Since the two triangles are similar, each triangles three corresponding sides must have the same ratio. 

The ratio of the triangle is:

Applying this ratio we are able to find the lengths of a similar triangle.

As we are establishing whether or not , then , , and  correspond respectively to , , and .

According to the Side-Angle-Side Similarity Theorem (SASS), if the lengths of two pairs of corresponding sides of two triangles are in proportion, and their included angles are congruent, then the triangles are similar. 

 and  are corresponding sides, as are  and ;  and  are their included angles. Substituting  

Therefore, , and corresponding sides are in proportion.

 and ; the included angles are congruent.

The conditions of SASS are met, and it follows that .

As we are establishing whether or not , then , , and  correspond respectively to , , and .

If , then corresponding sides must be in proportion; that is, it must hold that 

Substituting the lengths of the sides for the respective quantities:

The inequality of these two side ratios disproves the similarity of the triangles, so the correct answer is "false".

As we are establishing whether or not , then , , and  correspond respectively to , , and .

The sum of the measures of the interior angles of a triangle is . Therefore, 

Set  and , and solve for :

By the Angle-Angle Similarity Postulate (AA), two triangles are similar if two angles of the first triangle are congruent to those of their counterparts in the second.  and , so ;  and , so . The conditions of AA are met, so it follows that . The correct response is "true".

As we are establishing whether or not , then , , and  correspond respectively to , , and .

Let . 

Let 

The measures of the interior angles of a triangle total . Therefore, 

Substituting:

By the same reasoning,

Therefore, . 

By the Angle-Angle Similarity Postulate (AA), two triangles are similar if two angles of the first triangle are congruent to those of their counterparts in the second.   and , so the conditions of AA are met; it follows that .

As we are establishing whether or not , then , , and  correspond respectively to  , , and .

If two triangles are similar, then it must hold that corresponding sides are in proportion. Specifically, if , it must hold that

By the Segment Addition Postulate, 

and

Set , , ,  in the above proportion statement, which becomes

Reduce both ratios to lowest terms:

The corresponding sides are not proportional, so the statement  is false.

As we are establishing whether or not , then , , and  correspond respectively to  , , and .

By the Side-Angle-Side Similarity Theorem (SASS), two triangles are similar if two pairs of corresponding sides are in proportion and their included angles are congruent. 

Examine two pairs of corresponding sides:  and , and  and . In both cases, their included angle is ; by the Reflexive Property of Congruence, . 

It remains to be demonstrated that

By the Segment Addition Postulate, 

and

Set , , ,  in the above proportion statement, which becomes

Reduce both ratios to lowest terms:

The corresponding sides are proportional.

The conditions of SASS have been proved, so the statement  is true.

The two given angle congruences set up the conditions of the Angle-Angle Similarity Postulate - if two angles of one triangle are congruent to the two corresponding angles of another triangle, the two triangles are similar. It follows that 

.

However, congruence cannot be proved, since at least one side congruence is needed to prove this. This is not given in the problem. 

Therefore, statement (a) must hold, but not necessarily statement (b).