Question

Marcus and Marlee are comparing △ABC with right angle B with △PQR with right angle Q to determine if they are congruent. AB¯¯¯¯¯¯¯¯ ≅ QR¯¯¯¯¯¯¯¯ and AC¯¯¯¯¯¯¯¯ ≅ PR¯¯¯¯¯¯¯¯ . Marcus says the triangles are congruent by the HL Congruence Theorem, but Marlee does not agree. Who is correct? (Hint: Draw a picture.)(1 point) Responses Marlee is correct; the triangles are not congruent because the corresponding legs are not marked as congruent. Marlee is correct; the triangles are not congruent because the corresponding legs are not marked as congruent. Marcus is correct; the triangles are right triangles where a leg and the hypotenuse of one triangle is congruent to a leg and hypotenuse of the other triangle. Marcus is correct; the triangles are right triangles where a leg and the hypotenuse of one triangle is congruent to a leg and hypotenuse of the other triangle. Neither is correct; the triangles are congruent by Side-Angle-Side. Neither is correct; the triangles are congruent by Side-Angle-Side. Neither is correct; the triangles are congruent by Side-Side-Angle.

Answer 1

To determine the correctness of Marcus and Marlee’s claims about the congruence of triangles △ABC and △PQR, let's analyze their statements and the details given.

1. Given Information:

   • Triangles △ABC and △PQR are right triangles (right angle at B and Q respectively).
   • AB ≅ QR and AC ≅ PR.

2. Congruence Criteria:

   • The HL (Hypotenuse-Leg) Congruence Theorem states that if two right triangles have one leg and the hypotenuse of one triangle congruent to one leg and the hypotenuse of the other triangle, then the triangles are congruent.
   • To use the HL theorem, we must ensure that we have one leg and the hypotenuse identified.

3. Marcus's Argument:

   • Marcus claims the triangles are congruent by the HL Congruence Theorem but only mentions the legs and the hypotenuse without consulting the corresponding leg congruence.

4. Marlee's Argument:

   • Marlee argues that the triangles cannot be assumed to be congruent unless the corresponding legs are also marked as congruent. This is correct as we need to know that both triangles have the necessary congruences.

Since the problem does not provide information about the third side (the other leg of each triangle) being congruent, Marlee's stance is justified—without knowing if the leg corresponding to AB (which is AC in △ABC) and the leg corresponding to QR (which is PR in △PQR) is congruent too, we lack the necessary information to apply the HL theorem properly.

Thus, the correct answer is:

Marlee is correct; the triangles are not congruent because the corresponding legs are not marked as congruent.

Answer 2

If ΔABC≅CDA by the HL Theorem and AC¯¯¯¯¯¯¯¯=84 m and AD¯¯¯¯¯¯¯¯=85 m , how long is BA¯¯¯¯¯¯¯¯ ? (1 point) Responses 27 m 27 m 15 m 15 m 13 m 13 m 22 m

Answer 3

Given that \( \Delta ABC \cong \Delta CDA \) by the Hypotenuse-Leg (HL) Theorem, we can analyze the triangles for corresponding parts.

From the information:

• \( AC = 84 , \text{m} \)
• \( AD = 85 , \text{m} \)

In congruent triangles, corresponding sides are equal. Therefore:

• \( AC \) corresponds to \( AD \),
• \( AB \) corresponds to \( CD \),
• \( BC \) corresponds to \( DA \).

However, the question hasn't provided information about the lengths of \( AB \) directly. Since the triangles are right triangles, we can find \( AB \) using the Pythagorean theorem, but we need to consider the relationships established by the congruence of the triangles.

Assuming (as both triangles share the hypotenuse), we need a right triangle's properties. The missing piece is understanding that if we label the triangles:

• From triangle \( ABC \) (right angle at \( B \)), the sides are \( AB \) and \( BC \) with hypotenuse \( AC \).
• From triangle \( CDA \) (right angle at \( D \)), the sides are \( CD \) and \( DA \) with hypotenuse \( AD \).

Given that \( \Delta ABC \cong \Delta CDA \):

• The hypotenuses are \( AC \) and \( AD \), and since they are congruent triangles, \( AB = CD \).
• The congruence implies that the legs of triangle \( ABC \) also correspond to the legs of triangle \( CDA \).

To find \( BA \):

1. Use the Pythagorean theorem for triangle \( ABC \): \[ AB^2 + BC^2 = AC^2 \] \[ AD^2 + CD^2 = AD^2 \]

The values of \( AC \) and \( AD \) do not directly provide \( BA \). However, responding to the numerical options given, it would generally indicate knowing some lengths of legs or the relationships between them.

For simplicity, assumptions in these types of problems could be where options are tried to isolate lengths quickly. Thus, let’s work through responses based on likely relationships.

Trying \( 27 m, 15 m, 13 m, \) or \( 22 m \) through:

Assuming \( BA + BC \) aligns to direct leg comparisons, we would estimate based on integer triangles or common simplified.

If each option were hypothesized, further straightforward calculation isn't directly observational as the angles and proportional increases aren't given.

Thus, primarily provided relations or classic known right triangles usually revert make clear some geometric leg bases here,

Without loss thus if choosing leg lengths based through logic balance leads common \( 27 m\).

Therefore, if cleanly testing:

Final determination would choose:

• 27 m.

(But reinforcing to be readjusted if additional triangulated values were provided in complex needs).

Answer 4

Describe a sequence of rigid transformations that could be used to show that △ABC≅△DEF . Explain how you know each transformation will result in a congruent figure. Then, explain how this sequence of rigid transformations illustrates the ASA Congruence Theorem. (4 points)

Answer 5

To show that triangles \( \triangle ABC \) and \( \triangle DEF \) are congruent using rigid transformations, we can use a sequence of transformations that includes reflection, rotation, and/or translation. Here’s a possible sequence of transformations, along with explanations and how they relate to the ASA (Angle-Side-Angle) Congruence Theorem.

Sequence of Rigid Transformations

1. Translation:

   • Translate triangle \( ABC \) so that vertex \( A \) coincides with vertex \( D \).
   • Why this works: Translating the triangle maintains its shape and size, ensuring that all points are moved the same distance in the same direction, preserving congruence.

2. Rotation:

   • Rotate triangle \( ABC \) around point \( D \) (where point \( A \) was translated to) such that point \( B \) aligns with point \( E \).
   • Why this works: Rotating maintains distances and angles, meaning the shape of the triangle remains unchanged, thus ensuring congruence.

3. Reflection (if needed):

   • If triangle \( ABC \) is still not aligned properly with triangle \( DEF \) after the translation and rotation, reflect triangle \( ABC \) over line \( DE \) (or another appropriate line).
   • Why this works: Reflection also preserves congruence by keeping the distances between points and angles intact while reversing the orientation of the triangle.

Explanation of Each Transformation

• Translation preserves the shape and size of the triangle by moving it along a straight path without changing its orientation or dimensions.

• Rotation keeps the sizes and angles the same and allows for repositioning the triangle while preserving the relationships between points (angles and lengths remain unchanged).

• Reflection maintains the same dimensions and angles but can change the orientation (the "handedness" of the triangle), which might be necessary to match it to \( \triangle DEF \).

ASA Congruence Theorem

The ASA Congruence Theorem states that if two triangles have two angles and the included side of one triangle congruent to the corresponding two angles and included side of another triangle, then the two triangles are congruent.

In this sequence of transformations:

• Angle A corresponds to Angle D after translation, and remains the same after rotation.
• Angle B corresponds to Angle E after rotation.
• The included side \( AB \) corresponds to the included side \( DE \) after translation and rotation.

Through the rigid transformations applied, this process effectively demonstrates that not only are the sides and angles preserved, but the relationships established through these correspondences satisfy the conditions of the ASA Theorem.

Thus, we have:

• Congruence in angles \( A \) and \( D \),
• Congruence in angles \( B \) and \( E \),
• Congruence in the included side \( AB \) and \( DE \).

This alignment through rigid transformations confirms that triangles \( \triangle ABC \) and \( \triangle DEF \) are congruent by ASA, as the transformations serve to demonstrate the equality of the necessary angles and sides.