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¯¯¯¯¯¯¯¯≅XY¯¯¯¯¯¯¯¯ Modifying above upper B upper C 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 triangles ΔABC and ΔXYZ are congruent using the Hypotenuse-Leg (HL) Congruence Theorem, we need to establish two things:

1. The hypotenuses AC and XZ are already given to be congruent.
2. One pair of legs must also be congruent.

Since it’s given that angle B is congruent to angle Y (the angles at the right triangles), we can make use of this information. By the HL theorem, we can proceed as follows:

Given that:

• AC ≅ XZ (hypotenuses)
• ∠B ≅ ∠Y (corresponding angles in right triangles)

Now, we need to establish the congruence of one leg. Since we are not told directly which legs are congruent, we can check each option provided:

1. AB ≅ XY: This could be a suitable option if we were given additional information that these legs are equal.
2. BC ≅ XY: This would be invalid since BC is not the leg corresponding to XY.
3. BC ≅ XZ: This is invalid as we are comparing a leg with a hypotenuse.
4. AB ≅ YZ: Again, this could be valid but requires direct information.

Ultimately, since we know angle B and angle Y are congruent and the hypotenuses are congruent, we can directly infer that one of the legs must also correspond to an equality.

Since both triangles are right triangles, we would conclude that the other leg (the leg opposite the right angle) which is AB and XY must be corresponding.

Thus, in conclusion, the statement that must hold for ΔABC≅ΔXYZ by the HL Congruence Theorem is:

AB ≅ XY.

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

Answer 3

To determine if Malik can prove that the two right triangles are congruent, let's analyze the information given:

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

To determine the lengths of the other sides in each triangle, we can apply the Pythagorean Theorem, which states that in a right triangle, the sum of the squares of the legs equals the square of the hypotenuse:

\[ a^2 + b^2 = c^2 \]

For the first triangle, with one leg measuring 12 units and a hypotenuse of 13 units:

\[ 12^2 + b^2 = 13^2 \] \[ 144 + b^2 = 169 \] \[ b^2 = 169 - 144 = 25 \] \[ b = 5 \]

So, the first triangle has legs of lengths 12 and 5.

For the second triangle, with one leg measuring 5 units and a hypotenuse of 13 units, let's find the other leg:

\[ 5^2 + b^2 = 13^2 \] \[ 25 + b^2 = 169 \] \[ b^2 = 169 - 25 = 144 \] \[ b = 12 \]

So, the second triangle also has legs of lengths 5 and 12.

Now we see that both triangles have the same leg lengths of 5 and 12, along with the hypotenuse of 13.

Conclusion:

Malik can apply the Hypotenuse-Leg (HL) Congruence Theorem, as both triangles have a hypotenuse of 13 and one leg equal to 12.

Therefore, the correct response among the options given is:

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

Answer 4

Use the image to answer the question. Ignacio wants to prove that ΔABC≅ΔDCB. He says that he knows AC¯¯¯¯¯¯¯¯≅DB¯¯¯¯¯¯¯¯ because diagonals in rectangles are congruent. What rule can he use to prove that BC¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯? (1 point) Responses the HL Theorem the HL Theorem the Reflexive Property the Reflexive Property the Distributive Property the Distributive Property the Pythagorean Theorem

Answer 5

In the context of proving that triangles \( \Delta ABC \) and \( \Delta DCB \) are congruent, Ignacio has already established that \( AC \cong DB \) because the diagonals of a rectangle are congruent.

To show that \( BC \cong BC \), Ignacio can use the property that states any geometric figure is congruent to itself. This is known as the Reflexive Property of congruence.

Thus, the correct response is:

the Reflexive Property.

Answer 6

Use the image to answer the question. If C is the midpoint of AB¯¯¯¯¯¯¯¯ and DE¯¯¯¯¯¯¯¯ , what is the length of AB¯¯¯¯¯¯¯¯ ? (1 point) Responses 26 26 13 13 24 24 17

Answer 7

If \( C \) is the midpoint of \( AB \) and \( DE \), it means that the lengths \( AC \) and \( CB \) (as well as \( DE/2 \) if you have that context) are equal.

Without a diagram specifically indicating the lengths, we can use the concept of midpoints. If \( C \) is the midpoint of \( AB \), then the length of the entire segment \( AB \) is twice the length of \( AC \) (or \( CB \)).

If we assume \( AB \) is given in the answers provided, and the midpoint length is not explicitly stated, we could propose:

1. If \( AC \) or \( CB \) is stated to be \( 13 \), then \( AB = 2 \times 13 = 26 \).
2. If we assume \( AC \) or \( CB \) to be some other standard number provided in the choice, we would apply the same reasoning.

Thus, if \( C \) is indeed the midpoint and \( AB \) is twice that length,

Assuming \( C \) is known to be at \( AC = 13 \), the length of \( AB \) would be \( 26 \).

Hence, the answer is:

26.

Answer 8

Use the image to answer the question. In the diagram above, △ABC ≅△BAD . If AC=8 m and AB=17 m , how long is AD¯¯¯¯¯¯¯¯ ? (1 point) Responses 17 m 17 m 19 m 19 m 20 m 20 m 15 m

Answer 9

Given that the triangles \( \triangle ABC \) and \( \triangle BAD \) are congruent, we can conclude that their corresponding sides are also congruent.

From the problem, we know:

• \( AC = 8 , m \)
• \( AB = 17 , m \)

Since the triangles are congruent, we have:

• \( AB \) in triangle \( ABC \) corresponds to \( AD \) in triangle \( BAD \).
• Therefore, \( AD \) must also equal \( AB \).

Given that \( AB = 17 , m \), we conclude that:

\[ AD = AB = 17 , m \]

Thus, the length of \( AD \) is:

17 m.