Let's walk through the math exercises in this worksheet.

1. Developing Proof (Congruence in Right Triangles):

Given:

• ∠WVZ and ∠VWX are right angles.
• $WZ \equiv VX$

Prove:

• $\triangle WVZ \cong \triangle VWX$

Reasoning:

• Step 1: We are given that ∠WVZ and ∠VWX are right angles (this is already filled in).
• Step 2: We are given that $WZ \equiv VX$ (this is also given).
• Step 3: We can conclude that $WZ \equiv VX$ is a congruent side by the Reflexive Property of Congruence (shared side $WV$).
• Step 4: Finally, by the Hypotenuse-Leg (HL) Theorem, the triangles are congruent.

2. If m∠X = 54, what is m∠Z?

In Exercise 1, since $\triangle WVZ$ and $\triangle VWX$ are congruent right triangles, ∠X and ∠Z are complementary to the right angles. Therefore:

• m∠Z = 90° - 54° = 36°.

3. If m∠X = 54, what is m∠VWZ?

• Since $\triangle WVZ \cong \triangle VWX$, the corresponding angles must be equal. Therefore, m∠VWZ = m∠VWX = 54°.

4. Can you prove ΔWVZ and ΔVWX are congruent without the HL Theorem?

• Yes, you could use the Angle-Angle-Side (AAS) Postulate. Both triangles share the right angle (∠WVZ and ∠VWX), and two pairs of angles are equal (∠V and the 54° angle). One side (WZ and VX) is also congruent, which satisfies AAS.

5 & 6. Find the values of x and y so that the triangles are congruent by HL Theorem.

For question 5:

• $x - 4$ must be equal to $y + 3$.
• $2y$ must equal $x$.

Solving the system:

• From the second equation, $x = 2y$.
• Substitute into the first equation: $2y - 4 = y + 3$, which simplifies to $y = 7$.
• Substituting back: $x = 2(7) = 14$.

So, $x = 14$ and $y = 7$.

For question 6:

• $y + 5 = 2y - x$
• $y + x = x + 7$

From the second equation, $y = 7$. Substituting into the first equation:

• $7 + 5 = 2(7) - x$, which simplifies to $x = 9$.

So, $y = 7$ and $x = 9$.

7. Reasoning about the LL Theorem:

The Leg-Leg (LL) Theorem is essentially the Side-Side-Side (SSS) Postulate applied to right triangles. It states that if the two legs of one right triangle are congruent to the two legs of another right triangle, then the triangles are congruent. The proof of the LL Theorem relies on SSS because if both legs are congruent, the hypotenuse is automatically congruent by the Pythagorean theorem.

Do you have any questions or want more details on a particular problem? Here are 5 related questions to extend your understanding:

1. How does the Reflexive Property work in geometry?
2. Could you prove congruence using other theorems besides HL or AAS?
3. What are other situations where the Pythagorean theorem is used to prove congruence?
4. How do we know that complementary angles in a triangle add up to 90°?
5. Can you explain how SSS works differently than HL for proving triangle congruence?

Tip: For triangle proofs, always look for shared sides or angles first, as these are common starting points in proving congruence!