A quadrilateral abcd where sides ab=bc=ad as well as angle dab=90 degree and angle abc=60 degree find angle bcd

We are given a quadrilateral \( ABCD \) where the following conditions hold:

- \( AB = BC = AD \)
- \( \angle DAB = 90^\circ \)
- \( \angle ABC = 60^\circ \)

### Step-by-Step Solution:

#### Step 1: Identify the geometry of quadrilateral \( ABCD \)

- Since \( AB = AD \), triangle \( ABD \) is isosceles, and with \( \angle DAB = 90^\circ \), this makes \( \triangle ABD \) a right isosceles triangle.
- In a right isosceles triangle, the two base angles are equal. So, \( \angle ABD = \angle ADB = 45^\circ \).

#### Step 2: Use angle sum in triangle \( ABC \)

- \( \triangle ABC \) has \( AB = BC \), so it is also an isosceles triangle.
- From the given, \( \angle ABC = 60^\circ \), so the remaining two angles in \( \triangle ABC \) must sum to \( 120^\circ \) (since the angle sum in any triangle is \( 180^\circ \)).
- Let \( \angle BAC = \angle ACB = x \). Then,
  \[
  2x + 60^\circ = 180^\circ
  \]
  \[
  2x = 120^\circ \implies x = 60^\circ
  \]
  Hence, \( \angle BAC = \angle ACB = 60^\circ \).

#### Step 3: Find angle \( BCD \)

- Now, consider quadrilateral \( ABCD \). The sum of the interior angles of any quadrilateral is \( 360^\circ \). Therefore, we can write:
  \[
  \angle DAB + \angle ABC + \angle BCD + \angle ADB = 360^\circ
  \]
  Substituting the known angles:
  \[
  90^\circ + 60^\circ + \angle BCD + 45^\circ = 360^\circ
  \]
  Simplifying this:
  \[
  195^\circ + \angle BCD = 360^\circ
  \]
  \[
  \angle BCD = 360^\circ - 195^\circ = 165^\circ
  \]

Thus, the value of \( \angle BCD \) is \( 165^\circ \).

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Would you like more details on any of these steps or have any questions?

Here are 5 related questions for further practice:
1. Find the area of triangle \( ABD \).
2. If \( AB = 5 \) units, find the length of diagonal \( BD \).
3. What type of quadrilateral is \( ABCD \) based on the given information?
4. How does the sum of the angles in a quadrilateral help in solving problems?
5. Can you find the perimeter of \( ABCD \) given \( AB = 5 \) units?

**Tip:** In any isosceles triangle, the base angles are always equal, which can simplify the calculation of unknown angles.

A quadrilateral ABCD where sides AB = BC = AD as well as angle DAB = 90 degrees and angle ABC = 60 degrees. Find angle BCD.

Learn how to find angle BCD in a quadrilateral ABCD where AB = BC = AD, angle DAB = 90°, and angle ABC = 60°. This problem explores key geometry concepts, including isosceles triangles and the angle sum property of quadrilaterals. Suitable for students in Grades 9-11.

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