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: &â–³ WJZ is equilateral;
&∠ZWP ≅ ∠WJM ≅ ∠JZL
Prove:& WP≅ZL≅JM
We know that â–³ 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 ≅ ∠JZWThe 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 â–³ 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 â–³ WZL, â–³ ZJM, and â–³ 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.
