Below is the Isosceles triangle ABC (AC = AB):
In its interior, a line ED is drawn parallel to CB.
Is the triangle AED also an isosceles triangle?
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Below is the Isosceles triangle ABC (AC = AB):
In its interior, a line ED is drawn parallel to CB.
Is the triangle AED also an isosceles triangle?
To demonstrate that triangle AED is isosceles, we must prove that its hypotenuses are equal or that the opposite angles to them are equal.
Given that angles ABC and ACB are equal (since they are equal opposite bisectors),
And since ED is parallel to BC, the angles ABC and ACB alternate and are equal to angles ADE and AED (alternate and equal angles between parallel lines)
Opposite angles ADE and AED are respectively sides AD and AE, and therefore are also equal (opposite equal angles, the legs of triangle AED are also equal)
Therefore, triangle ADE is isosceles.
AED isosceles
Is the triangle in the drawing a right triangle?
Parallel lines create equal alternate angles! When ED is parallel to BC, the angles at B and C (which are equal in isosceles triangle ABC) become equal to angles at D and E respectively.
In an isosceles triangle, the base angles are always equal. Since AC = AB, triangle ABC has equal base angles at B and C.
The position doesn't matter as long as ED remains parallel to BC! The parallel line theorem creates equal alternate angles regardless of where the line intersects the triangle's sides.
Use the Isosceles Triangle Theorem: if two angles in a triangle are equal (∠ADE = ∠AED), then the sides opposite those angles must be equal (AD = AE).
Alternate angles are on opposite sides of the transversal and equal when lines are parallel. Corresponding angles are in the same relative position. Both types are equal with parallel lines!
No! As long as triangle ABC is isosceles and ED is parallel to BC, triangle AED must be isosceles due to the parallel line angle relationships.
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