Question

i can determine if multiple shapes are similar. (8.g.a.4)

1. choose whether each of the statements is true in all cases, in some cases, or in no cases.

explain why

a. if an angle is dilated with the center of dilation at its vertex, the angle measure
may change.

b. if two triangles are similar, then they are congruent.

c. if two triangles are congruent, then they are similar.

Explanation:

• For statement a:
• Dilation is a transformation that changes the size of a figure but not the shape. When an angle is dilated with the center of dilation at its vertex, the sides of the angle are stretched or compressed, but the amount of rotation (which determines the angle measure) remains the same. So, the angle measure does not change. This statement is false in all cases.
• For statement b:
• Similar triangles have the same shape (corresponding angles are equal) but not necessarily the same size (corresponding sides are in proportion). Congruent triangles have the same shape and size (corresponding angles and sides are equal). So, if two triangles are similar, they are not congruent unless the scale factor of similarity is \(1\). This statement is true in some cases (when the scale factor of similarity is \(1\)).
• For statement c:
• Congruent triangles have the same shape and size. Similar triangles have the same shape. Since congruent triangles satisfy the condition of having the same shape (as they have the same shape and size), if two triangles are congruent, they are similar (with a scale factor of \(1\)). This statement is true in all cases.

Answer:

• a. False in all cases.
• b. True in some cases.
• c. True in all cases.