Teach me ---Here is the extracted content from the image: --- **Question 9** **Question Stem:** A diagram of a flower garden is shown below. What is the total area, in square feet, of the flower garden? **Chart/Diagram Description:** * **Type:** Geometric figure, specifically an L-shaped polygon representing a flower garden. * **Main Elements:** * The shape has 6 vertices and is generally rectangular with a rectangular cutout. * **Dimensions/Labels:** * The top horizontal side has a length of 9 ft. * The leftmost vertical side (upper part) has a length of 3 ft. * The inner vertical side has a length of 4 ft. * The bottom horizontal side has a length of 6 ft. * Implicitly, the total height of the garden is 3 ft + 4 ft = 7 ft. * Implicitly, the total width of the garden is 9 ft. * The width of the cutout is 9 ft - 6 ft = 3 ft. * The height of the cutout is 7 ft - 3 ft = 4 ft. (This matches the labeled inner vertical side). **Options:** A 22 B 27 C 51 D 54 --- **Question 7** **Question Stem:** Ray KM divides straight angle JKL into two parts as shown below. Which equation represents how to find the measure, in degrees, of angle JKM ? **Chart/Diagram Description:** * **Type:** Geometric figure, specifically an angle diagram. * **Main Elements:** * A straight line is represented by a horizontal segment with points J and L at its ends, and an arrow extending from J to the left and from L to the right, indicating a line. * Point K is located on the straight line, between J and L, serving as the vertex. * Ray KM originates from point K and extends upwards to the right, with an arrow at point M. * **Angles:** * Angle LKM is explicitly labeled with a measure of 70°. * Angle JKM is labeled with a question mark (?), indicating it's the unknown value to be found. * Angle JKL is a straight angle, formed by the straight line JKL. **Options:** A 90 - 20 = ? B 90 - 70 = ? C 180 - 70 = ? D 180 - 110 = ? ---

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Welcome to our lesson on solving area problems! Today we'll learn how to find the area of composite shapes by using a powerful strategy called decomposition. The key idea is to break down complex shapes into simpler rectangles, then add or subtract their areas as needed. Remember, the area of a rectangle equals length times width. Let's see how this applies to more complex shapes like L-shaped figures. Now let's analyze the L-shaped flower garden problem step by step. We have a garden with specific dimensions: 9 feet across the top, 3 feet on the upper left side, 4 feet on the inner vertical side, and 6 feet across the bottom. There are two effective methods to find the total area. Method 1 treats this as a large rectangle minus a cutout. The large rectangle would be 9 by 7 feet, giving us 63 square feet. We subtract the cutout area of 3 by 4 feet, which is 12 square feet, leaving us with 51 square feet. Method 2 divides the garden into two separate rectangles. The top rectangle is 9 by 3 feet, giving 27 square feet. The bottom rectangle is 6 by 4 feet, giving 24 square feet. Adding these together: 27 plus 24 equals 51 square feet. Both methods give us the same answer of 51 square feet, which corresponds to option C. Now let's transition to angle geometry and explore straight angle fundamentals. A straight angle measures exactly 180 degrees and forms a straight line. When a ray divides a straight angle into two parts, we get adjacent angles that have a special relationship. The key principle is that adjacent angles on a straight line always add up to 180 degrees. In our diagram, we see ray KM dividing the straight angle JKL into two parts: angle JKM, which is unknown and marked with a question mark, and angle LKM, which measures 70 degrees. Since these are adjacent angles on a straight line, we can write the equation: angle JKM plus angle LKM equals 180 degrees. This fundamental property will help us solve for the unknown angle in the next step. Now let's solve the angle problem step by step. We're given that angle LKM equals 70 degrees, and we know that JKL is a straight angle measuring 180 degrees. We need to find the measure of angle JKM. Since angles JKM and LKM are adjacent angles on a straight line, they must add up to 180 degrees. So we can write the equation: angle JKM plus angle LKM equals 180 degrees. Substituting the known value: angle JKM plus 70 degrees equals 180 degrees. To solve for angle JKM, we subtract 70 degrees from both sides: angle JKM equals 180 degrees minus 70 degrees, which gives us 110 degrees. Looking at the multiple choice options, this corresponds to option C: 180 minus 70 equals question mark. The answer is 110 degrees. Let's review the key problem-solving strategies we used today. For area problems, we learned to identify given dimensions, use the decomposition method to break complex shapes into simple rectangles, and then add or subtract areas as needed. The L-shaped garden problem was solved using both the subtraction method and the addition method, both giving us 51 square feet, which corresponds to option C. For angle problems, we identified angle relationships and applied the straight angle property that adjacent angles sum to 180 degrees. In our angle problem, we found that angle JKM equals 180 degrees minus 70 degrees, which gives us 110 degrees, corresponding to option C. Both problems required systematic thinking: identifying given information, applying fundamental geometric principles, and performing careful calculations. These strategies will serve you well in solving similar geometry problems. Remember to always check your answers and understand why certain methods work for different types of problems.