Isosceles Triangle theorem (Proof, Converse, & Examples)

Isosceles triangles have equal legs (that"s what the word "isosceles" means). Yippee because that them, yet what carry out we know about their basic angles? exactly how do we know those are equal, too? we reach into our geometer"s toolbox and also take out the Isosceles Triangle Theorem. No should plug it in or recharge its battery -- it"s best there, in your head!

Isosceles Triangle

Here we have actually on display screen the majestic isosceles triangle, △DUK. You can attract one yourself, making use of △DUK together a model.

You are watching: The base angles of an isosceles triangle are congruent

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Hash marks show sides ∠DU ≅ ∠DK, i beg your pardon is her tip-off the you have actually an isosceles triangle. If these 2 sides, called legs, space equal, then this is one isosceles triangle. What else have actually you got?

Properties of an Isosceles Triangle

Let"s usage △DUK to discover the parts:

Like any type of triangle, △DUK has three interior angles: ∠D, ∠U, and also ∠KAll three internal angles are acuteLike any triangle, △DUK has three sides: DU, UK, and DK∠DU ≅ ∠DK, therefore we refer to those twins together legsThe 3rd side is dubbed the base (even as soon as the triangle is not sitting on that side)The 2 angles formed in between base and also legs, ∠DUK and also ∠DKU, or ∠D and also ∠K for short, are dubbed base angles:

Isosceles Triangle Theorem

Knowing the triangle"s parts, here is the challenge: just how do us prove the the base angles are congruent? the is the heart of the Isosceles Triangle Theorem, i m sorry is built as a conditional (if, then) statement:


To mathematically prove this, we require to present a median line, a line constructed from an interior angle to the midpoint of the contrary side. We uncover Point C on base UK and also construct heat segment DC:

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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 the the three sides the △DUC space congruent to △DCK, which method you have actually the Side next Side Postulate, which gives congruence. For this reason if the two triangles space congruent, then corresponding parts the congruent triangles are congruent (CPCTC), which method …

∠U ≅ ∠K

Converse of the Isosceles Triangle Theorem

The converse the a conditional explain is do by swapping the theory (if …) through the conclusion (then …). You might need to tinker v it come ensure it renders sense. So here once again is the Isosceles Triangle Theorem:

If two sides the a triangle room congruent, then angles opposite those sides are congruent.

To do its converse, us could precisely swap the parts, obtaining a little of a mish-mash:

If angle opposite those sides space congruent, then two sides that a triangle are congruent.

That is awkward, for this reason tidy increase the wording:


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Now it renders sense, yet is that true? no every converse explain of a conditional explain is true. If the initial conditional declare is false, then the converse will likewise be false. If the premise is true, then the converse might be true or false:

If I check out a bear, then I will lie down and remain still.If i lie down and remain still, then ns will see a bear.

For the converse declare to it is in true, resting in your bed would end up being a bizarre experience.

Or this one:

If I have honey, then i will tempt bears.If I tempt bears, then ns will have honey.

Unless the bears bring honeypots to share through you, the converse is unlikely ever to happen. And bears space famously selfish.

Proving the Converse Statement

To prove the converse, let"s construct another isosceles triangle, △BER.

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Given the ∠BER ≅ ∠BRE, we should prove the BE ≅ BR.

Add the angle bisector from ∠EBR down to base ER. Whereby the edge bisector intersects base ER, brand it Point A.

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Now we have actually two small, appropriate triangles where as soon as we had actually one big, isosceles triangle: △BEA and △BAR. Due to the fact that line segment BA is an angle bisector, this makes ∠EBA ≅ ∠RBA. Due to the fact that line segment BA is provided in both smaller appropriate triangles, that is congruent to itself. What carry out we have?

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∠BER ≅ ∠BRE (given)∠EBA ≅ ∠RBA (angle bisector)BA ≅ BA (reflexive property)

Let"s watch … that"s one angle, another angle, and also a side. That would certainly be the Angle Angle next Theorem, AAS:


With the triangles themselves confirmed congruent, their corresponding parts are congruent (CPCTC), which makes BE ≅ BR. The converse the the Isosceles Triangle theorem is true!

Lesson Summary

By working through these exercises, you now are able come recognize and also draw one isosceles triangle, mathematically prove congruent isosceles triangles making use of the Isosceles triangle Theorem, and mathematically prove the converse that the Isosceles triangle Theorem. You also should currently see the connection in between the Isosceles Triangle Theorem to the next Side next Postulate and the edge Angle side Theorem.

Next Lesson:

Alternate Exterior Angles


What you"ll learn:

After functioning your way through this lesson, you will certainly be maybe to:

Recognize and also draw one isosceles triangleMathematically prove congruent isosceles triangles making use of the Isosceles triangle TheoremMathematically prove the converse the the Isosceles triangle TheoremConnect the Isosceles Triangle Theorem come the next Side next Postulate and the angle Angle next Theorem