A triangle which has two of its sides equal in length. Show
Try this Drag the orange dots on each vertex to reshape the triangle. Notice it always remains an isosceles triangle, the sides AB and AC always remain equal in length The word isosceles is pronounced "eye-sos-ell-ease" with the emphasis on the 'sos'. It is any triangle that has two sides the same length.If all three sides are the same length it is called an equilateral triangle. Obviously all equilateral triangles also have all the properties of an isosceles triangle.
L is the length of a leg A is the altitude To find the leg length given the base and altitude, use the formula: where: B is the length of the base A is the altitude To find the altitude given the base and leg, use the formula: where: L is the length of a leg B is the base If you are given one interior angle of an isosceles triangle you can find the other two. For example, We are given the angle at the apex as shown on the right of 40°. We know that the interior angles of all triangles add to 180°. So the two base angles must add up to 180-40, or 140°. Since the two base angles are congruent (same measure), they are each 70°. If we are given a base angle of say 45°, we know the base angles are congruent (same measure) and the interior angles of any triangle always add to 180°. So the apex angle must be 180-45-45 or 90°. Other triangle topicsGeneralPerimeter / AreaTriangle typesTriangle centersCongruence and SimilaritySolving triangles
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Isosceles triangles have equal legs (that's what the word "isosceles" means). 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. No need to plug it in or recharge its batteries -- it's right there, in your head! Isosceles TriangleHere we have on display the majestic isosceles triangle, △DUK. You can draw one yourself, using △DUK as a model. Hash marks show sides ∠DU ≅ ∠DK, which is your tip-off that you have an isosceles triangle. If these two sides, called legs, are equal, then this is an isosceles triangle. What else have you got? Properties of an Isosceles TriangleLet's use △DUK to explore the parts:
Isosceles Triangle TheoremKnowing 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 Isosceles Triangle Theorem states: If two sides of a triangle are congruent, then angles opposite those sides are congruent. 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. We find Point C on base UK and construct line segment DC: There! That's just DUCKy! Look at the two triangles formed by the median. We are given:
We just showed that the three sides of △DUC are congruent to △DCK, which means you have the Side Side Side Postulate, which gives congruence. So if the two triangles are congruent, then corresponding parts of congruent triangles are congruent (CPCTC), which means … Converse of the Isosceles Triangle TheoremThe converse of a conditional statement is made by swapping the hypothesis (if …) with the conclusion (then …). You may need to tinker with it to ensure it makes sense. So here once again is the Isosceles Triangle Theorem: If two sides of a triangle are congruent, then angles opposite those sides 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. That is awkward, so tidy up the wording:
The Converse of the Isosceles Triangle Theorem states: If two angles of a triangle are congruent, then sides opposite those angles are congruent. Now it makes sense, but is it true? Not every converse statement of a conditional statement is true. If the original conditional statement is false, then the converse will also be false. 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 I lie down and remain still, then I will see a bear. For that converse statement to be true, sleeping in your bed would become a bizarre experience. Or this one: If I have honey, then I will attract bears. If I attract bears, then I will have honey. Unless the bears bring honeypots to share with you, the converse is unlikely ever to happen. And bears are famously selfish. Proving the Converse StatementTo prove the converse, let's construct another isosceles triangle, △BER. Given that ∠BER ≅ ∠BRE, we must prove that BE ≅ BR. Add the angle bisector from ∠EBR down to base ER. Where the angle bisector intersects base ER, label it Point A. Now we have two small, right triangles where once we had one big, isosceles triangle: △BEA and △BAR. Since line segment BA is an angle bisector, this makes ∠EBA ≅ ∠RBA. Since line segment BA is used in both smaller right triangles, it is congruent to itself. What do we have?
Let's see … that's an angle, another angle, and a side. That would be the Angle Angle Side Theorem, AAS:
The AAS Theorem states: If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. With the triangles themselves proved congruent, their corresponding parts are congruent (CPCTC), which makes BE ≅ BR. The converse of the Isosceles Triangle Theorem is true! Lesson SummaryBy working through these exercises, you now are able to recognize and draw an isosceles triangle, mathematically prove congruent isosceles triangles using the Isosceles Triangles Theorem, and mathematically prove the converse of the Isosceles Triangles Theorem. You also should now see the connection between the Isosceles Triangle Theorem to the Side Side Side Postulate and the Angle Angle Side Theorem. Next Lesson:Alternate Exterior Angles
Instructor: Malcolm M. |