Isosceles Triangle to organize (Proof, Converse, & Examples)

Isosceles triangles have equal foot (that"s what the word "isosceles" means). Yippee for them, yet what execute we know about their basic angles? exactly how do us know those space equal, too? us reach right into our geometer"s toolbox and also take the end the Isosceles Triangle Theorem. No must plug it in or recharge its batteries -- it"s ideal there, in your head!

Isosceles Triangle

Here we have actually on display the majestic isosceles triangle, △DUK. You can attract one yourself, using △DUK as a model.

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Hash marks show sides ∠DU ≅ ∠DK, which is your tip-off that you have actually an isosceles triangle. If these two sides, referred to as 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 kind of triangle, △DUK has three interior angles: ∠D, ∠U, and ∠KAll three internal angles are acuteLike any kind of triangle, △DUK has three sides: DU, UK, and also DK∠DU ≅ ∠DK, for this reason we describe those twins together legsThe third side is dubbed the base (even as soon as the triangle is no sitting on that side)The two angles formed between base and also legs, ∠DUK and also ∠DKU, or ∠D and ∠K for short, are called base angles:

Isosceles Triangle Theorem

Knowing the triangle"s parts, below is the challenge: just how do we prove that the base angles space congruent? the is the love of the Isosceles Triangle Theorem, i beg your pardon is developed as a conditional (if, then) statement:

To mathematically prove this, we need to introduce a typical line, a line constructed from an internal angle to the midpoint of the contrary side. We uncover Point C on basic UK and also construct heat segment DC:


There! That"s simply DUCKy! Look in ~ the 2 triangles formed by the median. We are given:

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

We simply showed the the 3 sides that △DUC space congruent to △DCK, which method you have the Side next Side Postulate, which provides congruence. So if the 2 triangles are congruent, then corresponding parts of congruent triangles room congruent (CPCTC), which means …

∠U ≅ ∠K

Converse that the Isosceles Triangle Theorem

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

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

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

If angles opposite those sides room congruent, then 2 sides that a triangle room congruent.

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


Now it provides sense, but is it true? not every converse explain of a conditional statement is true. If the original conditional explain 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 certainly lie down and remain still.If i lie down and also remain still, then ns will see a bear.

For the converse explain to be true, sleeping in her bed would come to be a bizarre experience.

Or this one:

If I have honey, then i will attract bears.If I entice bears, then ns will have honey.

Unless the bears bring honeypots come share with you, the converse is unlikely ever to happen. And bears room famously selfish.

Proving the Converse Statement

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


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

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

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Now we have two small, ideal 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 edge bisector, this provides ∠EBA ≅ ∠RBA. Due to the fact that line segment BA is supplied in both smaller best triangles, that is congruent come itself. What perform we have?


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

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

With the triangle themselves verified congruent, their matching parts are congruent (CPCTC), which provides BE ≅ BR. The converse that the Isosceles Triangle organize is true!

Lesson Summary

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

Next Lesson:

Alternate Exterior Angles

What you"ll learn:

After working your method through this lesson, you will be maybe to:

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