Given the problem, we need to find the angles $\angle AED$ and $\angle ABC$ in a rhombus $ABCD$, where point $E$ bisects side $BC$ and is connected to vertices $A$ and $D$.

Step 1: Analyze the Rhombus Properties

• In a rhombus, all sides are equal: $AB = BC = CD = DA$.
• Opposite angles are equal.
• The diagonals of a rhombus bisect each other at right angles (90°).

Step 2: Understanding the Point E

• Since $E$ bisects $BC$, we have $BE = EC$.
• This makes $E$ the midpoint of $BC$.

Step 3: Find $\angle ABC$

• Since $ABCD$ is a rhombus, and $AB = BC$, by the property of a rhombus, the diagonals bisect each other at right angles. Therefore: $\angle ABC = \angle DAB$
• In any rhombus, $\angle ABC$ can be calculated using the fact that diagonals bisect the interior angles. Thus: $\angle ABC = \frac{\text{interior angle}}{2} = \frac{180^\circ - \theta}{2}$ where $\theta$ is the angle formed by the diagonals with the sides.

Step 4: Find $\angle AED$

• Consider triangles $\triangle AEB$ and $\triangle DEC$.

• Since $E$ is the midpoint of $BC$ and $AB = CD$, $\triangle AEB$ and $\triangle DEC$ are congruent.

• Because $E$ is the midpoint and $ABCD$ is a rhombus, $AE$ and $DE$ are equal, making $\triangle AED$ an isosceles triangle.

  We can calculate $\angle AED$ using the exterior angle property or by considering the fact that in isosceles triangles, base angles are equal.

Calculations

For $\angle ABC$:

• In a rhombus, each interior angle $\angle ABC$ can be represented as: $\angle ABC = \frac{180^\circ - \theta}{2}$ Here, $\theta = 90^\circ$, because diagonals bisect angles: $\angle ABC = 90^\circ$

For $\angle AED$:

• $\triangle AED$ is isosceles with $AE = ED$.
• $\angle AED = 180^\circ - 2 \times \angle AEB$.
• Since $\angle AEB$ is a straight line, it's $180^\circ - \theta$ where $\theta$ is the angle at the intersection.

Thus, in a rhombus:

• $\angle AED$ = 90° (as calculated from the symmetry and isosceles triangle properties)
• $\angle ABC$ = 90°

Final Answers:

• $\angle ABC = 90^\circ$
• $\angle AED = 90^\circ$

Would you like more details on any step or have any further questions?

Related Questions

1. What are the properties of diagonals in a rhombus?
2. How do the angles change in a parallelogram that is not a rhombus?
3. What is the significance of bisecting a side in quadrilaterals?
4. How does the area of a rhombus relate to its diagonals?
5. What happens to the angles if $E$ was not the midpoint of $BC$?

Tip

In a rhombus, angles between sides are crucial to understanding diagonal properties and can often simplify the calculation of other angles when using symmetry.