Isosceles Triangle Theorem (Proof, Converse, & Examples)

Isosceles triangles have equal legs (that"s what the word "isosceles" means). Yippee for them, but what carry out we understand around their base angles? How perform 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 batteries -- it"s ideal tright here, in your head!

Isosceles Triangle

Here we have actually on screen the majestic isosceles triangle, △DUK. You can draw one yourself, making use of △DUK as a version.

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Hash marks display sides ∠DU ≅ ∠DK, which is your tip-off that you have actually an isosceles triangle. If these 2 sides, called legs, are equal, then this is an isosceles triangle. What else have you got?

Properties of an Isosceles Triangle

Let"s usage △DUK to check out the parts:

Like any triangle, △DUK has actually three inner angles: ∠D, ∠U, and ∠KAll 3 internal angles are acuteLike any type of triangle, △DUK has actually 3 sides: DU, UK, and DK∠DU ≅ ∠DK, so we refer to those twins as legsThe 3rd side is referred to as the base (even once the triangle is not sitting on that side)The 2 angles created in between base and legs, ∠DUK and also ∠DKU, or ∠D and also ∠K for brief, are called base angles:

Isosceles Triangle Theorem

Knowing the triangle"s parts, below is the challenge: exactly how carry out we prove that the base angles are congruent? That is the heart of the Isosceles Triangle Theorem, which is constructed as a conditional (if, then) statement:

To mathematically prove this, we have to introduce a median line, a line created from an inner angle to the midallude of the opposite side. We discover Point C on base UK and also construct line segment DC:


There! That"s simply DUCKy! Look at the two triangles created by the median. We are given:

UC ≅ CK (median)DC ≅ DC (reflexive property)DU ≅ DK (given)

We just confirmed that the 3 sides of △DUC are congruent to △DCK, which indicates you have actually the Side Side Side Postulate, which gives congruence. So if the 2 triangles are congruent, then corresponding components of congruent triangles are congruent (CPCTC), which means …

∠U ≅ ∠K

Converse of the Isosceles Triangle Theorem

The converse of a conditional statement is made by swapping the hypothesis (if …) through the conclusion (then …). You might need to tinker with it to encertain it provides sense. So here when again is the Isosceles Triangle Theorem:

If 2 sides of a triangle are congruent, then angles oppowebsite those sides are congruent.

To make its converse, we could precisely swap the parts, obtaining a little bit of a mish-mash:

If angles oppowebsite those sides are congruent, then two sides of a triangle are congruent.

That is awkward, so tidy up the wording:


Now it renders feeling, yet is it true? Not every converse statement of a conditional statement is true. If the original conditional statement is false, then the converse will certainly additionally be false. If the premise is true, then the converse might be true or false:

If I watch a bear, then I will certainly lie dvery own and reprimary still.If I lie down and also remajor still, then I will certainly watch a bear.

For that converse statement to be true, sleeping in your bed would certainly end up being a bizarre endure.

Or this one:

If I have honey, then I will certainly entice bears.If I attract bears, then I will have honey.

Unless the bears lug honeypots to share with you, the converse is unmost likely ever before to take place. 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 have to prove that BE ≅ BR.

Add the angle bisector from ∠EBR down to base ER. Wright here the angle bisector intersects base ER, label it Point A.

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Now we have two tiny, right triangles wbelow when we had one significant, isosceles triangle: △BEA and also △BAR. Due to the fact that line segment BA is an angle bisector, this makes ∠EBA ≅ ∠RBA. Since line segment BA is used in both smaller sized appropriate triangles, it is congruent to itself. What carry out we have?


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

Let"s view … that"s an angle, one more angle, and a side. That would certainly be the Angle Angle Side Theorem, AAS:

With the triangles themselves proved congruent, their corresponding components are congruent (CPCTC), which provides BE ≅ BR. The converse of the Isosceles Triangle Theorem is true!

Lesson Summary

By working via these exercises, you now are able to identify and also draw an isosceles triangle, mathematically prove congruent isosceles triangles utilizing the Isosceles Triangles Theorem, and mathematically prove the converse of the Isosceles Triangles Theorem. You also should now see the link in between the Isosceles Triangle Theorem to the Side Side Side Postulate and also the Angle Angle Side Theorem.

Next Lesson:

Alternate Exterior Angles

What you"ll learn:

After functioning your method via this lesboy, you will be able to:

Recognize and also attract an isosceles triangleMathematically prove congruent isosceles triangles making use of the Isosceles Triangles TheoremMathematically prove the converse of the Isosceles Triangles TheoremConnect the Isosceles Triangle Theorem to the Side Side Side Postulate and also the Angle Angle Side Theorem