Amy has a master's degree in secondary education and has taught math at a public charter high school.
Watch this video lesson and you will understand what the transitive property is and how it applies to similar triangles. Watch as we apply the transitive property to three similar triangles.
Triangles that are similar are triangles whose only difference is size. Similar triangles will look like they have either been shrunk or puffed up in size. Also, all the corresponding angles will be equal to each other. So, if the top angle of one triangle equals 45 degrees, then the top angles of all other similar triangles will also equal 45 degrees. If we compare the side lengths of one triangle to another, we will find that all the sides are proportional. So if the left side of one triangle compared to the left side of another triangle is twice as big, then all the other sides of the one triangle will also be twice as big as the corresponding sides of the other triangle. For example, if the left side of triangle A is 6 and the left side of triangle B is 3, then if the right side of triangle B is 4, then triangle A's right side must be 8 to maintain the proportion.
The transitive property states that if a=b and b=c, then a=c. So, by the transitive property, I can say that if Joe is a boy and boys are tall, then Joe is tall. The transitive property helps you connect pieces of information together.
In algebra the transitive property is used to help you solve problems. If you had two different expressions that both equaled the same thing, then you can use the transitive property to help you connect your different expressions. For example, if you had x = 3 + 4y and you had 3 + 4y = z, then you can apply the transitive property and make the connection that x = z.
Three Similar Triangles
In geometry, you can apply the transitive property to similar triangles to make connections. Let's see how this works. We start out with three triangles. We draw triangle A and then we draw a triangle B that is similar to triangle A. We then take triangle B and then we draw a triangle C that is similar to triangle B. So now we have three triangles: triangle A, triangle B, and triangle C. We know that triangle A is similar to triangle B and that triangle B is similar to triangle C.
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Applying the transitive property, we can say that triangle A is similar to triangle C. Why can we say that? Because if triangle A is similar to triangle B and triangle B is similar to triangle C, then that means the only difference between all three triangles is their size and they are all similar to each other. Therefore, I can make the connection that triangle A is also similar to triangle C.
This transitive property can help us to solve problems involving similar triangles. For example, if we had three triangles where triangle A is similar to triangle B and triangle B is similar to triangle C, we can use the transitive property to help us figure out the measurement of angles. If we wanted to find out the measurement of the top angle of triangle A and we are given that the top angle of triangle C is 50 degrees, we can use the transitive property and say that triangle A is also similar to triangle C and therefore the top angle of triangle A is also 50 degrees.
What have we learned? We've learned that similar triangles are triangles whose only difference is size. The transitive property helps build connections by saying that if a = b and b = c, then a = c. This transitive property can be applied to a group of similar triangles when we say if triangle A is similar to triangle B and triangle B is similar to triangle C, then triangle A is similar to triangle C. We can use this property to help us find angle measurements. If we wanted to find the angle measurements of triangle A, we can use the angle measurements of triangle C by applying the transitive property.
After completion of this lesson, you should be able to:
Define the transitive property
Describe the meaning of similar triangles
Determine how to use the transitive property to prove the only difference between similar triangles is size
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