Given: abcd is a rectangle. prove: abcd has congruent diagonals. identify the steps that complete the proof.
2 answers:
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Answer:
1. triangle TCA ~ triangle GOD
4. If TC/GD=3/2, then m<T/m<G=3/2.
Step-by-step explanation:
<u><em>The statements of the question are</em></u>
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
<u><em>Verify each statement</em></u>
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
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