Congruence And Similarity - GCSE Maths - Steps & Examples - Free Printable

Problem Analysis:

The problem involves two triangles, and we need to solve for a specific measurement or property. Let's break it down step by step.

#### Step 1: Analyze the First Triangle
The first triangle is given with the following information:
- One angle is \(45^\circ\).
- Another angle is \(35^\circ\).
- The side opposite the \(35^\circ\) angle is \(9 \, \text{cm}\).

To find the third angle of the triangle:
\[
\text{Sum of angles in a triangle} = 180^\circ
\]
\[
\text{Third angle} = 180^\circ - 45^\circ - 35^\circ = 100^\circ
\]

So, the angles of the first triangle are:
- \(45^\circ\)
- \(35^\circ\)
- \(100^\circ\)

#### Step 2: Analyze the Second Triangle
The second triangle is given with the following information:
- One angle is \(45^\circ\).
- Another angle is \(100^\circ\).
- The side opposite the \(45^\circ\) angle is \(9 \, \text{cm}\).

To find the third angle of the second triangle:
\[
\text{Sum of angles in a triangle} = 180^\circ
\]
\[
\text{Third angle} = 180^\circ - 45^\circ - 100^\circ = 35^\circ
\]

So, the angles of the second triangle are:
- \(45^\circ\)
- \(100^\circ\)
- \(35^\circ\)

#### Step 3: Compare the Triangles
Upon comparing the two triangles:
- The first triangle has angles \(45^\circ\), \(35^\circ\), and \(100^\circ\).
- The second triangle also has angles \(45^\circ\), \(35^\circ\), and \(100^\circ\).

Since both triangles have the same set of angles, they are similar triangles by the Angle-Angle (AA) similarity criterion.

#### Step 4: Use Similarity to Solve for the Unknown Side
In similar triangles, corresponding sides are proportional. From the problem, we know:
- The side opposite the \(35^\circ\) angle in the first triangle is \(9 \, \text{cm}\).
- The side opposite the \(45^\circ\) angle in the second triangle is \(9 \, \text{cm}\).

Let the side opposite the \(35^\circ\) angle in the second triangle be \(x \, \text{cm}\). Since the triangles are similar, the ratio of corresponding sides is equal. We can set up the proportion based on the sides opposite the \(35^\circ\) and \(45^\circ\) angles:

\[
\frac{\text{Side opposite } 35^\circ \text{ in first triangle}}{\text{Side opposite } 45^\circ \text{ in first triangle}} = \frac{\text{Side opposite } 35^\circ \text{ in second triangle}}{\text{Side opposite } 45^\circ \text{ in second triangle}}
\]

However, we notice that the side opposite the \(45^\circ\) angle in the second triangle is already given as \(9 \, \text{cm}\), and the side opposite the \(35^\circ\) angle in the first triangle is also \(9 \, \text{cm}\). This implies that the triangles are not only similar but also congruent by the Side-Angle-Side (SAS) congruence criterion.

Thus, the side opposite the \(35^\circ\) angle in the second triangle is also \(9 \, \text{cm}\).

#### Final Answer:
\[
\boxed{9}
\]