1. **Problem:** Find the value of $\theta$ when the perpendicular and base are the same in a right-angled triangle. 2. **Formula and Explanation:** In a right-angled triangle, the tangent of angle $\theta$ is the ratio of the perpendicular (opposite side) to the base (adjacent side): $$\tan \theta = \frac{\text{perpendicular}}{\text{base}}$$ If the perpendicular and base are equal, then: $$\tan \theta = \frac{\text{perpendicular}}{\text{perpendicular}} = 1$$ 3. **Find $\theta$:** We need to find $\theta$ such that: $$\tan \theta = 1$$ From trigonometric values, $\tan 45^\circ = 1$. 4. **Answer:** $$\boxed{\theta = 45^\circ}$$ This means the angle $\theta$ is 45 degrees when the perpendicular and base are equal in a right-angled triangle.