Question

what must be marked on the triangles below for them to be congruent by sss?
what must be marked on the triangles below for them to be congruent by asa?

Explanation:

Step1: Analyze the first pair of triangles

For the first pair of triangles, we have two sides marked equal and a non - included angle. But for congruence, the included angle is required for SAS. So, the answer is No for SSS, SAS, AAS, ASA, HL.

Step2: Analyze the second pair of triangles

In the second pair, we have two angles and a non - included side. For AAS (Angle - Angle - Side) congruence, we need two angles and a non - included side. Here, \(\angle ONH=\angle WNZ\) (vertically opposite angles), two sides \(NH = NZ\) and \(NO = NW\) (marked). But the given congruence criteria, we check:

• SSS: No (only two sides marked)
• SAS: No (the angle is not included)
• AAS: Yes (two angles \(\angle ONH=\angle WNZ\) and \(\angle H=\angle Z\) and a non - included side \(NH = NZ\))
• ASA: No (the side is not between the two angles)
• HL: No (not right - angled triangles)

Step3: Analyze the third pair of triangles

For the third pair of triangles \( \triangle QRS\) and \( \triangle HGK\), we have \(QR = HG = 31\), \(QS=HK = 40\), \(RS = GK=42\). By SSS (Side - Side - Side) congruence criterion (if three sides of one triangle are equal to three sides of another triangle, the triangles are congruent), the answer is Yes for SSS.

Step4: Analyze the SSS requirement for the fourth figure

For two triangles \( \triangle QRS\) and \( \triangle QDS\) to be congruent by SSS, since \(QR = QD\) (marked) and \(QS\) is common. We need \(RS = DS\) to be marked.

Step5: Analyze the ASA requirement for the fifth figure

For two triangles \( \triangle DEF\) and \( \triangle LJK\) to be congruent by ASA (Angle - Side - Angle), we know \(\angle D=\angle L\) (given as one angle) and \(\angle E=\angle K\) (given as another angle). We need the included side. So, \(DE = LK\) must be marked.

Answer:

1. No for SSS, SAS, AAS, ASA, HL.
2. Yes for AAS.
3. Yes for SSS.
4. \(RS = DS\) must be marked.
5. \(DE = LK\) must be marked.