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We are going to be talking about triangles. Triangles are a two dimensional shape with three sides and three angles. These sides may be different lengths and the angles may be different measures. If you and a friend were to each draw a triangle, they may not be the same!
There would be some features that are the same. Both of the triangles would have three sides and three angles. If you and your friend were to measure the angles and add them together, they would both equal 180°!
👉🏻 The sum of the interior angles of a triangle is equal to 180°
👉🏻 An equilateral triangle has three angles that are each 60°
Triangles can be classified, or named by either their sides or their angles. We will start with classifying triangles by their sides. Classifying by sides is really naming the triangles by their side lengths.
With each of the descriptions are diagrams. Take note of the marks and notations on them. The number of tick marks are helpful to indicate congruent lengths.
If all of the sides have the same length, this is known as an Equilateral Triangle. “Equi” means equal, and “lateral” means sides. Although it describes the angle measures, an equilateral triangle can also be called an equiangular triangle since all of the angles are the same measures, but that is a lesson for a different time.
These two triangles are both equilateral. One you can see by the labels of the side lengths. The second triangle shows only tick marks on the sides. Notice that each side has one tick mark.
This indicates that each of the side lengths is congruent, or have the same length.
The next triangle type of triangle is called an isosceles triangle. These triangles have two sides that have the same length. These two sides are referred to as “legs”.
The third side, which has a different length, is known as the base. These triangles also have two angles that have the same measure.
The last triangle that is classified by the side lengths is called a scalene triangle. A scalene triangle is a triangle where no sides have the same length.
Once more take notice of the tick marks on each of the sides. None of the sides have the same number of ticks. No sides are congruent.
Now that we have classified triangles by their sides, now we can take a look at their angles. There are three different types of angles that we are going to use to classify triangles, acute, right, and obtuse.
The first angle is an acute angle. An acute angle is an angle that is between 0° and 90°. When a triangle only has angles that are less than 90°, it is known as an acute triangle.
👉🏻 The hypotenuse is the side opposite the right angle of a triangle.
👉🏻 A triangle must have at least two acute angles
It does not matter if the angles are the same or different for classifying the triangle as an acute triangle. You may start to notice a pattern when there are congruent angles, there are congruent sides.
The second angle is a right angle. A right angle is an angle that has a measure of 90°. When a triangle has an angle measure of 90° it is known as a right triangle.
The right angle in a diagram may be noted by what looks like a box in the corner. This is a symbol that indicates 90° or a right angle.
A triangle can only have one right angle. Since the sum of the angles of the triangle is equal to 180°, two right angles would already hit that sum and can no longer create a triangle.
When a triangle has a measure of 90° the other two angles must be acute. If you only need to classify a triangle by the angle measures, once you know one angle is 90° you do not need to worry about the other two angles.
Right angles are used with many other theorems and equations in math and more information may be needed at that point.
The last angle, and the last type of triangle, is the obtuse angle. An obtuse angle has a measure between 90° and 180°. If a triangle has an angle that is obtuse, then it is classified as an obtuse triangle. Similar to right triangles, there can only be one obtuse angle in a triangle.
These two ways of classifying triangles can overlap. Just like people may have two names, a first name and the last name, triangles may have two names as well! Let’s look at a few different examples.
For this example, we will start by looking at the sides. Notice that two of the sides are the same length, indicated by the two tick marks. This means that it is an isosceles triangle.
👉🏻 There is an arrangement of numbers known as Pascal’s Triangle.
👉🏻 Pythagorean Theorem relates the sum of the squares of the legs to the square of the hypotenuse
Now if we look at the angles we can see that one of the angles has a measure of 90°. This is indicated by the square in the corner. This triangle is also a right triangle! A more precise name for this triangle is an isosceles right triangle!
Classify the following triangle by both sides and angles:
This triangle is an obtuse scalene triangle.
You can see by the tick marks that there are no sides that have the same measure. This makes it a scalene triangle. What about the angle measures? There are only two that are given. The given angles have a sum of 85°. This means that the missing angle must have a measure of 95° in order for the sum of the angles to equal 180°. Since 95° is an obtuse angle, this is also an obtuse triangle
What type has the following angle measures?
45°, 95°, and 40°
Sometimes a diagram is not given and just the angle measures or side lengths. If a triangle has angle measures of 45°, 95°, and 40°, what type of triangle is it? Since we are given the measures of the angles we can see if there are any angles that are right, 90°, or obtuse, greater than 90°. One of the angles is 95°. This makes it an obtuse angle and therefore an obtuse triangle.
The sides that have the same length are called legs and the third side is called the base.
Yes! If there are angles that are the same measure, then we know that there are side lengths that are the same. If there are two angle measures that are the same then we know that the two corresponding side lengths are the same measure. The angle and side that is opposite of it are known as corresponding.
Yes! We can use pythagoras Theorem to determine if a triangle is an acute, obtuse, or right triangle!
Make sure that the longest side is labeled as c.
$$a^2+b^2=c^2\;\;\;\;\;\rightarrow Right\;triangle\\a^2+b^2<c^2\;\;\;\;\;\rightarrow Obtuse\;triangle\\a^2+b^2>c^2\;\;\;\;\;\rightarrow Acute\;triangle\\$$
There is a tool called a compass that can be used to measure an angle’s degree. Most diagrams in problems are not to scale. This means that you may need to use formulas to find the measure of angles. A common formula that is using the angle sum being 180°. For example, if you are given two angle measures of 23° and 87° you can create an equation to solve for the unknown:
23° + 87° + x° = 180°
110° + x° = 180°
x° = 70°
If you study Trigonometry you will discover and learn other formulas to find the measure of triangles.
Rulers can be used to measure length of line segments. Most diagrams are not to scale. One way to find the length of the sides of a right triangle is by using the Pythagorean Theorem.
Given a triangle has two leg lengths with lengths of 9 cm and 12 cm.
a = 9 b = 12
In Trigonometry you will discover and learn more formulas to find the lengths of other types of triangles.
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