Chapter 14.1, Problem 10CE

To determine

To describe: The explanation for Corollary 2 that follows from Theorem $14-1$ .

Explanation of Solution

Given information: The theorem 14-1 states that an isometry maps a triangle to a congruent triangle.

Given: Isometry T : $\Delta ABC\to \Delta A^{\prime }{B}^{\prime }{C}^{\prime }$

Prove: $\Delta ABC\cong \Delta A^{\prime }{B}^{\prime }{C}^{\prime }$

  

Corollary 2 states that an isometry maps a polygon to a polygon with the same area. It is also termed as polygon isometry theorem corollary.

By the theorem 14-1 it can be stated that $\Delta ABC$ and $\Delta A^{\prime }{B}^{\prime }{C}^{\prime }$ are congruent to each other and isometric. If two triangles are congruent to each other as per the theorem, then their areas such as $\text{ar}\left(\Delta ABC\right)=\text{ar}\left(\Delta A\text{'}B\text{'}C\text{'}\right)$ , two triangle are mapped to congruent triangle thus have equal areas.

Hence, it verified that a polygon to a polygon of same area is mapped by isometry.