mathematically, this is because a 180-degree rotation matrix is its own inverse, so the direction of rotation does not affect the end result.
Why do all exterior angles equal 360?
The concept of exterior angles is a fundamental concept in geometry that helps to determine the properties of two-dimensional shapes such as triangles and polygons. An exterior angle of a shape is the angle formed by any one side of the shape and the adjacent side’s extension.
The reason why all exterior angles in a polygon add up to 360 degrees is rooted in geometry’s fundamental properties, including the angle sum property of polygons and the concepts of congruence and similarity.
The angle sum property of polygons states that the sum of all interior angles in a polygon with n sides equals (n-2) x 180 degrees. For instance, a triangle, which has three sides, has (3-2) x 180 = 180 degrees as the sum of its interior angles.
Now, when we talk about exterior angles, we can see that each exterior angle of a polygon is supplementary to the interior angle adjacent to it. That is, the exterior and interior angles, when added, form a straight line, which is 180 degrees. This concept applies to all the exterior angles in a polygon.
For instance, let’s take a regular pentagon (five-sided polygon). If we draw all the diagonals from one vertex and measure each exterior angle, we find that they all sum up to 360 degrees. This is because, for every vertex in a polygon, there is one exterior angle, and the sum of all exterior angles is 360 degrees.
This result can also be verified using the concept of similarity and congruence. The exterior angles of a polygon are congruent to each other if the polygon is regular (all its sides and angles are equal). If the polygon is not regular, the exterior angles are still similar to each other.
Therefore, the sum of all exterior angles in a polygon can be found by multiplying the number of exterior angles with which the angle size that each exterior angle has. Since all exterior angles in a polygon are similar or congruent, their angle size will be the same, and multiplying this angle size by the number of exterior angles will give us the total sum, which is 360 degrees.
The reason why all exterior angles in a polygon add up to 360 degrees is rooted in the angle sum property of polygons and the concepts of congruence and similarity. Each exterior angle of the polygon is supplementary to the adjacent interior angle, and the sum of all exterior angles is 360 degrees.
This fundamental concept is essential in solving problems in geometry involving angles and polygons.
