Answer:
D: I and IV only.
Step-by-step explanation:
We can go through each statement and examine its validity.
I) All congruent triangles are similar.
For congruent triangles, all of its corresponding sides and angles are congruent.
Since all angles are congruent, this fulfills the AA Similarity criterion. Thus, I is true.
II) All similar triangles are congruent.
Congruence guarantees similarity, but similarity does not guarantee congruence.
Two triangles need only two congruent angles to guarantee similarity, while two congruent angles does guarantee congruence. II is false.
III) All right triangles are similar.
Triangles are similar through the AA criterion. One angle of a right triangle must be 90, so this leaves 90 for the two remaining angles. Since these two angles can be anything that sum to 90, we are not guaranteed similarity. For example, 20 - 70 - 90 and 30 - 60 - 90 triangles. They are both right triangles, but are not similar. III is false.
IV) All isosceles right triangles are similar.
Again, since we have a right triangle, one angle is 90, which leaves two remaining angles that must sum to 90.
However, since it is isosceles, the two remaining angles must measure the same. Only one solution exists then: both angles must be 45.
Since all angles must be 45 - 45 - 90, it fulfills the AA Criterion. Thus, all isosceles right triangles are similar. IV is true.
The correct statements are I and IV.
