Chapter 4.6, Problem 45HP

To determine

To prove:

  $\overline{WP}\cong \overline{ZL}\cong \overline{JM}$

Explanation of Solution

Given Information:

  $\Delta WJZ$ is equilateral, $\angle ZWP\cong \angle WJM\cong \angle JZL$ .

  

Proof:

  $\Delta WJZ$ is equilateral, since an equilateral $\Delta$ is equiangular, $\angle ZWJ\cong \angle WJZ\cong \angle JZW$ .

So, $\angle ZWP\cong \angle WJM\cong \angle JZL$ , $m\angle ZWJ=m\angle WJZ=m\angle JZW$ bythe definition of congruence.

Since $\angle ZWP\cong \angle WJM\cong \angle JZL$ , $m\angle ZWP=\angle WJM=m\angle JZL$ by the definition of congruence.

By the AngleAddition Postulate,

  $m\angle ZWJ=m\angle ZWP+m\angle PWJ$

  $m\angle WJZ=m\angle WJM+m\angle MJZ$

  $m\angle JZW=m\angle JZL+m\angle LZW$

By substitution,

  $m\angle ZWP+m\angle PWJ=m\angle WJM+m\angle MJZ=m\angle JZL+m\angle LZW$

Again, by substitution,

  $m\angle ZWP+m\angle PWJ=m\angle ZWP+m\angle PJZ=m\angle ZWP+m\angle LZW$

Bythe Subtraction Property,

  $m\angle PWJ=m\angle PJZ=m\angle LZW$

By the definition of congruence,

  $\angle PWJ\cong \angle PJZ\cong \angle LZW$

So, by ASA,

  $\Delta WZL\cong \Delta ZLM\cong \Delta JWP$

By CPCTC,

  $\overline{WP}\cong \overline{ZL}\cong \overline{JM}$

Hence, it is proved that $\overline{WP}\cong \overline{ZL}\cong \overline{JM}$ .