Question

Which of the following pairs of triangles could you use to justify that the SAS Congruence Theorem is true?(1 point) Responses Image with alt text: An illustration shows two triangles side by side. The two triangles are labeled E F G and H I J. Side F G is congruent to side I J. Side E F is congruent to side H I. Image with alt text: An illustration shows two triangles side by side. The two triangles are labeled E F G and H I J. Side E F is congruent to side H I. Angle F is congruent to angle I. Angle E is congruent to angle H. Image with alt text: An illustration shows two triangles side by side. The two triangles are labeled E F G and H I J. Side E G is congruent to side H J. Side E F is congruent to side H I. Angle E is congruent to angle H.

Answer 1

To justify that the SAS (Side-Angle-Side) Congruence Theorem is true, we need to show that if two sides of one triangle are congruent to two sides of another triangle, and the included angle (the angle between those two sides) is also congruent, then the triangles are congruent.

Among the pairs of triangles you provided:

1. First Image: This indicates that two sides are congruent (FG ≅ IJ, EF ≅ HI), but there’s no information about the included angle. Thus, this pair does not justify SAS.

2. Second Image: Here, two angles are congruent (∠F ≅ ∠I, ∠E ≅ ∠H) but only one angle and no sides are provided for SAS’s requirements. This does not justify SAS.

3. Third Image: This shows two sides that are congruent (EG ≅ HJ, EF ≅ HI) along with one included angle (∠E ≅ ∠H). This precisely matches the SAS theorem conditions.

Therefore, the Third Image represents a situation that could be used to justify that the SAS Congruence Theorem is true.

Answer 2

The SAS Congruence Theorem Quick Check 4 of 54 of 5 Items Question Use the image to answer the question. If you wanted to explain why △ABD≅CBD by the SAS Congruence Theorem, which of the following would be helpful to know? (1 point) Responses AC¯¯¯¯¯¯¯¯ is the angle bisector of ∠BCD. Modifying above upper A upper C with bar is the angle bisector of angle upper B upper C upper D . BD¯¯¯¯¯¯¯¯ is the angle bisector of ∠BCD. Modifying above upper B upper D with bar is the angle bisector of angle upper B upper C upper D . AC¯¯¯¯¯¯¯¯ is the angle bisector of ∠BAD. Modifying above upper A upper C with bar is the angle bisector of angle upper B upper A upper D . BD¯¯¯¯¯¯¯¯ is the angle bisector of ∠ABC.

Answer 3

To apply the SAS (Side-Angle-Side) Congruence Theorem and justify that triangles \( \triangle ABD \) and \( \triangle CBD \) are congruent, you need to establish the following:

1. The two sides that are included between the angles: You need congruent sides \( AB \) and \( CB \) that are adjacent to the angle in question.
2. The included angle: You need to show that angle \( \angle ABD \) is congruent to angle \( \angle CBD \).

Therefore, knowing that one of the segments serves as an angle bisector that helps confirm the congruence of the included angles is critical to applying the SAS theorem.

From your options, the following would be helpful:

• AC is the angle bisector of ∠BAD. This implies that \( \angle BAD \) is split into two equal parts, thus allowing you to show that \( \angle ABD \) is congruent to \( \angle CBD \) if \( D \) is positioned appropriately.

Since we're focusing on establishing the angle between the two sides of each triangle, the option saying that AC is the angle bisector of ∠BAD is the one that would be most helpful for your explanations regarding the SAS Congruence Theorem to justify \( \triangle ABD \cong \triangle CBD \).