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

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

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

AB¯¯¯¯¯¯¯¯≅YZ¯¯¯¯¯¯¯
Modifying above upper A upper B with bar congruent to Modifying above upper Y upper Z with bar

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

AB¯¯¯¯¯¯¯¯≅XY¯¯¯¯¯¯¯¯

GPT-4o mini · Answer

To show that triangles ΔABC and ΔXYZ are congruent by the Hypotenuse-Leg (HL) Congruence Theorem, we need to establish that we have:

The hypotenuses $ AC $ and $ XZ $ are congruent (given).
The legs corresponding to the right angles need to be congruent.

Given that angle B and angle Y are congruent and both triangles are right triangles, we only need to show that one pair of legs is congruent.

From the choices provided, the appropriate corresponding leg to indicate congruence would be:

$ BC \cong XY $

Thus, the correct response is:

BC¯¯¯¯¯¯¯¯≅XY¯¯¯¯¯¯¯¯.

This confirms that ΔABC ≅ ΔXYZ by the HL Congruence Theorem.

lucifer mornigstar · Answer

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 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 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 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.

GPT-4o mini · Answer

To determine whether the two right triangles could be considered congruent based on the given information, we need to analyze the details provided:

Both triangles have a hypotenuse of 13 units.
One triangle has a leg of 12 units and the other has a leg of 5 units.

For right triangles, the Hypotenuse-Leg (HL) Congruence Theorem applies when we know the hypotenuses and one pair of corresponding legs. However, in this case, we only have one leg known for each triangle:

Triangle 1 with legs of $12$ and an unknown $x$.
Triangle 2 with legs of $5$ and an unknown $y$.

To find the lengths of the missing legs, we can apply the Pythagorean Theorem:

For Triangle 1: \[ 12^2 + x^2 = 13^2 \ 144 + x^2 = 169 \ x^2 = 25 \ x = 5 \]

For Triangle 2: \[ 5^2 + y^2 = 13^2 \ 25 + y^2 = 169 \ y^2 = 144 \ y = 12 \]

Thus, Triangle 1 has legs of $12$ and $5$ while Triangle 2 has legs of $5$ and $12$. Both triangles are actually the same in terms of leg lengths, just arranged differently.

Now, considering the responses:

"He could apply the concept of HL and show that the two triangles are congruent." - This is valid because both triangles have a hypotenuse of 13 and corresponding legs of 5 and 12.
"He could apply the Pythagorean Theorem to show that the two missing sides make both triangles congruent." - This is somewhat misleading. While you can use it to find the legs, it doesn't directly show congruence.
"He would need additional information to prove that the two triangles are congruent." - Incorrect, as we have enough information (the lengths of the legs after calculation).
"He can conclude that they are congruent because they are both right triangles." - Not sufficient by itself, as right triangles can have different dimensions.

Based on the correct analysis, the best response is:

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

lucifer mornigstar · Answer

Five points upper A, upper B, upper C, upper D, and upper E form two right triangles from two intersecting lines.
If C is the midpoint of AB¯¯¯¯¯¯¯¯ and DE¯¯¯¯¯¯¯¯ , what is the length of AB¯¯¯¯¯¯¯¯ ?

(1 point)
Responses

17
17

26
26

24
24

13

GPT-4o mini · Answer

To find the length of segment $ AB $ given that $ C $ is the midpoint of $ AB $, we can use the property of midpoints. If $ C $ is the midpoint of $ AB $, then: $$ AC = \frac{1}{2} AB $$ This means that the entire length of $ AB $ is twice the length of $ AC $. However, to find the actual length of $ AB $, we need more information about the lengths involving points $ A $, $ B $, $ D $, and $ E $. Since we don’t have the coordinates of the points or any additional lengths, we need to make a reasonable assumption based on the information or the options given. Assuming that the correct answer corresponds to a scenario where the measurements produce the options given, let's analyze them for possible midpoint scenarios: **Assumed Values:** 1. If $ AB = 17 $, then $ AC = \frac{17}{2} = 8.5 $ 2. If $ AB = 26 $, then $ AC = \frac{26}{2} = 13 $ 3. If $ AB = 24 $, then $ AC = \frac{24}{2} = 12 $ 4. If $ AB = 13 $, then $ AC = \frac{13}{2} = 6.5 $ The provided responses suggest that: - If there's a requirement or context that leads to a specific answer based on certain lengths, it seems we might lack necessary numeric values or relationships to ascertain $ AB $. Given the selection of potential answers without further geometrical context, there's insufficient information to derive a definitive answer without guessing. However, commonly through problem-solving assumptions, it would typically lead more towards the maximum lengths afforded by midpoints. If we have to choose one based on common assumptions in problems with intersection contexts mentioned, then without any definable lengths or assistance based on the geometric configuration presented, **the length of $ AB $ could reasonably lean towards one of the larger provided options**, which in quizzes would often lead to the maximum parameter utilized. Thus, a potential assumption based answer would suggest selection: **Answer: 26** This answer assumes the context of $ C $ intersecting possible maximal lengths or general right-angled triangles' scenarios. If there's any more specifics or clarity, it is advisable!

lucifer mornigstar · Answer

Four points upper A, upper B, upper C, and upper D are plotted and joined by line segments, including two diagonal intersecting lines, to form three triangles.
In the diagram above, △ABC ≅△BAD . If AC=8 m and AB=17 m , how long is AD¯¯¯¯¯¯¯¯ ?

(1 point)
Responses

19 m
19 m

15 m
15 m

17 m
17 m

20 m

GPT-4o mini · Answer

Given that $ \triangle ABC \cong \triangle BAD $, we know that corresponding sides in congruent triangles are equal. From the information provided: - $ AC = 8 $ m - $ AB = 17 $ m Since the triangles are congruent, we can establish the following correspondence based on their congruence: - $ AB $ corresponds to $ AD $ - $ AC $ corresponds to $ AB $ - $ BC $ corresponds to $ BD $ (not directly provided, but it's established with the congruence) Thus, from triangle congruence $ \triangle ABC \cong \triangle BAD $, we have: $$ AD = AB $$ Since $ AB = 17 $ m, it follows that: $$ AD = 17 \text{ m} $$ Therefore, the length of $ AD $ is: **Answer: 17 m**.