Chapter 5.6, Problem 19PSC

To determine

To prove:JMPis anequilateral triangle.

Explanation of Solution

Given information:

ΔKOR is an equilateral triangle.

KOPR is a parallelogram.

KMOR is a parallelogram.

Proof:

  

It is given that,

ΔKOR is an equilateral triangle.

KOPR is a parallelogram.

KMOR is a parallelogram.

Opposite sides of parallelogram are parallel.

  $\begin{array}{l}\overline{JM }\parallel \overline{RO }\\ \overline{JP }\parallel \overline{KO }\end{array}$

Equilateral triangle has its all sides congruent.

  $\overline{KO }\cong \overline{OR }\cong \overline{KR }$

Opposite sides of parallelogram are congruent.

  $\begin{array}{l}\overline{KR }\cong \overline{MO }\\ \overline{KM }\cong \overline{RO }\end{array}$

Opposite sides of parallelogram are congruent.

  $\begin{array}{l}\overline{RP }\cong \overline{KO }\\ \overline{KR }\cong \overline{OP }\end{array}$

By substitution property, we get

  $\begin{array}{l}\overline{KR }\cong \overline{MO }\cong \overline{KM }\cong \overline{RO }\\ \overline{RP }\cong \overline{KO }\cong \overline{KR }\cong \overline{OP }\end{array}$

Two pairs of opposite sides are parallel.

JKOR is a parallelogram.

Opposite sides of parallelogram are congruent.

  $\begin{array}{l}\overline{JK }\cong \overline{OR }\\ \overline{KO }\cong \overline{JR }\end{array}$

By substitution property, we get

  $\overline{JK }\cong \overline{JR }$

By addition property, we get

  $\overline{JM }\cong \overline{JP }\cong \overline{MP}$

Triangle is equilateral triangle if all sides congruent.

JMP is an equilateral triangle.