1. **Problem Statement:** Given a rectangle CMXZ, determine whether each of the following statements is always, sometimes, or never true: - $CM \cong MX$ - $ZX \cong CM$ - $\angle Z \cong \angle M$ - $ZM \cong CX$ - $CM \perp ZX$ - $\angle CMZ \cong \angle XZM$ 2. **Recall properties of rectangles:** - Opposite sides are congruent and parallel. - All angles are right angles ($90^\circ$). - Diagonals are congruent. 3. **Analyze each statement:** **(a) $CM \cong MX$** - $CM$ and $MX$ are adjacent sides of the rectangle. - Adjacent sides in a rectangle can be different lengths. - So, $CM \cong MX$ is **sometimes true** (only if the rectangle is a square). **(b) $ZX \cong CM$** - $ZX$ and $CM$ are opposite sides. - Opposite sides in a rectangle are always congruent. - So, $ZX \cong CM$ is **always true**. **(c) $\angle Z \cong \angle M$** - All angles in a rectangle are right angles. - So, $\angle Z \cong \angle M$ is **always true**. **(d) $ZM \cong CX$** - $ZM$ and $CX$ are diagonals of the rectangle. - Diagonals in a rectangle are always congruent. - So, $ZM \cong CX$ is **always true**. **(e) $CM \perp ZX$** - $CM$ and $ZX$ are opposite sides, so they are parallel, not perpendicular. - So, $CM \perp ZX$ is **never true**. **(f) $\angle CMZ \cong \angle XZM$** - $\angle CMZ$ and $\angle XZM$ are angles formed by the diagonals intersecting at $M$ and $Z$ respectively. - In a rectangle, diagonals bisect each other but do not necessarily form congruent angles. - So, $\angle CMZ \cong \angle XZM$ is **sometimes true** (only if the rectangle is a square). 4. **Summary:** - $CM \cong MX$: sometimes - $ZX \cong CM$: always - $\angle Z \cong \angle M$: always - $ZM \cong CX$: always - $CM \perp ZX$: never - $\angle CMZ \cong \angle XZM$: sometimes