To prove that WP≅ZL≅JM, we will write a two-column proof. To begin, we will state the given information and the statement to be proven.
Given: &△ WJZ is equilateral;
&∠ ZWP ≅ ∠ WJM ≅ ∠ JZL
Prove:& WP≅ZL≅JM
We know that △ WJZ is equilateral. Since an equilateral triangle is equiangular, the interior angles of the triangle are congruent to each other.
2. Definition of Equilateral Triangle
∠ ZWJ ≅ ∠ WJZ ≅ ∠ JZW
The angles are congruent to each other if and only if they have the same measure.
3. Definition of Congruence
m∠ ZWJ = m∠ WJZ = m∠ JZW
Notice that each interior angle of △ WJZ consists of two adjacent angles. In this case, we can write the interior angles as the sum of the adjacent angles that form them.
4. Angle Addition Postulate
m∠ ZWJ = m∠ ZWP+m∠ PWJ
m∠ WJZ = m∠ WJM+m∠ MJZ
m∠ JZW = m∠ JZL+m∠ LZW
By the second step, we will substitute m∠ ZWP+m∠ PWJ for m∠ ZWJ, m∠ WJM+m∠ MJZ for m∠ WJZ, and m∠ JZL+m∠ LZW for m∠ JZW.
5. Substitution Property
m∠ ZWP+m∠ PWJ = m∠ WJM+m∠ MJZ = m∠ JZL+m∠ LZW
By the given information, ∠ ZWP ≅ ∠ WJM ≅ ∠ JZL. Thus, the angles have the same angle measure by the definition of congruence.
6. Definition of Congruence
m∠ ZWP =m∠ WJM =m∠ JZL
Now we will substitute m∠ ZWP for m∠ WJM and m∠ JZL.
7. Substitution Property
m∠ ZWP+m∠ PWJ = m∠ ZWP+m∠ MJZ = m∠ ZWP+m∠ LZW
Since each side has m∠ ZWP as a term, we can simplify the equation by subtracting m∠ ZWP from each side.
8. Subtraction Property
m∠ PWJ = m∠ MJZ = m∠ LZW
By the definition of congruence, we can conclude that the angles in the eighth step are congruent.
9. Definition of Congruence
∠ PWJ ≅ ∠ MJZ ≅ ∠ LZW
Looking at the figure, we can see that two angles and the included side of each triangle are congruent to each other. With this, we can conclude that △ WZL, △ ZJM, and △ JWP are congruent by the Angle-Side-Angle Theorem.
10. Angle-Side-Angle Theorem
△ WZL≅ △ ZJM≅ △ JWP
Since the corresponding parts of the congruent triangles are congruent, we can finish the proof by stating that WP≅ZL≅JM.
11. CPCTC
WP≅ZL≅JM
Combining these steps, let's construct the two-column proof.
