An isosceles triangle is a triangle that has at least 2 congruent sides. The congruent sides of the isosceles triangle are dubbed the legs. The other side is dubbed the base. The angles between the base and the legs are referred to as base angles. The edge made through the two legs is called the vertex angle. One of the vital properties of isosceles triangle is the their base angles are always congruent. This is referred to as the Base angles Theorem.

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

For \(\DeltaDEF, if \(\overlineDE\cong \overlineEF\), then \(\angle D\cong \angle F\).

*
Figure \(\PageIndex1\)

Another essential property that isosceles triangles is the the angle bisector of the vertex angle is additionally the perpendicular bisector the the base. This is dubbed the Isosceles Triangle Theorem. (Note this is just true that the vertex angle.) The converses of the Base angles Theorem and the Isosceles Triangle Theorem room both true as well.

Base angles Theorem Converse: If two angles in a triangle space congruent, then the political parties opposite those angles are likewise congruent. So for \(\Delta DEF\), if \(\angle D\cong \angle F\), climate \(\overlineDE\cong \overlineEF\).

*
Figure \(\PageIndex2\)

Isosceles Triangle organize Converse: The perpendicular bisector that the basic of an isosceles triangle is also the angle bisector that the peak angle. So because that isosceles \(\DeltaDEF\), if \(\overlineEG\perp \overlineDF\) and \(\overlineDG\cong \overlineGF\), climate \(\angle DEG\cong \angle FEG\).

*


Example \(\PageIndex3\)

Which 2 angles room congruent?

*
Figure \(\PageIndex5\)

Solution

This is one isosceles triangle. The congruent angles space opposite the congruent sides. From the arrows we see that \(\angle S\cong \angle U\).

*
api/deki/files/1358/f-d_de914e240e0cf6a0e6476970d4e50f195a1fadcf9ef72d46d58ebc2c%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450" /> figure \(\PageIndex13\)

Determine if the following statements are true or false.

Base angles of one isosceles triangle space congruent. Base angles of one isosceles triangle space complementary. Base angles of one isosceles triangle deserve to be equal to the crest angle. Base angle of one isosceles triangle space acute.

See more: What Slimline Form Factor Is Similar To Atx But Not Compatible

Fill in the proofs below.

Given: Isosceles \(\DeltaCIS\), through base angle \(\angle C\) and \(\angle S\). \overlineIO\) is the edge bisector of \(\angle CIS\) Prove: \(\overlineIO\) is the perpendicular bisector that \(\overlineCS\)f-d_86710951ccd60fc1e610a6006a9b85087731b60338b197b133e18029+IMAGE_TINY+IMAGE_TINY.pngthe base angles of an isosceles triangle are congruent