Isosceles triangles have equal legs (that's what the word "isosceles" means). Yippee for them, but what do we know about their base angles? How do we know those are equal, too? We reach into our geometer's toolbox and take out the Isosceles Triangle Theorem. No need to plug it in or recharge its batteries -- it's right there, in your head!
Isosceles Triangle
Here we have on display the majestic isosceles triangle, △DUK. You can draw one yourself, using △DUK as a model.
Hash marks show sides ∠DU ≅ ∠DK, which is your tip-off that you have an isosceles triangle. If these two sides, called legs, are equal, then this is an isosceles triangle. What else have you got?
Properties of an Isosceles Triangle
Let's use △DUK to explore the parts:
- Like any triangle, △DUK has three interior angles: ∠D, ∠U, and ∠K
- All three interior angles are acute
- Like any triangle, △DUK has three sides: DU, UK, and DK
- ∠DU ≅ ∠DK, so we refer to those twins as legs
- The third side is called the base (even when the triangle is not sitting on that side)
- The two angles formed between base and legs, ∠DUK and ∠DKU, or ∠D and ∠K for short, are called base angles:
Isosceles Triangle Theorem
Knowing the triangle's parts, here is the challenge: how do we prove that the base angles are congruent? That is the heart of the Isosceles Triangle Theorem, which is built as a conditional (if, then) statement:
The Isosceles Triangle Theorem states: If two sides of a triangle are congruent, then angles opposite those sides are congruent.
To mathematically prove this, we need to introduce a median line, a line constructed from an interior angle to the midpoint of the opposite side. We find Point C on base UK and construct line segment DC:
There! That's just DUCKy! Look at the two triangles formed by the median. We are given:
- UC ≅ CK (median)
- DC ≅ DC (reflexive property)
- DU ≅ DK (given)
We just showed that the three sides of △DUC are congruent to △DCK, which means you have the Side Side Side Postulate, which gives congruence. So if the two triangles are congruent, then corresponding parts of congruent triangles are congruent (CPCTC), which means …
Converse of the Isosceles Triangle Theorem
The converse of a conditional statement is made by swapping the hypothesis (if …) with the conclusion (then …). You may need to tinker with it to ensure it makes sense. So here once again is the Isosceles Triangle Theorem:
If two sides of a triangle are congruent, then angles opposite those sides are congruent.
To make its converse, we could exactly swap the parts, getting a bit of a mish-mash:
If angles opposite those sides are congruent, then two sides of a triangle are congruent.
That is awkward, so tidy up the wording:
The Converse of the Isosceles Triangle Theorem states: If two angles of a triangle are congruent, then sides opposite those angles are congruent.
Now it makes sense, but is it true? Not every converse statement of a conditional statement is true. If the original conditional statement is false, then the converse will also be false. If the premise is true, then the converse could be true or false:
If I see a bear, then I will lie down and remain still.
If I lie down and remain still, then I will see a bear.
For that converse statement to be true, sleeping in your bed would become a bizarre experience.
Or this one:
If I have honey, then I will attract bears.
If I attract bears, then I will have honey.
Unless the bears bring honeypots to share with you, the converse is unlikely ever to happen. And bears are famously selfish.
Proving the Converse Statement
To prove the converse, let's construct another isosceles triangle, △BER.
Given that ∠BER ≅ ∠BRE, we must prove that BE ≅ BR.
Add the angle bisector from ∠EBR down to base ER. Where the angle bisector intersects base ER, label it Point A.
Now we have two small, right triangles where once we had one big, isosceles triangle: △BEA and △BAR. Since line segment BA is an angle bisector, this makes ∠EBA ≅ ∠RBA. Since line segment BA is used in both smaller right triangles, it is congruent to itself. What do we have?
- ∠BER ≅ ∠BRE (given)
- ∠EBA ≅ ∠RBA (angle bisector)
- BA ≅ BA (reflexive property)
Let's see … that's an angle, another angle, and a side. That would be the Angle Angle Side Theorem, AAS:
The AAS Theorem states: If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
With the triangles themselves proved congruent, their corresponding parts are congruent (CPCTC), which makes BE ≅ BR. The converse of the Isosceles Triangle Theorem is true!
Lesson Summary
By working through these exercises, you now are able to recognize and draw an isosceles triangle, mathematically prove congruent isosceles triangles using the Isosceles Triangles Theorem, and mathematically prove the converse of the Isosceles Triangles Theorem. You also should now see the connection between the Isosceles Triangle Theorem to the Side Side Side Postulate and the Angle Angle Side Theorem.
Next Lesson:
Alternate Exterior Angles
Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.
Triangles can be similar or congruent. Similar triangles will have congruent angles but sides of different lengths. Congruent triangles will have completely matching angles and sides. Their interior angles and sides will be congruent. Testing to see if triangles are congruent involves three postulates, abbreviated SAS, ASA, and SSS.
Congruence Definition
Two triangles are congruent if their corresponding sides are equal in length and their corresponding interior angles are equal in measure.
We use the symbol ≅ to show congruence.
Corresponding sides and angles mean that the side on one triangle and the side on the other triangle, in the same position, match. You may have to rotate one triangle, to make a careful comparison and find corresponding parts.
How can you tell if triangles are congruent?
You could cut up your textbook with scissors to check two triangles. That is not very helpful, and it ruins your textbook. If you are working with an online textbook, you cannot even do that.
Geometricians prefer more elegant ways to prove congruence. Comparing one triangle with another for congruence, they use three postulates.
Postulate Definition
A postulate is a statement presented mathematically that is assumed to be true. All three triangle congruence statements are generally regarded in the mathematics world as postulates, but some authorities identify them as theorems (able to be proved).
Do not worry if some texts call them postulates and some mathematicians call the theorems. More important than those two words are the concepts about congruence.
Triangle Congruence Theorems
Testing to see if triangles are congruent involves three postulates. Let's take a look at the three postulates abbreviated ASA, SAS, and SSS.
- Angle Side Angle (ASA)
- Side Angle Side (SAS)
- Side Side Side (SSS)
ASA Theorem (Angle-Side-Angle)
The Angle Side Angle Postulate (ASA) says triangles are congruent if any two angles and their included side are equal in the triangles. An included side is the side between two angles.
In the sketch below, we have △CAT and △BUG. Notice that ∠C on △CAT is congruent to ∠B on △BUG, and ∠A on △CAT is congruent to ∠U on △BUG.
See the included side between ∠C and ∠A on △CAT? It is equal in length to the included side between ∠B and ∠U on △BUG.
The two triangles have two angles congruent (equal) and the included side between those angles congruent. This forces the remaining angle on our △CAT to be:
180° - ∠C - ∠A
This is because interior angles of triangles add to 180°. You can only make one triangle (or its reflection) with given sides and angles.
You may think we rigged this, because we forced you to look at particular angles. The postulate says you can pick any two angles and their included side. So go ahead; look at either ∠C and ∠T or ∠A and ∠T on △CAT.
Compare them to the corresponding angles on △BUG. You will see that all the angles and all the sides are congruent in the two triangles, no matter which ones you pick to compare.
SAS Theorem (Side-Angle-Side)
By applying the Side Angle Side Postulate (SAS), you can also be sure your two triangles are congruent. Here, instead of picking two angles, we pick a side and its corresponding side on two triangles.
The SAS Postulate says that triangles are congruent if any pair of corresponding sides and their included angle are congruent.
Pick any side of △JOB below. Notice we are not forcing you to pick a particular side, because we know this works no matter where you start. Move to the next side (in whichever direction you want to move), which will sweep up an included angle.
For the two triangles to be congruent, those three parts -- a side, included angle, and adjacent side -- must be congruent to the same three parts -- the corresponding side, angle and side -- on the other triangle, △YAK.
SSS Theorem (Side-Side-Side)
Perhaps the easiest of the three postulates, Side Side Side Postulate (SSS) says triangles are congruent if three sides of one triangle are congruent to the corresponding sides of the other triangle.
This is the only postulate that does not deal with angles. You can replicate the SSS Postulate using two straight objects -- uncooked spaghetti or plastic stirrers work great. Cut a tiny bit off one, so it is not quite as long as it started out. Cut the other length into two distinctly unequal parts. Now you have three sides of a triangle. Put them together. You have one triangle. Now shuffle the sides around and try to put them together in a different way, to make a different triangle.
Guess what? You can't do it. You can only assemble your triangle in one way, no matter what you do. You can think you are clever and switch two sides around, but then all you have is a reflection (a mirror image) of the original.
So once you realize that three lengths can only make one triangle, you can see that two triangles with their three sides corresponding to each other are identical, or congruent.
You can check polygons like parallelograms, squares and rectangles using these postulates.
Introducing a diagonal into any of those shapes creates two triangles. Using any postulate, you will find that the two created triangles are always congruent.
Suppose you have parallelogram SWAN and add diagonal SA. You now have two triangles, △SAN and △SWA. Are they congruent?
You already know line SA, used in both triangles, is congruent to itself. What about ∠SAN? It is congruent to ∠WSA because they are alternate interior angles of the parallel line segments SW and NA (because of the Alternate Interior Angles Theorem).
You also know that line segments SW and NA are congruent, because they were part of the parallelogram (opposite sides are parallel and congruent).
So now you have a side SA, an included angle ∠WSA, and a side SW of △SWA. You can compare those three triangle parts to the corresponding parts of △SAN:
- Side SA ≅ Side SA (sure hope so!)
- Included angle ∠WSA ≅ ∠NAS
- Side SW ≅ Side NA
Lesson Summary
After working your way through this lesson and giving it some thought, you now are able to recall and apply three triangle congruence postulates, the Side Angle Side Congruence Postulate, Angle Side Angle Congruence Postulate, and the Side Side Side Congruence Postulate. You can now determine if any two triangles are congruent!
Next Lesson:
Conditional Statements and Their Converse