Question

1. help your classmate

a. john claims the triangles are congruent by ssa, jack says there is not enough information. who is correct?
b. emily needs to prove the triangles are congruent. what third piece of information would she need?

Explanation:

Part a

Step1: Recall triangle congruence criteria

Triangle congruence criteria are SSS (Side - Side - Side), SAS (Side - Angle - Side), ASA (Angle - Side - Angle), AAS (Angle - Angle - Side), and HL (Hypotenuse - Leg) for right triangles. SSA (Side - Side - Angle) is not a valid congruence criterion because it can lead to two non - congruent triangles (the ambiguous case in the Law of Sines).

Step2: Analyze John's and Jack's claims

John claims congruence by SSA, but since SSA is not a valid congruence criterion, we cannot conclude that the triangles are congruent just based on SSA. Jack says there is not enough information, which is correct because SSA does not guarantee congruence.

Step1: Identify the given information

From the diagram (with the marked sides), we can see that two sides of the triangles are marked as equal (probably vertical angles or some other pair of angles are equal? Wait, looking at the diagram with the two triangles and the marked sides, if we assume that we have two sides and we need a third piece of information for congruence. The common congruence criteria that use two sides are SAS (which needs the included angle) or SSS (which needs the third side). Also, if there are vertical angles (the angles formed by the intersecting lines), those are equal. So, if we have two sides and the included angle (SAS) or the vertical angles (which would give us AAS or ASA if combined with the sides) or the third side (SSS). But looking at the diagram with the two triangles that seem to share a vertical angle (the angle at the intersection), the most likely third piece of information needed is that the vertical angles are equal (so we can use ASA or AAS) or the third side (for SSS) or the included angle (for SAS). But given the diagram with the two triangles and the marked sides, a common case is that the vertical angles (the angles formed by the intersecting lines) are equal. So if we know that the vertical angles are congruent, or the included angle between the two sides is congruent, or the third side is congruent. A typical answer here would be that the vertical angles are equal (so we can use ASA or AAS) or the included angle (for SAS) or the third side (for SSS). But more precisely, if we have two sides, to prove congruence, we can use SAS (so we need the included angle between the two sides to be equal) or SSS (the third side equal) or AAS/ASA (if we have an angle). Given the diagram with the two triangles that appear to have a pair of vertical angles, the third piece of information could be that the vertical angles are congruent (so we can use ASA or AAS) or the included angle between the two marked sides is congruent (for SAS) or the third side is congruent (for SSS). A common answer is that the vertical angles are equal (so we can use ASA or AAS) or the included angle (for SAS) or the third side (for SSS). But let's think again. If we have two sides, for SAS, we need the included angle. So if the two sides are, say, side \(a\), side \(b\), and the included angle \(\theta\) between them, then SAS would apply. Alternatively, if there are vertical angles (angles opposite each other when two lines intersect), those are equal. So if we have two sides and the vertical angle (which is not the included angle, but could be part of AAS or ASA), but more accurately, the most probable third piece of information is that the vertical angles are congruent (so we can use ASA or AAS) or the included angle (for SAS) or the third side (for SSS). But given the diagram, the most likely answer is that the vertical angles are equal (so we can use ASA or AAS) or the included angle (for SAS) or the third side (for SSS). A typical answer here is that the vertical angles are congruent (so we can use ASA or AAS) or the included angle (for SAS) or the third side (for SSS). But to be more specific, if we have two sides, the third piece of information needed for congruence could be:

• The included angle between the two sides is congruent (for SAS).
• The third side is congruent (for SSS).
• The vertical angles (the angles formed by the intersecting lines) are congruent (for ASA or AAS).

Answer:

Jack is correct because SSA is not a valid triangle congruence criterion, so there is not enough information to conclude the triangles are congruent.

Part b