Question

2
drag the tiles to the boxes to form correct pairs.
match each set of conditions with the corresponding relationship between △abc and △xyz and the criterion (if any) that proves the relationship.
ab = xy, bc = yz, and angle b is congruent to angle y.
ab = xy, and angles a and b are congruent to angles x and y, respectively.
angles a, b, and c are congruent to angles x, y, and z, respectively.
ab = xy, bc = yz, and ca = zx.
△abc and △xyz are congruent by the sss criterion.
△abc and △xyz are not necessarily congruent.
△abc and △xyz are congruent by the sas criterion.
△abc and △xyz are congruent by the asa criterion.

Explanation:

Step1: Recall congruence criteria

Side - Angle - Side (SAS) criterion: Two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle.

Step2: Analyze first condition

Given $AB = XY$, $BC = YZ$, and angle $B$ is congruent to angle $Y$. This satisfies the SAS criterion. So, $\triangle ABC$ and $\triangle XYZ$ are congruent by the SAS criterion.

Step3: Recall ASA criterion

Angle - Side - Angle (ASA) criterion: Two angles and the included side of one triangle are congruent to two angles and the included side of another triangle.

Step4: Analyze second condition

Given $AB = XY$, and angles $A$ and $B$ are congruent to angles $X$ and $Y$ respectively. This satisfies the ASA criterion. So, $\triangle ABC$ and $\triangle XYZ$ are congruent by the ASA criterion.

Step5: Recall AA (not a congruence criterion)

Two triangles with all - angle congruence (angles $A$, $B$, and $C$ are congruent to angles $X$, $Y$, and $Z$ respectively) are similar but not necessarily congruent.

Step6: Recall SSS criterion

Side - Side - Side (SSS) criterion: All three sides of one triangle are congruent to all three sides of another triangle. Given $AB = XY$, $BC = YZ$, and $CA = ZX$, $\triangle ABC$ and $\triangle XYZ$ are congruent by the SSS criterion.

Answer:

$AB = XY$, $BC = YZ$, and angle $B$ is congruent to angle $Y$ $\longleftrightarrow$ $\triangle ABC$ and $\triangle XYZ$ are congruent by the SAS criterion.
$AB = XY$, and angles $A$ and $B$ are congruent to angles $X$ and $Y$ respectively $\longleftrightarrow$ $\triangle ABC$ and $\triangle XYZ$ are congruent by the ASA criterion.
Angles $A$, $B$, and $C$ are congruent to angles $X$, $Y$, and $Z$ respectively $\longleftrightarrow$ $\triangle ABC$ and $\triangle XYZ$ are not necessarily congruent.
$AB = XY$, $BC = YZ$, and $CA = ZX$ $\longleftrightarrow$ $\triangle ABC$ and $\triangle XYZ$ are congruent by the SSS criterion.