Question

state what additional information is required in order to know that the triangles are congruent for the reason given. 11) sas 12) aas 13) sas 14) asa 15) asa

Explanation:

11) SAS

Step1: Recall SAS criterion

The Side - Angle - Side (SAS) congruence criterion states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.
In the given triangles \( \triangle BCD\) and \( \triangle SDC\), we already have \(BC = SC\) (marked) and \(CD=CD\) (common side).

Step2: Identify the missing angle

The included angle for \(BC\) and \(CD\) in \( \triangle BCD\) is \( \angle BCD\), and the included angle for \(SC\) and \(CD\) in \( \triangle SCD\) is \( \angle SCD\). So, we need \( \angle BCD=\angle SCD\)

12) AAS

Step1: Recall AAS criterion

The Angle - Angle - Side (AAS) congruence criterion 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, then the two triangles are congruent.
We can assume that \( \angle C=\angle S\) (by vertical angles or some other un - marked relation, but for AAS, if we consider the non - included sides.

Step2: Identify the missing side

In \( \triangle CDE\) and \( \triangle SLT\), if we assume \( \angle D=\angle L\) (from the figure's angle markings), and we know that one side (say \(CD = SL\) (marked as equal in length conceptually for AAS, since AAS requires a non - included side. So we need \(DE = LT\)

13) SAS

Step1: Recall SAS criterion

For \( \triangle GHJ\) and \( \triangle XWV\), we know that \(HJ = WV\) (marked).

Step2: Identify the missing sides and angle

The SAS criterion requires two sides and the included angle. If we assume the angle \( \angle H=\angle W\) (marked as equal in the angle - like markings), then we need \(GH = XW\)

14) ASA

Step1: Recall ASA criterion

The Angle - Side - Angle (ASA) congruence criterion states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.
In \( \triangle QSR\) and \( \triangle VRX\), we know that \( \angle Q=\angle V\) (marked) and \( \angle SRQ=\angle XRV\) (vertically opposite angles).

Step2: Identify the missing side

The included side for \( \angle Q\) and \( \angle SRQ\) in \( \triangle QSR\) is \(SR\), and the included side for \( \angle V\) and \( \angle XRV\) in \( \triangle VRX\) is \(RX\). So we need \(SR = RX\)

15) ASA

Step1: Recall ASA criterion

In \( \triangle TUV\) and \( \triangle JUV\), we have \(UV = UV\) (common side) and \( \angle TUV=\angle JUV\) (marked).

Step2: Identify the missing angle

The ASA criterion requires two angles and the included side. The included side is \(UV\). The other angle for \( \triangle TUV\) is \( \angle TVU\) and for \( \triangle JUV\) is \( \angle JVU\). So we need \( \angle TVU=\angle JVU\)

Answer:

1. \( \angle BCD=\angle SCD\)
2. \(DE = LT\)
3. \(GH = XW\)
4. \(SR = RX\)
5. \( \angle TVU=\angle JVU\)