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How Many Triangles Can You Make? (Given Conditions)
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 We’ll learn that when you are given certain side lengths and angles of triangles, you can either form 1 triangle, no triangle, or many triangles!
 We will also learn that the 3 angles in a triangle add up to 180 degrees.
 And we'll see that this knowledge can help us make jewelry, design clothes, and even work on spacecraft!

Discussion Questions

Before VideoWhat is the name of a threesided polygon?ANSWER

A triangle.

Three sides, three angles. Three sides, three corners or vertices.

In degrees.

Right, acute, obtuse, isosceles, scalene, equilateral.

A triangle in which all three sides are the same length.


After VideoWhat is the sum of the three angles of a triangle?ANSWER

180 degrees.

The length of two sides combined must be greater than the third side. Example: 2 cm, 3 cm, 4 cm

The lengths of two sides combined must be less than the third side. Example: 2 in., 2 in., 5 in.

Many triangles (an infinite number of triangles)

No triangles; the angles do not add to 180 degrees.



Vocabulary

Triangle
DEFINE
A threesided polygon.

Angle
DEFINE
A figure formed by two rays or two line segments with a common endpoint called the vertex of the angle.

Ray
DEFINE
A part of a line starting at an endpoint and continuing forever in one direction.

Vertex
DEFINE
The point at which the rays of an angle meet.

Scale
DEFINE
The relative size of something. It is the same as a scale factor.

Protractor
DEFINE
A tool used for measuring or drawing angles.

Triangle
DEFINE

Reading Material
Download as PDF Download PDF View as Separate PageWHAT ARE THE PROPERTIES OF A TRIANGLE?A triangle is a closed shape with three sides. This means that all three sides must connect. If a set of three side lengths cannot form a 3sided closed shape (two sides can’t touch), then those lengths cannot make a triangle. A triangle also has three angles whose measures add to 180°. If a set of three angle measures do not add to 180°, they cannot form a triangle.
To better understand triangles you can make (given conditions)…
WHAT ARE THE PROPERTIES OF A TRIANGLE?. A triangle is a closed shape with three sides. This means that all three sides must connect. If a set of three side lengths cannot form a 3sided closed shape (two sides can’t touch), then those lengths cannot make a triangle. A triangle also has three angles whose measures add to 180°. If a set of three angle measures do not add to 180°, they cannot form a triangle. To better understand triangles you can make (given conditions)…LET’S BREAK IT DOWN!
When you are given three sides, they can form a maximum of one unique triangle.
How many different triangles can you construct with sides lengths of 6, 8, and 10 inches? You can put the side lengths together in different orientations, but the triangle always has the same size and shape. All of the triangles you make are the same triangle. So if your three side lengths can form a triangle, they can only form exactly 1 unique triangle! Try this one yourself: How many unique triangles can you make with side lengths of 4, 6, and 8 inches?
When you are given three sides, they can form a maximum of one unique triangle. How many different triangles can you construct with sides lengths of 6, 8, and 10 inches? You can put the side lengths together in different orientations, but the triangle always has the same size and shape. All of the triangles you make are the same triangle. So if your three side lengths can form a triangle, they can only form exactly 1 unique triangle! Try this one yourself: How many unique triangles can you make with side lengths of 4, 6, and 8 inches?When you are given three side lengths, sometimes you cannot form any triangles at all.
Next, let’s look at how many triangles you can form with side lengths of 6, 8, and 16 inches. If you put the longest side across the bottom, and the shortest side on the left, you can see that the other side is too short to connect the first two sides to form a triangle. That is because, in order to form a triangle, the two shorter sides together must be longer than the longest side. Mathematically speaking, the sum of the lengths of the two shorter sides of a triangle must be greater than the length of the longest side. Otherwise, they cannot form a triangle. Try this yourself: Given side lengths of 4, 8, and 12 meters, how many unique triangles can you make?
When you are given three side lengths, sometimes you cannot form any triangles at all. Next, let’s look at how many triangles you can form with side lengths of 6, 8, and 16 inches. If you put the longest side across the bottom, and the shortest side on the left, you can see that the other side is too short to connect the first two sides to form a triangle. That is because, in order to form a triangle, the two shorter sides together must be longer than the longest side. Mathematically speaking, the sum of the lengths of the two shorter sides of a triangle must be greater than the length of the longest side. Otherwise, they cannot form a triangle. Try this yourself: Given side lengths of 4, 8, and 12 meters, how many unique triangles can you make?When you are given three angles, you can form either no triangles or infinitely many triangles.
How many different triangles can you make with angle measures of 30 degrees, 60 degrees, and 90 degrees? To draw a triangle, use a protractor to make one of the angles. Make the side lengths as long or short as you want. At the end of one of the sides, make another angle, and join the sides together. You've made a triangle! Notice that you decided the side lengths. This means you could make a different decision. Because of this, you could make infinitely many different triangles with the same angles. Try this yourself: Given angle measures 100 degrees, 45 degrees, and 35 degrees, can you form one triangle, no triangles, or an infinite number of triangles?
When you are given three angles, you can form either no triangles or infinitely many triangles. How many different triangles can you make with angle measures of 30 degrees, 60 degrees, and 90 degrees? To draw a triangle, use a protractor to make one of the angles. Make the side lengths as long or short as you want. At the end of one of the sides, make another angle, and join the sides together. You've made a triangle! Notice that you decided the side lengths. This means you could make a different decision. Because of this, you could make infinitely many different triangles with the same angles. Try this yourself: Given angle measures 100 degrees, 45 degrees, and 35 degrees, can you form one triangle, no triangles, or an infinite number of triangles?When you are given three angles, you may not be able to form any triangles at all.
Remember that the angles inside a triangle must add to 180°. 30° + 60° + 90° = 180°. What if you are given angles 45°, 20°, and 30°? 45° + 20° + 30° = 95°. These angles do not sum to 180°, so they cannot be used to form a triangle. Try this yourself: Given angle measures 60 degrees, 55 degrees, and 35 degrees, can you form one triangle, no triangles, or an infinite number of triangles?
When you are given three angles, you may not be able to form any triangles at all. Remember that the angles inside a triangle must add to 180°. 30° + 60° + 90° = 180°. What if you are given angles 45°, 20°, and 30°? 45° + 20° + 30° = 95°. These angles do not sum to 180°, so they cannot be used to form a triangle. Try this yourself: Given angle measures 60 degrees, 55 degrees, and 35 degrees, can you form one triangle, no triangles, or an infinite number of triangles? 
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Science & Math$_{/yr}

Science Only$_{/yr}
Given side lengths 3 cm, 4 cm, and 6 cm, can you draw one triangle, no triangles, or many triangles?
Given side lengths 5 inches, 7 inches, and 14 inches, can you draw one triangle, no triangles, or many triangles?
Given angle measures 27°, 58°, and 95°, can you draw one triangle, no triangle, or many triangles?
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