We are tasked with calculating the size of angle $\angle CDA$ in the trapezium $ABCD$, where the given dimensions are:

• $AB = 25 \, \text{cm}$
• $BC = 24 \, \text{cm}$
• $CD = 10 \, \text{cm}$
• $\angle ABC = \angle BCD = 90^\circ$

Step 1: Analyze the Shape

Since $ABCD$ is a trapezium with $\angle ABC$ and $\angle BCD$ being $90^\circ$, $AB$ and $CD$ are parallel, and $BC$ is perpendicular to both $AB$ and $CD$.

This makes triangles $\triangle ABC$ and $\triangle BCD$ right-angled.

Step 2: Calculate the Height $AD$ Using Pythagoras

In $\triangle ABC$: $AD^2 = AB^2 - BC^2$ $AD^2 = 25^2 - 24^2 = 625 - 576 = 49$ $AD = \sqrt{49} = 7 \, \text{cm}.$

Step 3: Determine Angle $\angle CDA$

In $\triangle CDA$:

• The height $AD = 7 \, \text{cm}$,
• The base $CD = 10 \, \text{cm}$.

Using the tangent function: $\tan(\angle CDA) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AD}{CD} = \frac{7}{10}.$

$\angle CDA = \tan^{-1} \left( \frac{7}{10} \right).$

Step 4: Calculate $\angle CDA$

Using a calculator: $\angle CDA = \tan^{-1}(0.7) \approx 34.99^\circ.$

To three significant figures: $\angle CDA = 35.0^\circ.$

Final Answer:

The size of $\angle CDA$ is 35.0°.

Let me know if you have questions or want further clarification!

Related Questions:

1. How do you find the area of trapezium $ABCD$?
2. What is the total perimeter of trapezium $ABCD$?
3. Can you find the diagonal $AC$ of the trapezium?
4. What is the angle $\angle DAC$ in triangle $\triangle DAC$?
5. How can trigonometry help in finding unknown angles in different quadrilaterals?

Tip:

Always verify if a shape has perpendicular sides when using trigon