🧠:<think>Let ( M ) represent Maria's current age and ( A ) represent Alice's current age. Since Maria is eight years older than Alice, we have:[ M = A + 8 ]We're also told that ten years from now, Maria's age will be three times Alice's age six years ago:[ M + 10 = 3(A - 6) ]Substitute ( M ) in terms of ( A ) into the equation:[ (A + 8) + 10 = 3(A - 6) ]Expanding and simplifying:[ A + 18 = 3A - 18 ][ 36 = 2A ][ A = 18 ]Thus, Alice is currently 18 years old, and Maria, being 8 years older, is:[ 18 + 8 = 26 ]Therefore, the sum of their current ages is:[ 18 + 26 = boxed{44} ]</think>

🧠:<think>First, I need to determine the number of days each student packs their lunch based on the information provided.For Aliyah at School A, she packs her lunch 3/5 of the time. Since School A has 'x' school days, the expression for P(A) is straightforward:[ P(A) = frac{3}{5}x ]Next, Becky goes to School B and packs her lunch 1/4 the time Aliyah does. This means Becky's packing rate is:[ frac{1}{4} times frac{3}{5} = frac{3}{20} ]Since School B has 'y' school days, the expression for P(B) is:[ P(B) = frac{3}{20}y ]Charlie attends School C and packs his lunch twice as much as Becky. Therefore, Charlie's packing rate is:[ 2 times frac{3}{20} = frac{3}{10} ]With School C having 'z' school days, the expression for P(C) is:[ P(C) = frac{3}{10}z ]Lastly, Dana goes to School A and packs her lunch every third day. This means Dana's packing rate is:[ frac{1}{3} ]Given that School A has 'x' school days, the expression for P(D) is:[ P(D) = frac{1}{3}x ]By carefully analyzing each student's packing habits and the number of school days at their respective schools, I have derived the expressions for P(A), P(B), P(C), and P(D) in terms of x, y, and z.</think>

🧠:<think>Alright, so I've got this question here that asks which of the statements A, B, C, or D is false. Let me go through each one step by step to figure out which one is not true.Starting with statement A: "The shortest distance between two points is a line segment." Hmm, I remember learning in geometry that a straight line is indeed the shortest distance between two points. So, this seems true. I don't think there's any doubt here; it's a fundamental concept.Moving on to statement B: "Vertical angles are equal." Okay, vertical angles are the angles opposite each other when two lines intersect. I recall that vertical angles are always equal in measure. So, this also seems true. No issues here.Now, statement C: "The complement of a right angle is still a right angle." Wait a minute, let's unpack this. A right angle is 90 degrees. The complement of an angle is what you add to it to get 90 degrees. So, if I have a right angle, what would its complement be? If I add something to 90 degrees to get 90 degrees, that something would have to be 0 degrees. But 0 degrees isn't an angle we usually talk about in this context. So, is the complement of a right angle still a right angle? That doesn't seem right. The complement should be zero degrees, which isn't a right angle. So, this statement might be false.Finally, statement D: "Same-side interior angles are supplementary." Same-side interior angles are the angles on the same side of a transversal line that intersects two other lines. I remember that if the two lines are parallel, then same-side interior angles are supplementary, meaning they add up to 180 degrees. But the statement doesn't specify that the lines are parallel. If the lines aren't parallel, then same-side interior angles don't have to be supplementary. So, this statement is only true under certain conditions, which aren't specified here. Therefore, it might be false in a general sense.So, both statements C and D seem to have issues. Statement C because the complement of a right angle isn't another right angle, and statement D because same-side interior angles are only supplementary if the lines are parallel, which isn't stated. But I need to determine which one is false according to the question.Looking back, statement C is definitely false because the complement of a right angle is zero degrees, not a right angle. Statement D is a bit tricky because it depends on the context, but since it's not specified that the lines are parallel, it's not universally true. However, in many geometry problems, when talking about same-side interior angles, it's often assumed that the lines are parallel. So, maybe statement D is considered true in that context.Given that, I think statement C is the one that's clearly false because it's a straightforward definition issue. The complement of a right angle isn't another right angle.</think>

🧠:<think>Okay, so I need to figure out how many unique pairs of parallel edges a regular octahedron has. Hmm, let me start by recalling what a regular octahedron looks like. It's one of the Platonic solids, right? It has eight triangular faces, twelve edges, and six vertices. I think it's like two square pyramids stuck together at their bases.Alright, so edges in a regular octahedron are all the same length, and each face is an equilateral triangle. Now, the question is about parallel edges. I remember that in a cube, which is another Platonic solid, edges can be parallel if they don't intersect and are in the same direction. But an octahedron is different from a cube, although they are duals of each other.Let me visualize the octahedron. It has six vertices, and each vertex is connected to four others. Wait, no, actually, each vertex is connected to four edges, right? Because each vertex is part of four triangular faces. So, each vertex has four edges coming out of it.Now, to find parallel edges, I need to identify pairs of edges that don't intersect and are in the same direction. Since the octahedron is regular, it's symmetric, so the number of parallel edge pairs should be consistent.I think a good approach is to consider the axes of the octahedron. A regular octahedron has several axes of symmetry: three axes that go through opposite vertices, and four axes that go through the centers of opposite faces. But I'm not sure if that's directly helpful for finding parallel edges.Wait, maybe I should think about the edges in terms of their directions. In a regular octahedron, edges can be grouped based on their directions. Each direction corresponds to a pair of opposite edges. So, if I can figure out how many distinct directions there are, that should give me the number of unique pairs of parallel edges.Let me try to count them. If I pick one edge, how many edges are parallel to it? In a regular octahedron, each edge has exactly one other edge that's parallel to it. So, for each edge, there's one parallel edge. Since there are twelve edges in total, that would suggest six pairs of parallel edges. But wait, that doesn't seem right because the answer choices don't include six as the correct answer.Hold on, maybe I'm missing something. Let me think again. If each edge has one parallel edge, and there are twelve edges, then the number of unique pairs would be twelve divided by two, which is six. But the answer choices include six as option A, but I have a feeling the correct answer is higher.Perhaps I'm not considering all possible directions. Let me try another approach. Let's consider the octahedron as two square pyramids glued together. Each pyramid has a square base, and when glued together, the bases form an octagon? Wait, no, actually, when you glue two square pyramids at their bases, you get an octahedron with eight triangular faces.In this structure, the edges can be categorized into two types: the edges of the square bases and the edges connecting the apexes of the pyramids to the base vertices. The edges of the square bases are all in the same plane, and the edges connecting the apexes are along the vertical axis.So, in this case, the edges of the square bases are all in the same plane, but are they parallel? In a square, opposite edges are parallel. So, in each square base, there are two pairs of parallel edges. Since there are two square bases (one from each pyramid), that would give us four pairs of parallel edges.Additionally, the edges connecting the apexes to the base vertices are all in different directions. Each of these edges is connected to one apex and one base vertex. Since there are four base vertices, each apex is connected to four edges. But are any of these edges parallel?Hmm, in the octahedron, the edges connecting the apexes to the base vertices are not parallel to each other because they all emanate from the apexes in different directions. However, each edge from one apex is parallel to the corresponding edge from the other apex.So, for each of the four edges connected to one apex, there is a corresponding edge connected to the other apex that is parallel. That would give us four more pairs of parallel edges.Adding these together, we have four pairs from the square bases and four pairs from the vertical edges, totaling eight pairs of parallel edges.Wait, but earlier I thought it might be six. Maybe I need to double-check. Let's see, in each square base, there are two pairs of parallel edges, and since there are two bases, that's four pairs. Then, for the vertical edges, each of the four edges from one apex has a parallel counterpart from the other apex, giving another four pairs. So, in total, that's eight pairs.Looking back at the answer choices, eight is option B. So, I think the correct answer is B) 8.But just to be thorough, let me consider another perspective. Maybe using coordinates. If I place the octahedron in a coordinate system with vertices at (±1, 0, 0), (0, ±1, 0), and (0, 0, ±1), then the edges can be represented by the lines connecting these points.Each edge connects two vertices that differ in exactly two coordinates. For example, the edge from (1, 0, 0) to (0, 1, 0) has a direction vector of (-1, 1, 0). Similarly, the edge from (-1, 0, 0) to (0, -1, 0) has a direction vector of (1, -1, 0), which is a scalar multiple of the first vector, hence parallel.By examining all edges, I can see that each direction vector has exactly one other edge with the same direction, confirming that there are six pairs of parallel edges. Wait, now I'm confused because this contradicts my earlier conclusion of eight pairs.Hmm, maybe I made a mistake in my coordinate analysis. Let me list out all the edges and their direction vectors:1. (1, 0, 0) to (0, 1, 0): direction (-1, 1, 0)2. (1, 0, 0) to (0, -1, 0): direction (-1, -1, 0)3. (1, 0, 0) to (0, 0, 1): direction (-1, 0, 1)4. (1, 0, 0) to (0, 0, -1): direction (-1, 0, -1)5. (-1, 0, 0) to (0, 1, 0): direction (1, 1, 0)6. (-1, 0, 0) to (0, -1, 0): direction (1, -1, 0)7. (-1, 0, 0) to (0, 0, 1): direction (1, 0, 1)8. (-1, 0, 0) to (0, 0, -1): direction (1, 0, -1)9. (0, 1, 0) to (0, 0, 1): direction (0, -1, 1)10. (0, 1, 0) to (0, 0, -1): direction (0, -1, -1)11. (0, -1, 0) to (0, 0, 1): direction (0, 1, 1)12. (0, -1, 0) to (0, 0, -1): direction (0, 1, -1)Now, let's group these direction vectors:- (-1, 1, 0) and (1, -1, 0) are negatives of each other, so they are parallel.- (-1, -1, 0) and (1, 1, 0) are negatives of each other, so they are parallel.- (-1, 0, 1) and (1, 0, -1) are negatives of each other, so they are parallel.- (-1, 0, -1) and (1, 0, 1) are negatives of each other, so they are parallel.- (0, -1, 1) and (0, 1, -1) are negatives of each other, so they are parallel.- (0, -1, -1) and (0, 1, 1) are negatives of each other, so they are parallel.So, there are six pairs of parallel edges. This contradicts my earlier conclusion of eight pairs. It seems like the coordinate method gives six pairs, while my earlier geometric reasoning gave eight pairs. I must have made a mistake in my geometric approach.Let me revisit the geometric reasoning. I considered the octahedron as two square pyramids glued together. In this structure, the square bases have edges that are parallel in pairs. Each square has two pairs of parallel edges, and since there are two squares, that's four pairs. Then, I considered the vertical edges connecting the apexes to the base vertices, thinking that each of these four edges has a parallel counterpart, adding four more pairs, totaling eight.But according to the coordinate method, there are only six pairs. Where did I go wrong?Ah, I see now. In the coordinate method, the vertical edges (those along the z-axis) are actually not parallel to each other. Each vertical edge connects a base vertex to an apex, but these edges are not parallel because they are in different directions. Wait, no, in the coordinate system, the vertical edges are along the z-axis, so they should be parallel.Wait, no, in the coordinate system, the vertical edges are from (1,0,0) to (0,0,1) and (0,0,-1), etc. So, their direction vectors are (-1,0,1) and (-1,0,-1), which are not parallel. So, actually, the vertical edges are not parallel to each other. That means my earlier assumption that each vertical edge has a parallel counterpart was incorrect.Therefore, in the geometric approach, I mistakenly counted four pairs of parallel edges among the vertical edges, but in reality, they are not parallel. So, only the edges in the square bases are parallel, giving four pairs, and the vertical edges are not parallel to each other or to any other edges.But according to the coordinate method, there are six pairs of parallel edges, which suggests that there are more pairs than just those in the square bases.Wait, perhaps I need to consider that in the octahedron, edges can be parallel even if they are not in the same square base. For example, an edge from the top apex to a base vertex might be parallel to an edge from the bottom apex to another base vertex.Looking back at the coordinate method, edges like (1,0,0) to (0,0,1) have direction (-1,0,1), and edges like (-1,0,0) to (0,0,-1) have direction (1,0,-1), which are negatives and thus parallel. Similarly, other edges have their parallel counterparts.So, in total, there are six pairs of parallel edges:1. (-1,1,0) and (1,-1,0)2. (-1,-1,0) and (1,1,0)3. (-1,0,1) and (1,0,-1)4. (-1,0,-1) and (1,0,1)5. (0,-1,1) and (0,1,-1)6. (0,-1,-1) and (0,1,1)Therefore, the correct number of unique pairs of parallel edges is six, which is option A.But wait, earlier I thought the answer was eight, and now I'm getting six. I need to resolve this discrepancy.Let me try to visualize the octahedron again. If I consider the edges in the square bases, each square has two pairs of parallel edges, so two squares give four pairs. Then, considering the edges connecting the apexes to the base vertices, each of these edges has a parallel counterpart in the opposite direction, giving another two pairs. So, in total, six pairs.Yes, that makes sense. So, the four pairs from the square bases and two pairs from the vertical edges, totaling six pairs.Therefore, the correct answer should be six, which is option A.But wait, in the coordinate method, I found six pairs, and in the geometric approach, considering both square bases and vertical edges, I also get six pairs. So, why did I initially think it was eight?I think I confused the number of edges with the number of pairs. Each pair consists of two edges, so if there are six pairs, that accounts for twelve edges, which is correct because the octahedron has twelve edges.So, to summarize, there are six unique pairs of parallel edges in a regular octahedron.</think>