Question

determine whether the hypothesis of each given conditional is true from the given information.
given: if a, b, c, and d are collinear, then they lie in the same plane.
a, b, c, and d lie in the same plane.
conclusion: a, b, c, and d are collinear.
given: if ( overline{bd} ) bisects ( angle abc ), then ( d ) lies in the interior of ( angle abc ).
( d ) lies in the interior of ( angle abc ).
conclusion: ( overline{bd} ) bisects ( angle abc ).
table with \yes\ /
o\ columns for each conclusion

Explanation:

To solve this, we analyze each conditional statement by identifying the hypothesis (the "if" part) and checking if it's true based on the given information.

First Conditional:

Given: If \( A, B, C, \) and \( D \) are collinear, then they lie in the same plane.
Information: \( A, B, C, \) and \( D \) lie in the same plane.
Hypothesis: \( A, B, C, \) and \( D \) are collinear.

The conclusion (they lie in the same plane) is true, but the hypothesis (they are collinear) is not confirmed by the given information (lying in the same plane does not imply collinearity). So the hypothesis is false.

Second Conditional:

Given: If \( \overrightarrow{BD} \) bisects \( \angle ABC \), then \( D \) lies in the interior of \( \angle ABC \).
Information: \( D \) lies in the interior of \( \angle ABC \).
Hypothesis: \( \overrightarrow{BD} \) bisects \( \angle ABC \).

The conclusion ( \( D \) is in the interior) is true, but the hypothesis ( \( \overrightarrow{BD} \) bisects \( \angle ABC \)) is not confirmed by the given information (being in the interior does not imply bisecting the angle). So the hypothesis is false.

Filling the Table:

Given Statement	Yes	No
Second Conditional Hypothesis	☐	☑

(For each row, mark "Yes" if the hypothesis is true, "No" if false. Here, both hypotheses are false, so mark "No" for both.)

Answer:

To solve this, we analyze each conditional statement by identifying the hypothesis (the "if" part) and checking if it's true based on the given information.

First Conditional:

Given: If \( A, B, C, \) and \( D \) are collinear, then they lie in the same plane.
Information: \( A, B, C, \) and \( D \) lie in the same plane.
Hypothesis: \( A, B, C, \) and \( D \) are collinear.

The conclusion (they lie in the same plane) is true, but the hypothesis (they are collinear) is not confirmed by the given information (lying in the same plane does not imply collinearity). So the hypothesis is false.

Second Conditional:

Given: If \( \overrightarrow{BD} \) bisects \( \angle ABC \), then \( D \) lies in the interior of \( \angle ABC \).
Information: \( D \) lies in the interior of \( \angle ABC \).
Hypothesis: \( \overrightarrow{BD} \) bisects \( \angle ABC \).

The conclusion ( \( D \) is in the interior) is true, but the hypothesis ( \( \overrightarrow{BD} \) bisects \( \angle ABC \)) is not confirmed by the given information (being in the interior does not imply bisecting the angle). So the hypothesis is false.

Filling the Table:

Given Statement	Yes	No
Second Conditional Hypothesis	☐	☑

(For each row, mark "Yes" if the hypothesis is true, "No" if false. Here, both hypotheses are false, so mark "No" for both.)