Chapter 6.6, Problem 29PPS

To determine

To prove:

The following statement with paragraph proof:

Theorem 6.22: Trapezoid $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is an isosceles.

Explanation of Solution

Given information:

  $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is trapezoid

  $\angle \mathrm{D}\cong \angle \mathrm{C}$

Proof:

Here, it is given that $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is trapezoid. Draw auxiliary line $\overline{\mathrm{E}\mathrm{B}}\parallel \overline{\mathrm{A}\mathrm{D}}$ . According to the corollary angles theorem: $\angle \mathrm{D}\cong \mathrm{B}\mathrm{E}\mathrm{C}$ . Use a transitive property $\mathrm{B}\mathrm{E}\mathrm{C}\cong \angle \mathrm{C}$ since $\angle \mathrm{D}\cong \angle \mathrm{C}$ . Therefore, $\mathrm{\Delta }\mathrm{E}\mathrm{B}\mathrm{C}$ is isosceles and $\overline{\mathrm{E}\mathrm{B}}\cong \overline{\mathrm{B}\mathrm{C}}$ .

  

As per the definition of trapezoid $\overline{\mathrm{A}\mathrm{B}}\parallel \overline{\mathrm{D}\mathrm{E}}$ . $\mathrm{A}\mathrm{B}\mathrm{E}\mathrm{D}$ is a parallelogram as both pairs of opposite sides are parallel. So, get that $\overline{\mathrm{A}\mathrm{D}}\cong \overline{\mathrm{E}\mathrm{B}}$ . According to transitive property $\overline{\mathrm{B}\mathrm{C}}\cong \overline{\mathrm{A}\mathrm{D}}$ . Thus, $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is an isosceles which is trapezoid in original.

Hence proved