12. The triangle is congruent by the SAS theorem.
13. The triangle is congruent by the ASA theorem.
14. The triangles are congruent by the HL theorem.
15. The triangle is congruent by the SAS theorem.
16. The triangles are congruent by the ASA theorem.
17. The triangle is not congruent.
18. The triangles are congruent by the SAS theorem.
19. The triangles are congruent by the ASA theorem.
20. The triangles are congruent by the SSS theorem.
In general, to determine whether two triangles are congruent, we need to compare their corresponding side lengths and angles. If all three side lengths or all three angle measures are equal, then the triangles are congruent.
Here is a table summarizing the different congruence theorems:
Theorem Condition
SSS All three side lengths are equal.
SAS Two side lengths and the included angle are equal.
ASA Two angles and the included side length are equal.
AAS Two angles and a non-included side length are equal.
HL The hypotenuse and a leg of one triangle are equal to the hypotenuse and a leg of the other triangle.