answer:To determine whether a triangle can have two right angles, let's start by considering the properties of triangles and right angles. A right angle is an angle whose measure is 90 degrees. One fundamental property of triangles is that the sum of the measures of their angles is always 180 degrees. Now, if a triangle had two right angles, their combined measure would be 180 degrees, because 90 + 90 equals 180. However, this would leave no remaining degrees for the third angle, as the total of the angles in a triangle cannot exceed 180 degrees. A triangle by definition has three angles, so having two right angles would mean that the third angle would have a measure of zero degrees. A zero-degree angle doesn't make sense in the context of a triangle, as it would essentially mean that the two sides adjacent to the zero-degree angle would overlap or be the same line. In this case, the shape would no longer be a triangle. Therefore, based on the properties of triangles and angles, it is not possible to have a triangle with two right angles.

answer:To find the two marbles you gave away, let's compare the original set of marbles with the ones you have left. The original set consisted of marbles numbered 1 through 6. You still have the marbles numbered 1, 3, 4, and 6. This means that the marbles you gave away are the ones that are missing from this set. Looking at the sequence, we can see that the missing marbles are numbered 2 and 5. These are the two marbles you gave away to your friend.

answer:1. **Sort the groups**: There are two distinct groups: - Group A: 25 people who all know each other. - Group B: 15 people who know no one at the conference. 2. **Handshakes between Group A and Group B**: - Every individual in Group B will shake hands with every individual in Group A because they do not know each other. - The number of such handshakes is: [ 25 times 15 = 375. ] 3. **Handshakes within Group B**: - Since no one in Group B knows each other, each pair will shake hands. - The number of pairs chosen from 15 people is: [ binom{15}{2} = frac{15 times 14}{2} = 105. ] 4. **Total handshakes**: - The total number is the sum of handshakes between Group A and Group B, plus within Group B: [ 375 + 105 = 480. ] 5. **Conclusion**: - The total number of handshakes that occur at the conference is 480. The final answer is boxed{textbf{(D)}: 480}.

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