Question

for the following problems, complete the triangle - congruence statement and name the postulate that justifies the statement.

1. y is the mid - point of xz, ay = by, and ∠axy = ∠byz.

△axy≅____ by ____

1. △rts is isosceles with legs rt and ts, q is the mid - point of rs.

△rtq≅____ by ____

1. ∠p = ∠n and m is the mid - point of pn.

△pmk≅____ by ____

1. ac = bd and ad = bc.

△abd≅____ by ____

Explanation:

Step1: Analyze problem 10

Given \(Y\) is the mid - point of \(XZ\), so \(XY = YZ\), \(AY=BY\) and \(\angle AYX=\angle BYZ\). By the Side - Angle - Side (SAS) congruence postulate, \(\triangle AYX\cong\triangle BYZ\).

Step2: Analyze problem 11

Since \(\triangle RTS\) is isosceles with legs \(RT\) and \(ST\) and \(Q\) is the mid - point of \(RS\), then \(RQ = SQ\), \(RT = ST\) and \(QT=QT\). By the Side - Side - Side (SSS) congruence postulate, \(\triangle RTQ\cong\triangle STQ\).

Step3: Analyze problem 12

Given \(\angle P=\angle N\) and \(M\) is the mid - point of \(PN\), so \(PM = NM\), and \(\angle PMK=\angle NMQ\) (vertically opposite angles). By the Angle - Side - Angle (ASA) congruence postulate, \(\triangle PMK\cong\triangle NMQ\).

Step4: Analyze problem 13

Given \(AC = BD\) and \(AD = BC\), and \(AB = AB\). By the Side - Side - Side (SSS) congruence postulate, \(\triangle ABD\cong\triangle BAC\).

Answer:

1. \(\triangle AYX\cong\triangle BYZ\) by SAS
2. \(\triangle RTQ\cong\triangle STQ\) by SSS
3. \(\triangle PMK\cong\triangle NMQ\) by ASA
4. \(\triangle ABD\cong\triangle BAC\) by SSS