ClassifyingTriangles
By SIDES:
Equilateral triangle-all sides are congruent (have equal measures)
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Isosceles triangle-at least 2 sides are congruent
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Scalene triangle-no sides are congruent
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By ANGLES:
Acute triangle-all angles have measures that are less than 90 degrees
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Obtuse triangle-one angle is greater than 90 degrees (can only be one)
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Right triangle-there is a 90 degree angle (can only be one)
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TRIANGLE ANGLE SUM THEOREM
The Triangle Angle Sum Theorem states that the measures of the three angles in every triangle add up to 180 degrees. This is a picture of the scaffolded proof we did in class for the Theorem.
Corollaries of the Triangle Angles Sum Theorem include:
Corollaries of the Triangle Angles Sum Theorem include:
- In a triangle, no more than one angle can be greater than or equal to 90 degrees (this is proof by contradiction)
- In a right triangle, the two non-right angles are complementary
- If two angles of one triangle are congruent to two angles of another, then the third angles are also congruent.
Remote Interior Angles TheoremThe two remote interior angles in a triangle are the farthest from the exterior angle, which is on the same line as the base and outside the triangle. You can think of how a remote for a TV connects really far away, the same way the remote interior angles are farthest away from the exterior angle.
The sum of the two interior angles is equal to the exterior angle (in this picture, angle 3+ angle 4= angle 1). |
The bigger a triangle's angle is, the longer the opposite side will need to be. If you look at the picture of the alligator, and picture a line drawn from the top of it's mouth to the bottom of it's mouth, you can understand this better. If the alligator closed it's mouth more, the line would be shorter and if it opened up more, the line would be longer. We can apply this to the blue triangle above. The smallest angle is angle B, so the side opposite angle B, line AC, is the shortest. Using this, you can list angles in order from smallest to biggest (descending order) or biggest to smallest (ascending order).
You can do the same thing with sides, for example, the biggest side is side BC, and therefore the opposite angle is the biggest, angle A. If you were to list the sides in descending order, it would go: line BC, line AB, line AC.
You can do the same thing with sides, for example, the biggest side is side BC, and therefore the opposite angle is the biggest, angle A. If you were to list the sides in descending order, it would go: line BC, line AB, line AC.
How to tell if Triangles are Congruent
There needs to be at least three pieces of information to tell if a triangle is congruent. But they have to be the right pieces of information. A great video about this is: https://www.youtube.com/watch?v=PJ6TVVdIHpg
We use A to symbolize an angle and S to symbolize a side. Here are the four cases where we can tell that two triangles with these same corresponding parts are definitely congruent:
We use A to symbolize an angle and S to symbolize a side. Here are the four cases where we can tell that two triangles with these same corresponding parts are definitely congruent:
One other way we can tell if triangles are congruent is by RHL:
We can use the Pythagorean Theorem to say two triangles are congruent. It's important to remember that this isn't ASS, another form that is NOT correct in defining congruent triangles.
In RHL,
-the R stands for right angle, meaning there has to be a right angle in both triangles
-the H stands for hypotenuse, which has to be in the same place on both triangles
-the L stands for a leg, there has to be a corresponding (in the same place on both triangles) leg, which is a side that isn't the hypotenuse
In RHL,
-the R stands for right angle, meaning there has to be a right angle in both triangles
-the H stands for hypotenuse, which has to be in the same place on both triangles
-the L stands for a leg, there has to be a corresponding (in the same place on both triangles) leg, which is a side that isn't the hypotenuse
NOT able to tell if they're congruent: ASS (called this, not SSA), AAS, and AAA
Triangle Congruence Statement
When we know that two or more triangles are congruent, this is important to remember if you want to write a statement that says they're congruent:
-write the triangle symbol, then the name (ex: triangle ABC)
-it doesn't matter what order you write the first triangle
-for the second triangle, write each corresponding side in the same order you wrote the first one (ex: A corresponds to D, and B to E, and so on so it would be DEF)
-write a congruent symbol in between each triangle name
Triangle ABC is congruent to triangle DEF
-write the triangle symbol, then the name (ex: triangle ABC)
-it doesn't matter what order you write the first triangle
-for the second triangle, write each corresponding side in the same order you wrote the first one (ex: A corresponds to D, and B to E, and so on so it would be DEF)
-write a congruent symbol in between each triangle name
Triangle ABC is congruent to triangle DEF
Simple Proof Showing 2 Triangles are Congruent:
To show that two triangles are congruent, we can use a proof. You can simply show that three parts are congruent, by the given (or reflexive property, vertical angles, etc.), then saying that the two triangles are congruent by whatever you just listed (AAS, SAS, etc.). If the proof requires that you show that one part is congruent, you can prove this by CPCTC. This means: Corresponding Parts in Congruent Triangles are Congruent. In other words, if we know two triangles are congruent, then all the corresponding parts are sure to be congruent, also.