1. triangle TCA ~ triangle GOD
4. If TC/GD=3/2, then m<T/m<G=3/2.
Step-by-step explanation:
The statements of the question are
1. triangle TCA ~ triangle GOD
2. AT: OG= AC:OD
3. If TC/GD=3/2, then AC/OD=3/2.
4. If TC/GD=3/2, then m<T/m<G=3/2.
5. If the scale factor of triangle CAT to triangle DOG is 3 to 2, then the scale factor of triangle DOG to triangle CAT is 2 to 3.
6. If AC is twice as long as AT, then OD is twice as long as OG
Verify each statement
1) triangle TCA ~ triangle GOD
we know that
If two triangles are similar, then the ratio of its corresponding sides is proportional and its corresponding angles are congruent
In this problem
triangle CAT ~ triangle DOG -----> is given
so
Reorder
Corresponding sides are named using pairs of letters in the same position on either side of the congruence statement
triangle TCA ~ triangle GDO
therefore
The statement is not correct
2). AT: OG= AC:OD
we know that
If two triangles are similar, then the ratio of its corresponding sides is proportional and its corresponding angles are congruent
In this problem
triangle CAT ~ triangle DOG -----> is given
so
Applying proportion
The statement is true
3). If TC/GD=3/2, then AC/OD=3/2.
we have that
---> see part 2)
so
therefore
The statement is true
4). If TC/GD=3/2, then m<T/m<G=3/2
Remember that
In two similar triangles, corresponding angles are congruent
so
therefore
The statement is false
5). If the scale factor of triangle CAT to triangle DOG is 3 to 2, then the scale factor of triangle DOG to triangle CAT is 2 to 3
we know that
If the scale factor of triangle CAT to triangle DOG is a/b. then  the scale factor of triangle DOG to triangle CAT is b/a
The scale factor will be the reciprocal
therefore
The statement is true Â
6). If AC is twice as long as AT, then OD is twice as long as OG
we know that
Reorder
If AC is twice as long as AT ----> AC=2AT
If OD is twice as long as OG ----> OD=2OG
substitute
simplify
----> is true
therefore
The statement is true
