This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here's the solution to your questions. Part 1: Similar Quadrilaterals Given that quadrilateral ABCD is similar to quadrilateral EFGH (ABCD EFGH). This means their corresponding angles are equal, and the ratio of their corresponding sides is constant. From the image: Side AB = 4 cm Side EF = 6 cm Side FG = 4.5 cm Step 1: Find the common ratio of their corresponding sides. The common ratio (scale factor) k is the ratio of a side in EFGH to its corresponding side in ABCD. k = (EF)/(AB) k = 6 cm4 cm k = (3)/(2) = 1.5 The common ratio is 1.5. Step 2: Find the length of side BC. Since the quadrilaterals are similar, the ratio of FG to BC must also be equal to the common ratio k. (FG)/(BC) = k Substitute the known values: 4.5 cmBC = 1.5 To find BC, rearrange the equation: BC = 4.5 cm1.5 BC = 3 cm The length of BC is 3 cm. --- Part 2: Exercise 7.7 - Determine whether each statement is true or false. a. All equilateral triangles are similar. True*. All equilateral triangles have three 60^ angles. Since all corresponding angles are equal, and the ratio of sides is constant, they are similar. b. All isosceles triangles are similar. False*. Isosceles triangles can have different angle measures (e.g., 80^, 80^, 20^ vs. 50^, 50^, 80^), so they are not necessarily similar. c. All isosceles right triangles are similar. True*. All isosceles right triangles have angles 45^, 45^, 90^. Since all corresponding angles are equal, they are similar. d. All rectangles are similar. False*. While all rectangles have four 90^ angles, their side ratios (length to width) can differ (e.g., a 2 × 4 rectangle vs. a 2 × 2 square). e. All rhombuses are similar. False*. While all rhombuses have four equal sides, their angles can differ (e.g., a square vs. a rhombus with 60^ angles). f. All squares are similar. True*. All squares have four 90^ angles and four equal sides. Their corresponding angles are equal, and the ratio of their sides is constant. g. All congruent polygons are similar. True*. Congruent polygons have the same shape and size. This means their corresponding angles are equal, and the ratio of their corresponding sides is 1:1, which satisfies the conditions for similarity. h. All similar polygons are congruent. False*. Similar polygons have the same shape but not necessarily the same size. For example, a 2 × 2 square and a 4 × 4 square are similar but not congruent. i. All regular pentagons are similar. True*. All regular pentagons have five equal sides and five equal interior angles of 108^. Since all corresponding angles are equal, and the ratio of sides is constant, they are similar.