Consider isosceles triangle ABC in Figure 1.įigure 1 An isosceles triangle with a median. Isosceles triangles are special and because of that there are unique relationships that involve their internal line segments.

Summary of Coordinate Geometry Formulas.

Slopes: Parallel and Perpendicular Lines.

Similar Triangles: Perimeters and Areas.

Proportional Parts of Similar Triangles.

Formulas: Perimeter, Circumference, Area.

Proving that Figures Are Parallelograms.

Triangle Inequalities: Sides and Angles.

Special Features of Isosceles Triangles.

Classifying Triangles by Sides or Angles.Lines: Intersecting, Perpendicular, Parallel.Let's see … that's an angle, another angle, and a side. Since line segment B A is used in both smaller right triangles, it is congruent to itself. Since line segment B A is an angle bisector, this makes ∠ E B A ≅ ∠ R B A. Now we have two small, right triangles where once we had one big, isosceles triangle: △ B E A and △ B A R. Where the angle bisector intersects base E R, label it P o i n t A.

Given that ∠ B E R ≅ ∠ B R E, we must prove that B E ≅ B R.Īdd the angle bisector from ∠ E B R down to base E R. To prove the converse, let's construct another isosceles triangle, △ B E R. Unless the bears bring honeypots to share with you, the converse is unlikely ever to happen. If I attract bears, then I will have honey. Or this one: If I have honey, then I will attract bears. If I lie down and remain still, then I will see a bear.įor that converse statement to be true, sleeping in your bed would become a bizarre experience. 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 the original conditional statement is false, then the converse will also be false. Now it makes sense, but is it true? Not every converse statement of a conditional statement is true. The Converse of the Isosceles Triangle Theorem states: If two angles of a triangle are congruent, then sides opposite those angles 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. So here once again is the Isosceles Triangle Theorem: If two sides of a triangle are congruent, then angles opposite those sides are congruent. You may need to tinker with it to ensure it makes sense. The converse of a conditional statement is made by swapping the hypothesis (if …) with the conclusion (then …). So if the two triangles are congruent, then corresponding parts of congruent triangles are congruent (CPCTC), which means …Ĭonverse of the Isosceles Triangle Theorem We just showed that the three sides of △ D U C are congruent to △ D C K, which means you have the Side Side Side Postulate, which gives congruence. There! That's just D U C K y! Look at the two triangles formed by the median. We find P o i n t C on base U K and construct line segment D C: 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. The Isosceles Triangle Theorem states: If two sides of a triangle are congruent, then angles opposite those sides are congruent. 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 two angles formed between base and legs, ∠ D U K and ∠ D K U, or ∠ D and ∠ K for short, are called base angles:.The third side is called the base (even when the triangle is not sitting on that side).∠ D U ≅ ∠ D K, so we refer to those twins as legs.Like any triangle, △ D U K has three sides: D U, U K, and D K.Like any triangle, △ D U K has three interior angles: ∠ D, ∠ U, and ∠ K.What else have you got? Properties of an Isosceles Triangle If these two sides, called legs, are equal, then this is an isosceles triangle. Hash marks show sides ∠ D U ≅ ∠ D K, which is your tip-off that you have an isosceles triangle. You can draw one yourself, using △ D U K as a model. Here we have on display the majestic isosceles triangle, △ D U K. No need to plug it in or recharge its batteries - it's right there, in your head! 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. Isosceles triangles have equal legs (that's what the word 'isosceles' means). Isosceles Triangle Theorem (Proof, Converse, & Examples)

Properties of isosceles triangles