Consider right triangles ΔABC and ΔXYZ . Angle B and angle Y are congruent. The hypotenuse of each triangle is AC¯¯¯¯¯¯¯¯ and XZ¯¯¯¯¯¯¯¯ , and they are congruent. What other corresponding parts must be congruent to show that ΔABC≅ΔXYZ by the HL Congruence Theorem?(1 point)

Question

Responses

BC¯¯¯¯¯¯¯¯≅XY¯¯¯¯¯¯¯¯
Modifying above upper B upper C with bar congruent to Modifying above upper X upper Y with bar

AB¯¯¯¯¯¯¯¯≅XY¯¯¯¯¯¯¯¯
Modifying above upper A upper B with bar congruent to Modifying above upper X upper Y with bar

BC¯¯¯¯¯¯¯¯≅XZ¯¯¯¯¯¯¯¯
Modifying above upper B upper C with bar congruent to Modifying above upper X upper Z with bar

AB¯¯¯¯¯¯¯¯≅YZ¯¯¯¯¯¯¯

Answer 1

To show that \( \Delta ABC \cong \Delta XYZ \) by the Hypotenuse-Leg (HL) Congruence Theorem, we need to confirm that:

1. The hypotenuses \( AC \) and \( XZ \) are congruent (given).
2. The legs of the triangles must also be congruent.

Given that angle B and angle Y are congruent, the corresponding sides we need to demonstrate as congruent (in addition to the hypotenuses) can be either leg \( BC \) and \( XY \) or leg \( AB \) and \( YZ \).

Therefore, the correct response to show that the triangles are congruent using HL Theorem would be:

\( BC \cong XY \) (which corresponds to the legs of the triangles).

This implies that since we already know the hypotenuses are congruent and one pair of legs is congruent, the triangles are congruent by the HL theorem.

Answer 2

Malik is comparing 2 right triangles. Both triangles have a hypotenuse of 13 units. The first triangle has a leg of 12 units and the other triangle has a leg of 5 units. How could Malik prove that the two triangles are congruent?(1 point)
Responses

He could apply the Pythagorean Theorem to show that the two missing sides make both triangles congruent.
He could apply the Pythagorean Theorem to show that the two missing sides make both triangles congruent.

He can conclude that they are congruent because they are both right triangles.
He can conclude that they are congruent because they are both right triangles.

He could apply the concept of HL and show that the two triangles are congruent.
He could apply the concept of HL and show that the two triangles are congruent.

He would need additional information to prove that the two triangles are congruent.
He would need additional information to prove that the two triangles are congruent.

Answer 3

To prove that the two triangles are congruent, we need to consider the information given:

1. Both triangles have a hypotenuse of 13 units.
2. The first triangle has a leg of 12 units.
3. The second triangle has a leg of 5 units.

We can find the missing leg of each triangle by applying the Pythagorean Theorem \(a^2 + b^2 = c^2\) (where \(c\) is the hypotenuse):

For the first triangle:

• Let \(a = 12\) and \(c = 13\).
• Then, \(b^2 = c^2 - a^2 = 13^2 - 12^2 = 169 - 144 = 25\).
• Thus, \(b = \sqrt{25} = 5\).

For the second triangle:

• Let \(a = 5\) and \(c = 13\).
• Then, \(b^2 = c^2 - a^2 = 13^2 - 5^2 = 169 - 25 = 144\).
• Thus, \(b = \sqrt{144} = 12\).

Now we have:

• First triangle: legs are 12 and 5 (and hypotenuse 13).
• Second triangle: legs are 5 and 12 (and hypotenuse 13).

These triangles, with the legs 12 and 5, and hypotenuse 13, correspond to the same side lengths and configuration.

Therefore, Malik can conclude that the two triangles are congruent based on the Hypotenuse-Leg (HL) Theorem:

He could apply the concept of HL and show that the two triangles are congruent.

Answer 4

Consider right triangles ΔABC and ΔXYZ . Angle B and angle Y are congruent. The hypotenuse of each triangle is AC¯¯¯¯¯¯¯¯ and XZ¯¯¯¯¯¯¯¯ , and they are congruent. What other corresponding parts must be congruent to show that ΔABC≅ΔXYZ by the HL Congruence Theorem?(1 point) Responses AB¯¯¯¯¯¯¯¯≅XY¯¯¯¯¯¯¯¯ Modifying above upper A upper B with bar congruent to Modifying above upper X upper Y with bar BC¯¯¯¯¯¯¯¯≅XZ¯¯¯¯¯¯¯¯ Modifying above upper B upper C with bar congruent to Modifying above upper X upper Z with bar BC¯¯¯¯¯¯¯¯≅XY¯¯¯¯¯¯¯¯ Modifying above upper B upper C with bar congruent to Modifying above upper X upper Y with bar AB¯¯¯¯¯¯¯¯≅YZ¯¯¯¯¯¯¯