Isosceles Triangle ABC with Parallel Line ED: Investigating Triangle Properties

Question

Below is the Isosceles triangle ABC (AC = AB):

AAABBBCCCDDDEEE

In its interior, a line ED is drawn parallel to CB.

Is the triangle AED also an isosceles triangle?

Video Solution

Solution Steps

00:03 According to the given, the triangle is isosceles
00:06 Equal angles in an isosceles triangle
00:11 Let's mark them with the letter alpha
00:18 ED is parallel to CB according to the given
00:23 Corresponding angles are equal between parallel lines
00:33 We'll mark these also with the letter alpha
00:37 In a triangle, equal sides lie opposite equal angles
00:41 And this is the solution to the question

Step-by-Step Solution

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.

Answer

AED isosceles

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Related Subjects

Types of Triangles Angles In Parallel Lines Collateral angles Adjacent angles Vertically Opposite Angles Alternate angles Corresponding angles Equilateral triangle Isosceles triangle Acute triangle Obtuse Triangle Scalene triangle Triangle Identification of an Isosceles Triangle

Question Types

Types of Triangles: Identifying and defining elements Angles in Parallel Lines: Identifying and defining elements