answer:To rewrite the proposition "vertical angles are congruent" in the form "if ldots then ldots", we start by identifying the hypothesis and the conclusion based on the given proposition. 1. **Identify the Hypothesis**: The initial condition or given situation is that two angles are vertical angles. This is our "if" part of the statement. 2. **Identify the Conclusion**: The result or conclusion drawn from the hypothesis is that these two angles are congruent. This forms our "then" part of the statement. 3. **Combine Hypothesis and Conclusion**: By combining the hypothesis and conclusion, we form the statement in the desired format: "if two angles are vertical angles, then these two angles are congruent." Therefore, the proposition "vertical angles are congruent" rewritten in the form "if ldots then ldots" is: boxed{text{if two angles are vertical angles, then these two angles are congruent.}}

answer:First, let's calculate the total time Ellen spent painting the lilies, roses, and orchids. For the lilies: 17 lilies * 5 minutes per lily = 85 minutes For the roses: 10 roses * 7 minutes per rose = 70 minutes For the orchids: 6 orchids * 3 minutes per orchid = 18 minutes Now, let's add up the time spent on lilies, roses, and orchids: 85 minutes (lilies) + 70 minutes (roses) + 18 minutes (orchids) = 173 minutes Ellen spent a total of 213 minutes painting, so the time spent painting vines is: 213 minutes (total) - 173 minutes (lilies, roses, orchids) = 40 minutes Now, we know that it takes 2 minutes to paint a vine. To find out how many vines she painted, we divide the time spent on vines by the time it takes to paint one vine: 40 minutes / 2 minutes per vine = 20 vines Ellen painted boxed{20} vines.

answer:Let the equation of line AB be y = mx + n. Since the line passes through points A (-4, 0) and B (0, 2): begin{cases} -4m + n = 0 n = 2 end{cases} Solving the system, we get: begin{cases} m = frac{1}{2} n = 2 end{cases} Hence, the equation of line AB is y = frac{1}{2}x + 2. Translating line AB 2 units to the right, we obtain the equation y = frac{1}{2}(x - 2) + 2, which simplifies to y = frac{1}{2}x + 1. Then, rotating the line y = frac{1}{2}x + 1 180° around the origin, we get the equation -y = -frac{1}{2}x + 1, which simplifies to y = frac{1}{2}x - 1. Thus, the equation of line l is boxed{y = frac{1}{2}x - 1}. To solve this problem, first find the equation of line AB using the method of undetermined coefficients, then find the equation after translating the line to the right by 1 unit, and finally rotate the resulting equation 180° around the origin. This problem tests understanding of linear function graphs, geometric transformations, finding linear equations using the method of undetermined coefficients, as well as rules for line translation and rotation, and symmetry with respect to the origin.

answer:To analyze the given equation x^{2}+y^{2}+4x-2y-5c=0 and determine the range of values for c, we follow these steps: 1. **Completing the square** for both x and y terms: - For x: x^{2} + 4x can be rewritten as (x+2)^{2} - 4. - For y: y^{2} - 2y can be rewritten as (y-1)^{2} - 1. 2. **Substitute** the completed squares into the original equation: [ (x+2)^{2} - 4 + (y-1)^{2} - 1 + 4 - 2 - 5c = 0 ] Simplifying, we get: [ (x+2)^{2} + (y-1)^{2} = 5 + 5c ] 3. **Analyze** the equation (x+2)^{2} + (y-1)^{2} = 5 + 5c: - This equation represents a circle with a radius squared equal to 5+5c. 4. **Determine the condition** for the circle to have a real radius: - For the circle to exist in real space, its radius squared must be greater than 0, hence 5+5c > 0. 5. **Solve the inequality** for c: [ 5+5c > 0 implies 5c > -5 implies c > -1 ] 6. **Conclude** the range of values for c: - Since c > -1, the range of values for c is (-1, +infty). Therefore, the correct choice is boxed{A}.