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△GHK  and  △LMN  are given in the image. Use the slope formula and the distance formula to determine if  △GHK  is congruent to  △LMN  by the SAS Congruence Theorem.Select the four correct statements below.(2 points)ResponsesGH¯¯¯¯¯¯¯¯≅LM¯¯¯¯¯¯¯¯¯line segment cap g cap h is congruent to line segment cap l cap mGH¯¯¯¯¯¯¯¯≆LM¯¯¯¯¯¯¯¯¯line segment cap g cap h is not congruent to line segment cap l cap m∠G≅∠Langle cap g is congruent to angle cap L∠G≆∠Langle cap g is congruent to angle cap LGK¯¯¯¯¯¯¯¯≅LN¯¯¯¯¯¯¯¯line segment cap g cap k is congruent to line segment cap l cap nGK¯¯¯¯¯¯¯¯≆LN¯¯¯¯¯¯¯¯line segment cap g cap k is not congruent to line segment cap l cap n△GHK≅△LMN△GHK≅△LMN△GHK≆△LMN△GHK≆△LMN

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To determine if triangles \( \triangle GHK \) and \( \triangle LMN \) are congruent by the SAS (Side-Angle-Side) Congruence Theorem, we can use the slope formula to find the slopes of the sides and the distance formula to find the lengths of the sides.

Here are the steps you would typically take in this scenario:

  1. Calculate the lengths of the sides of both triangles using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

  2. Calculate the slopes of the sides to determine if the angles are congruent: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]

  3. Compare the lengths of corresponding sides and the measures of corresponding angles:

    • For the SAS theorem, you need at least two sides of one triangle to be equal to two sides of the other triangle, and the included angle between those two sides must also be equal.

Suppose you've performed these calculations and determined the following:

  • If \( GH \) is congruent to \( LM \) (i.e., \( GH \cong LM \))
  • If \( HK \) is congruent to \( LN \) (i.e., \( HK \cong LN \))
  • If \( \angle G \) is equal to \( \angle L \) (i.e., \( \angle G \cong \angle L \))

Given this information, we can select the correct statements regarding the congruence of the triangles:

  1. If you found \( GH \cong LM \), select the statement that indicates that \( \overline{GH} \) is congruent to \( \overline{LM} \).
  2. If \( \angle G \cong \angle L \), select the statement indicating that angles are congruent.
  3. If \( HK \cong LN \), select the statement that indicates that \( \overline{GK} \) is congruent to \( \overline{LN} \).
  4. Conclude \( \triangle GHK \cong \triangle LMN \) if the conditions of the SAS theorem are met.

From these details, you would select:

  • \( \overline{GH} \cong \overline{LM} \)
  • \( \angle G \cong \angle L \)
  • \( \overline{GK} \cong \overline{LN} \)
  • \( \triangle GHK \cong \triangle LMN \)

If you found any segment or angle measurements that were not congruent, choose the opposite statements accordingly. Always double-check your calculations to ensure accuracy in your conclusions.