Chapter 14.1, Problem 9CE

To determine

To describe: The explanation for Corollary 1 follows from the 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 1 states that an isometry maps an angle to a congruent angle. It is also termed as angle isometry theorem corollary.

According to theorem the two triangles $\Delta ABC$ and $\Delta A\text{'}B\text{'}C\text{'}$ are congruent to each other. They are also isometric. So,

  $\begin{array}{l}\angle A=\angle A\text{'}\\ \angle B=\angle B\text{'}\\ \angle C=\angle C\text{'}\end{array}$

Thus, all the three angles are mapped to congruent triangle.

Therefore, it verified that a triangle is mapped to a congruent triangle by isometry.