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Geometry - Similarity

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    Study Guide Geometry – Similarity 1. Properties of Proportions Let’s make proportions simple and clear! A proportion is an equation that says two ratios are equal. For example: Both fractions represent the same value, so this is a proportion. There are four important properties of proportions. These rules help us solve problems quickly and correctly. 1.1 Means – Extremes (Cross Products) Property If then In other words, multiply across diagonally . The product of the extremes equals the product of the means. If ( ad = bc ) and none of the numbers are zero, then the ratios form a proportion:

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    Study Guide Example 1: Solve Find ( a ) if Step 1: Cross multiply. Step 2: Solve. Example 2: Solve Is ( 3:4 = 7:8 ) a proportion? If it were true, cross products would match: This is not true, so it is not a proportion. Example 3: Means – Extremes Switching Property If is a proportion, then switching the means or the extremes still gives a true proportion. For example, if

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    Study Guide then we can switch: Both new statements are still proportions. Example 4: Solve If find the ratio (x/y). Switch the means (5 and ( y ) ): So the ratio of ( x ) to ( y ) is 5 to 4 . 1.2 Upside - Down Property If then you can flip both fractions: If a proportion is true, its reciprocal is also true.

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    Study Guide Example 5: Solve If find (a/b). Step 1: Rewrite using ratios. Step 2: Flip both sides (Property 3). So, 1.4 Denominator Addition/Subtraction Property If then and This property allows you to add or subtract the numerator and denominator in a special way.

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    Study Guide Example 6: Solve If find ( 13/8 ). Since ( 13 = 5 + 8 ) , use Property 4: Example 7: Geometry Application Figure 1 Using the Segment Addition Postulate. Suppose in a figure: We want to find (AC/BC).

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    Study Guide Using the Segment Addition Postulate : Now apply Property 4: Example 8: Real - World Application A map scale says: 3 cm on the map = 5 actual miles If two cities are 10 cm apart on the map, how far apart are they in real life? Let ( x ) = actual distance. Set up the proportion: Cross multiply: Solve: So, the cities are 16⅔ miles apart .

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    Study Guide Summary Proportions are powerful tools. They help us: • Solve equations • Compare ratios • Work with geometry • Solve real - world problems like maps and scaling 2. Similar Polygons 2.1 What Are Similar Polygons? Figure 1 Similar quadrilaterals. Two polygons that have the same shape are called similar polygons . We use the symbol ~ to show similarity. For example: Notice this is different from the congruence symbol ( ≅ ). • Congruent figures are the same shape and size. • Similar figures are the same shape, but not necessarily the same size.

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    Study Guide 2.2 What Must Be True? When two polygons are similar, both of these facts must be true: 1. Corresponding angles are equal. 2. Corresponding sides are proportional. (Their ratios are equal.) Both conditions are required. If even one fails, the polygons are not similar . Example 1: Similar Quadrilaterals Figure 2 Quadrilaterals that are not similar to one another. Even though the ratios of corresponding sides are equal, corresponding angles are not equal (90° ≠ 120°, 90° ≠ 60°). Figure 3 Quadrilaterals that are not similar to one another. Suppose: This tells us something very important.

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    Study Guide • Corresponding Angles Are Equal • ∠ A = ∠ E • ∠ B = ∠ F • ∠ C = ∠ G • ∠ D = ∠ H The order of the letters matters! It tells you which angles match. • Corresponding Sides Are Proportional Each side in the first quadrilateral matches a side in the second one. 2.3 When Polygons Are NOT Similar It’s possible for one condition to be true without the other. But remember — both must be true for similarity. Case 1: Side Ratios Are Equal, but Angles Are Not Even if corresponding sides have equal ratios, the polygons are not similar if the angles don’t match. For example: • One angle is 90° • The corresponding angle is 120° Since 90° ≠ 120°, the polygons are not similar . Case 2: Angles Are Equal, but Side Ratios Are Not Sometimes all corresponding angles are equal, but the side lengths are not proportional.

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    Study Guide For example: Since the side ratios are not equal, the polygons are not similar , even though the angles match. Example 1: Solve Figure 4 Similar quadrilaterals. In Figure 4, suppose: We are asked to: (a) Find m ∠ E (b) Find x (a) Finding m ∠ E Angle E corresponds to angle A. Since corresponding angles of similar polygons are equal: If ∠ A = 90°, then:

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