Question

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

1. sas
2. asa
3. hl
4. aas

Explanation:

21) SAS

Step1: Recall SAS congruence

SAS (Side - Angle - Side) congruence states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
We have \(AB = WB\) (marked) and \(CB=UB\) (marked). The included angle for \(\triangle ABC\) and \(\triangle WB U\) is \(\angle ABC\) and \(\angle WB U\). Since \(\angle ABC\) and \(\angle WB U\) are vertical angles, if \(\angle A=\angle W\), we can use SAS.

22) ASA

Step1: Recall ASA congruence

ASA (Angle - Side - Angle) congruence states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
We know \(IH = IH\) (common side). We have one angle (\(\angle IHG\) and \(\angle FHI\) related). If \(\angle G=\angle F\), we can use ASA.

23) HL

Step1: Recall HL congruence

HL (Hypotenuse - Leg) congruence is for right - triangles. We have \(GH = GH\) (common side) and \(\angle GHR=\angle GHF = 90^{\circ}\). If \(RG = FG\), we can use HL.

24) AAS

Step1: Recall AAS congruence

AAS (Angle - Angle - Side) congruence states that if two angles and a non - included side of one triangle are equal to two angles and the corresponding non - included side of another triangle, the triangles are congruent.
We have \(\angle FGH=\angle LGH\) (marked). If \(\angle F=\angle L\), we can use AAS.

Answer:

1. \(\angle A=\angle W\)
2. \(\angle G=\angle F\)
3. \(RG = FG\)
4. \(\angle F=\angle L\)