Chapter 6.6, Problem 30PPS

To determine

To prove:

The following statement with paragraph proof:

Theorem 6.23: 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

  $\overline{\mathrm{A}\mathrm{C}}\cong \overline{\mathrm{B}\mathrm{D}}$

Proof:

Here, it is given that $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is trapezoid. Draw auxiliary line $\overline{\mathrm{A}\mathrm{E}}\perp \overline{\mathrm{D}\mathrm{C}}$ and $\overline{\mathrm{B}\mathrm{F}}\perp \overline{\mathrm{D}\mathrm{C}}$ . Know that right angle is formed by the perpendicular lines so, $\angle \mathrm{A}\mathrm{E}\mathrm{F}$ and $\angle \mathrm{B}\mathrm{F}\mathrm{E}$ are right angles. As per the definition of the triangle $\mathrm{\Delta }\mathrm{A}\mathrm{E}\mathrm{C}$ and $\mathrm{\Delta }\mathrm{B}\mathrm{F}\mathrm{D}$ are right angle triangles. As opposite side of a trapezoid are congruent $\overline{\mathrm{E}\mathrm{B}}\cong \overline{\mathrm{B}\mathrm{C}}$ .

  

According to hypotenuse leg theorem $\mathrm{\Delta }\mathrm{A}\mathrm{E}\mathrm{C}\cong \mathrm{\Delta }\mathrm{B}\mathrm{F}\mathrm{D}$ . Know that the corresponding parts of congruent triangles are congruent so: $\angle \mathrm{A}\mathrm{C}\mathrm{D}\cong \angle \mathrm{B}\mathrm{D}\mathrm{C}$ . According to the reflexive property of congruence $\mathrm{\Delta }\mathrm{A}\mathrm{E}\mathrm{C}\cong \mathrm{\Delta }\mathrm{B}\mathrm{F}\mathrm{D}$ since $\overline{\mathrm{D}\mathrm{C}}\cong \overline{\mathrm{D}\mathrm{C}}$ . By CPCTC $\overline{\mathrm{A}\mathrm{D}}\cong \overline{\mathrm{B}\mathrm{C}}$ . Thus, $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is an isosceles.

Hence proved