Question

state what additional information is required in order to know that the triangles are congruent for the reason given.

1. asa
2. aas
3. sas
4. sss
5. aas

Explanation:

Problem 6 (ASA)

Step1: Recall ASA Criterion

ASA (Angle - Side - Angle) congruence criterion states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
In the given diagram for problem 6, we have $\angle I\cong\angle D$ (marked angles) and $\angle KEI\cong\angle FED$ (vertical angles, which are congruent). For ASA, we need the included side between the two angles to be congruent. So we need $EI\cong ED$ (or $KI\cong FD$, but more directly the included side between the two angles at $E$ and the marked angles at $I$ and $D$ is $EI$ and $ED$).

Step2: Determine Additional Info

To apply ASA, we need the included side between the two angles (the angle at $I$ and the vertical angle at $E$, and the angle at $D$ and the vertical angle at $E$) to be congruent. So the additional information required is $EI = ED$ (or $KI = FD$ but the most direct included side is $EI$ and $ED$).

Step1: Recall AAS Criterion

AAS (Angle - Angle - Side) congruence criterion states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the triangles are congruent.
In the diagram for problem 7, we know that $AB = PB$ (marked? Wait, no, the diagram shows some markings. Wait, the triangles are $\triangle ABC$ and $\triangle PBR$? Wait, actually, the vertical angles at $B$ are congruent. Let's assume we have two angles. For AAS, we need another pair of angles and a non - included side. Wait, looking at the diagram, we have one side marked as equal (maybe $AC = PR$? Wait, no, the diagram has a mark on $AC$ and $PR$? Wait, actually, to apply AAS, we need either another angle or a side. Wait, the triangles are $\triangle ABC$ and $\triangle PBR$ (with $B$ as the common vertex). We know that $\angle ABC=\angle PBR$ (vertical angles). If we have $\angle A=\angle P$ and we need a non - included side, or $\angle C=\angle R$ and a non - included side. Wait, the additional information required could be $\angle A=\angle P$ (or $\angle C=\angle R$) and a side, but more precisely, since we have a side? Wait, no, let's re - examine. The AAS requires two angles and a non - included side. So if we have $\angle ABC=\angle PBR$ (vertical angles), and we need another pair of angles (say $\angle A=\angle P$) and a non - included side (like $BC = BR$? No, wait, the non - included side. Wait, maybe the additional information is $\angle A=\angle P$ (or $\angle C=\angle R$) and the side $AC = PR$? Wait, no, the correct additional information: Let's say in $\triangle ABC$ and $\triangle PBR$, $\angle ABC=\angle PBR$ (vertical angles). For AAS, we need either $\angle A=\angle P$ and $AC = PR$, or $\angle C=\angle R$ and $BC = BR$. But the most likely additional information is $\angle A=\angle P$ (or $\angle C=\angle R$) and the corresponding non - included side. Wait, actually, the additional information required is $\angle A=\angle P$ (or $\angle C=\angle R$) and the side $AC = PR$ (or $BC = BR$). But more simply, if we consider the triangles, the additional information could be $\angle A=\angle P$ (or $\angle C=\angle R$) and the side $AC = PR$ (the non - included side with respect to the two angles).

Step2: Determine Additional Info

To apply AAS, we need another pair of congruent angles (e.g., $\angle A\cong\angle P$ or $\angle C\cong\angle R$) and the corresponding non - included side (e.g., $AC\cong PR$ or $BC\cong BR$). So the additional information required is $\angle A=\angle P$ (or $\angle C=\angle R$) and $AC = PR$ (or $BC = BR$). A more precise answer: If we assume the triangles are $\triangle ABC$ and $\triangle PBR$, the additional information is $\angle A=\angle P$ (or $\angle C=\angle R$) and $AC = PR$ (the non - included side for the two angles $\angle ABC$ and $\angle A$ (or $\angle C$)).

Step1: Recall SAS Criterion

SAS (Side - Angle - Side) congruence criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
In the diagram for problem 8, we have one side marked as equal (e.g., $UV = KL$? Wait, the triangles are $\triangle VWU$ and $\triangle KLM$? Wait, $\triangle VWU$ has a marked side $UV$ and $\triangle KLM$ has a marked side $ML$. The included angle is $\angle V$ and $\angle K$. So we have one side equal (maybe $UV = KL$) and we need the included angle $\angle V=\angle K$ and the other side $VW = KM$ (or $VU = KL$ and $VW = KM$ with included angle $\angle V=\angle K$). Wait, actually, the triangles are $\triangle VWU$ and $\triangle KLM$. We have a side marked on $\triangle VWU$ (say $UV$) and a side marked on $\triangle KLM$ (say $ML$). The included angle is at $V$ and $K$. So to apply SAS, we need $UV = KL$, $\angle V=\angle K$, and $VW = KM$. So the additional information required is $\angle V=\angle K$ (the included angle) and $VW = KM$ (or $VU = KL$ and $\angle V=\angle K$ and $VW = KM$). But more precisely, since we have one side (maybe $UV = KL$) marked, we need the included angle $\angle V=\angle K$ and the other side $VW = KM$.

Step2: Determine Additional Info

To apply SAS, we need the included angle between the two sides to be congruent. So the additional information required is $\angle V=\angle K$ (the included angle) and $VW = KM$ (the other pair of sides).

Answer:

$EI = ED$ (or the included side between the two angles to be congruent, e.g., $KI = FD$ can also be considered but $EI = ED$ is more direct)

Problem 7 (AAS)