Question

ABCD-ПАРАЛЕЛОГРАМ AB=BE , E є BC Довести що AE БІСЕКТРИСАИ

Answer 1

1. Рассмотрим треугольник АВЕ – по условию он равнобедренный (АВ=ВЕ) значит ∠ВАЕ=АЕВ (как углы при основании равнобедренного треугольника).

2.  Так как АВСД – параллелограмм то ∠ДАЕ=АЕВ – как внутренние накрест лежащие при ВС||АД и АЕ – секущей.

Так как ∠ВАЕ=АЕВ и ∠ДАЕ=АЕВ то ВАЕ=ДАЕ, что означает, что АЕ делит ∠ВАД на равные части, то есть является биссектрисой угла ВАД

Answer 2

ABCD is a parallelogram with AB = BE and E lies on BC. We need to prove that AE is the bisector of angle BAC.

To prove that AE is the bisector of angle BAC, we can use the property of parallelograms and the fact that AB = BE.

Property of Parallelograms:

In a parallelogram, opposite sides are equal in length and opposite angles are equal in measure.

Proof:

Since ABCD is a parallelogram, we know that AB = CD and BC = AD.

Given that AB = BE, we can conclude that BE = CD.

Now, let's consider triangle ABE and triangle CDE.

In triangle ABE: - AB = BE (given) - Angle ABE = Angle CDE (opposite angles in a parallelogram are equal)

In triangle CDE: - CD = BE (proved above) - Angle CDE = Angle ABE (opposite angles in a parallelogram are equal)

From the above information, we can conclude that triangle ABE is congruent to triangle CDE by the Side-Angle-Side (SAS) congruence criterion.

Since the triangles are congruent, their corresponding angles are equal. Therefore, Angle BAE = Angle DCE.

Since Angle BAE and Angle DCE are corresponding angles, we can conclude that AE is the bisector of angle BAC.

Hence, we have proved that AE is the bisector of angle BAC in parallelogram ABCD, where AB = BE and E lies on BC.

Please let me know if you need any further clarification.