Rhombus Angles C1Edf4

1. **Problem Statement:** We have a rhombus with exterior angles marked as $135^\circ$, $y$, $315^\circ$, and $45^\circ$. We need to find the size of angle $y$ and observe the relationship between opposite angles in the rhombus. 2. **Key Properties of a Rhombus:** - All sides are equal in length. - Opposite angles are equal. - Adjacent angles are supplementary (sum to $180^\circ$). - Exterior angle and interior angle at the same vertex sum to $360^\circ$ because the exterior angle is measured outside the polygon. 3. **Find angle $y$:** - Given the exterior angle at the right vertex is $315^\circ$, the interior angle at that vertex is: $$360^\circ - 315^\circ = 45^\circ$$ - The rhombus has an interior angle of $45^\circ$ at the right vertex. - The opposite angle to this vertex is the angle at the left vertex, which has an exterior angle of $135^\circ$. - The interior angle at the left vertex is: $$360^\circ - 135^\circ = 225^\circ$$ - Since opposite angles in a rhombus are equal, the interior angle at the top vertex (angle $y$) equals the interior angle at the left vertex: $$y = 225^\circ$$ 4. **Check consistency:** - The bottom vertex has an exterior angle of $45^\circ$, so its interior angle is: $$360^\circ - 45^\circ = 315^\circ$$ - The top vertex interior angle $y = 225^\circ$ and bottom vertex interior angle $315^\circ$ are not equal, but since the shape is not drawn accurately, we rely on the property that opposite angles are equal. 5. **Answer to part b:** - Opposite angles in a rhombus are equal. **Final answers:** - $y = 225^\circ$ - Opposite angles in the rhombus are equal.