Which sides of ABCD are involved in each statement? Answer whether these sides are congruent or not.
Which is Different: Is there enough information to prove that AB≅ DC? Both Answers: Yes and no.
Practice makes perfect
To find which of the statements is different, we will study them one at a time. Also, we will answer the corresponding question.
In each statement, we will refer the trapezoid ABCD shown above.
Statement 1
Is there enough information to prove that trapezoid ABCD is isosceles?
By definition, an isosceles trapezoid has congruent legs. Therefore, the given statement can be rewritten as follows.
Question: & Are AD and BC congruent?
Since ∠ C ≅ ∠ D, by the Isosceles Trapezoid Base Angles Converse, we have that AD ≅ BC. Thus, the trapezoid ABCD is isosceles.
Answer: & Yes
Statement 2
Is there enough information to prove that AB≅DC?
We can rewrite this statement by using an equivalent question.
Question: & Are AB and DC congruent?
Let's assume temporarily that if AB ≅ DC. Then, by the Opposite Sides Parallel and Congruent Theorem, we have that ABCD is a parallelogram, which contradicts the fact that ABCD is a trapezoid. Thus, the mentioned segments are not congruent.
Answer: & No
Statement 3
Is there enough information to prove that the non-parallel sides of trapezoid ABCD are congruent?
The non-parallel sides of trapezoid ABCD are its legs. Then, we can rewrite the given question as shown below.
Question: & Are AD and BC congruent?
Proceeding as in Statement 1, we conclude that AD≅ BC.
Answer: & Yes
Statement 4
Is there enough information to prove that the legs of trapezoid ABCD are congruent?
Notice that this statement, like in the previous statement, is asking about the sides AD and BC. Thus, we can rewrite this statement as follows.
Question: & Are AD and BC congruent?
By Statement 1, we can conclude that AD≅ BC.
Answer: & Yes
Conclusion
To make a conclusion, we will make a table where we will write the sides of ABCD involved in each statement and the answers we got for each one.
Statement
Sides Involved
Answer
1
AD and BC
Yes
2
AB and DC
No
3
AD and BC
Yes
4
AD and BC
Yes
As we can see, the different statement is the second one and the two answers wanted are yes and no.
