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Two segments with a common endpoint form an angle. If the free endpoints of each segment are connected, then three angles are formed, one at each endpoint. The plane figure formed by these three angles and segments is called a triangle. In this lesson, the definition and different classifications of triangles are studied.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Explore
Studying the Angle Type
Consider three segments AB, BC and AC that form three angles. Click and drag point C to change the shape of the figure.
What types of angle are formed at vertex C?
Challenge
Matchstick Riddles
Dylan and Zosia love to solve riddles and decided to have a competition to see who can solve them faster. In one riddle, they have to form a triangle with three matchsticks that are 4, 4.5, and 8.8 centimeters long.
Dylan says that such a triangle cannot be formed, while Zosia says it can. Who is right?
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Discussion
Figures With Three Angles
Discussion
Triangles With Three Acute Angles
Discussion
Triangles With a Right Angle
The second type of triangle groups all those triangles that have one right angle.
Concept
Right Triangle
A right triangle is a triangle that has one right angle. The side opposite the right angle is always the longest and is known as the hypotenuse. The other sides are commonly called legs. Notice that in a right triangle, the legs are perpendicular to each other.
Discussion
Triangles With an Obtuse Angle
The next type of triangle includes those triangles that have one obtuse angle.
Concept
Obtuse Triangle
An obtuse triangle is a triangle that has one obtuse angle. In other words, one of the angles measures more than 90∘.
Discussion
Triangles With Three Congruent Angles
The last classification of triangles according to the measure of their angles consists of those triangles that have three angles of the same measure.
Concept
Equiangular Triangle
An equiangular triangle is a triangle in which the three angles have the same measure. In other words, the three angles are congruent. In fact, the three angles measure 60∘ each. 
Notice that an equiangular triangle is also an acute triangle. In an equiangular triangle, the three sides have the same measure.
Pop Quiz
Classifying Triangles
Classify the given triangle according to its angle measures as an acute, right, or obtuse triangle.
Discussion
Segments With the Same Length
The same way two angles with the same measure are called congruent angles, there is a special term for two segments that have the same length.
Concept
Congruent Segments
Discussion
Triangles With Three Congruent Sides
In addition to their angle measures, triangles can also be classified by comparing the lengths of their three sides. There are three different types. The first includes those triangles in which all three sides have the same length.
Concept
Equilateral Triangle
An equilateral triangle is a triangle in which the three sides have the same length. In other words, the three sides are congruent.
In an equilateral triangle, the three angles have the same measure. Therefore, equilateral triangles are also equiangular triangles.
Discussion
Triangles With Only Two Congruent Sides
The second way of classifying a triangle according to its side lengths includes all the triangles in which only two sides have the same length.
Concept
Isosceles Triangle
An isosceles triangle is a triangle that has exactly two sides of the same length. In other words, it has two congruent sides. The congruent sides are called legs while the third side is called the base. The angle between the legs is called the vertex angle and the other two angles are called base angles.
In an isosceles triangle, the base angles are congruent angles — they have the same measure.
Discussion
Triangles With Three Different Side Lengths
The last classification of triangles based on their side lengths is when the three sides have different lengths.
Concept
Scalene Triangle
A scalene triangle is a triangle in which all three sides have different lengths. In other words, it has no congruent sides.
Additionally, in a scalene triangle, the three angles have different measures. In other words, scalene triangles have no congruent angles.
Pop Quiz
Classifying Triangles From Their Side Lengths
Classify the given triangle according to its side lengths as an equilateral, isosceles, or scalene triangle.
Discussion
Triangle Inequality Theorem
In any triangle, the sum of the lengths of any two sides is greater than the length of the third side.
Therefore, given a triangle ABC, three inequalities hold true.
AB+BC>ACBC+AC>ABAC+AB>BC
Example
Challenging Lunch
At lunch, Ali challenged Dylan and Zosia to build a triangle with the objects they had on hand. He offered a free dessert to whoever did it successfully.
Since Zosia brought Chinese food and a soft drink, she decided to use the two chopsticks and the straw to build her triangle. The straw is 18 centimeters long and the chopsticks are each 24 centimeters long.
For his part, Dylan decided to use a pen, a short pencil, and a small crayon from his backpack. The pen, pencil, and crayon are 15, 9, and 5 centimeters long, respectively.
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Hint
Check if the lengths of the objects satisfy the Triangle Inequality Theorem.
Solution
In order for three segments to form a triangle, their lengths must satisfy the Triangle Inequality Theorem. In other words, the sum of the lengths of any two segments must be greater than the length of the third segment. Consider Zosia's and Dylan's attempts separately.
Zosia
Start by writing the lengths of the objects that Zosia chose.
| Object | Length (cm) |
|---|---|
| Straw | 18 |
| Chopstick 1 | 24 |
| Chopstick 2 | 24 |
Now, calculate the sums of the lengths of the possible pairs of objects and check if they are greater than the length of the third object.
| Pair of Objects | Sum of Lengths | Object 3 | Comparison |
|---|---|---|---|
| Straw and Chopstick 1 | 18+24=42 | Chopstick 2 | 42>24✓ |
| Straw and Chopstick 2 | 18+24=42 | Chopstick 1 | 42>24✓ |
| Chopstick 1 and Chopstick 2 | 24+24=48 | Straw | 48>18✓ |
As shown, the sum of the lengths of any two objects is always greater than the length of the third object. Therefore, Zosia can form a triangle with the objects she chose, so she will get a free dessert!
Notice that the triangle has two sides of the same length, which means that Zozia built an isosceles triangle.
Dylan
Dylan can build a triangle with the objects he chose only if their lengths meet the Triangle Inequality Theorem. Follow the same process to see if he can earn a free dessert. Begin by writing the lengths of the objects.
| Object | Length (cm) |
|---|---|
| Pen | 15 |
| Pencil | 9 |
| Crayon | 5 |
Next, calculate the sums of the lengths of the possible pairs of objects and verify if they are greater than the length of the third object.
| Pair of Objects | Sum of Lengths | Object 3 | Comparison |
|---|---|---|---|
| Pen and Pencil | 15+9=24 | Crayon | 24>5✓ |
| Pen and Crayon | 15+5=20 | Pencil | 20>9✓ |
| Pencil and Crayon | 9+5=14 | Pen | 14>15× |
The sum of the lengths of the pencil and the crayon in the last row of the table is not greater than the length of the pen, which means that the object's lengths do not meet the Triangle Inequality Theorem. Dylan cannot build a triangle with the objects he chose, so he will not get a free dessert. Bummer!
Pop Quiz
Building a Triangle
Determine whether the given segments can form a triangle.
Recall that three segments can form a triangle if their lengths satisfy the Triangle Inequality Theorem.
Extra
TipThere is no need to verify all three inequalities. Just check whether the sum of the lengths of the two shorter segments is greater than the length of the longer segment.
Closure
Solving the Riddle
At the beginning of the lesson, Dylan and Zosia were challenged to form a triangle using three matchsticks. However, they came to opposite conclusions. Dylan claimed that no triangle could be formed with the given matches, while Zosia said that a triangle could be formed.
The riddle can be solved with the information given in this lesson. The three matches can form a triangle if they satisfy the Triangle Inequality Theorem. In other words, check whether the sum of the lengths of any pair of matches is greater than the length of the third match.
| Pair of Matches | Sum of Lengths | Third Match | Comparison |
|---|---|---|---|
| Match 1 & Match 2 | 8.8+4.5=13.3 | 4.0 | 13.3>4.0 ✓ |
| Match 1 & Match 3 | 8.8+4.0=12 | 4.5 | 12>4.5 ✓ |
| Match 2 & Match 3 | 4.5+4.0=8.5 | 8.8 | 8.5>8.8 × |
As shown, the sum of the lengths of the two shorter matches is not greater than the length of the longest match. Therefore, the given matchsticks cannot form a triangle. Dylan was right!
