To construct the quadrilateral ABCD with the given measurements, follow these steps:

1. Draw Line Segment AB:

   • Draw a line segment $AB$ of length 6 cm.

2. Construct Angle $\angle A = 55^\circ$:

   • At point $A$, use a protractor to construct an angle of $55^\circ$.
   • Draw a ray $AX$ from point $A$ along this angle.

3. Draw Line Segment BC:

   • From point $B$, draw an arc with a radius of 5 cm to intersect the ray $AX$. Label the intersection point as $C$.

4. Construct Angle $\angle C = 110^\circ$:

   • At point $C$, use a protractor to construct an angle of $110^\circ$.
   • Draw a ray $CY$ from point $C$ along this angle.

5. Construct Angle $\angle D = 90^\circ$:

   • At point $D$, use a protractor to construct a right angle ($90^\circ$).
   • Draw a perpendicular line to $CD$ that intersects $CY$. Label this intersection point as $D$.

6. Complete the Quadrilateral:

   • Connect points $D$ and $A$ with a straight line.

Summary of the Construction Steps:

1. Draw $AB = 6$ cm.
2. Construct $\angle A = 55^\circ$ and draw ray $AX$.
3. Draw arc from $B$ with radius 5 cm to intersect $AX$ at $C$.
4. Construct $\angle C = 110^\circ$ and draw ray $CY$.
5. Construct $\angle D = 90^\circ$ at point $D$ and draw perpendicular to $CD$.
6. Connect points $D$ and $A$ to complete the quadrilateral.

Additional Details:

• Verify that all sides and angles match the given measurements.
• Check that the sum of the angles $\angle A + \angle B + \angle C + \angle D = 360^\circ$.

If there are any questions or you need further details, please let me know.

Follow-Up Questions:

1. How would you verify that the constructed quadrilateral is accurate?
2. What tools are necessary to construct this quadrilateral?
3. Can you explain the importance of each angle in determining the shape of the quadrilateral?
4. How would the construction change if $\angle A$ was different?
5. What is the sum of the interior angles of any quadrilateral?
6. How can you ensure that the right angle at $\angle D$ is precise?
7. What other properties of quadrilaterals can be derived from the given angles and sides?
8. How would you construct this quadrilateral if only a ruler and compass were available?

Tip:

Always double-check your measurements and angle constructions to ensure accuracy in geometric constructions.