Geometry - Similarity - Quick Guides | Mathematics

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Study GuideGeometry–Similarity1.Properties of ProportionsLet’s make proportions simple and clear!Aproportionis 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 andcorrectly.1.1Means–Extremes (Cross Products) PropertyIfthenIn 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 GuideExample 1:SolveFind(a)ifStep 1:Cross multiply.Step 2:Solve.Example 2:SolveIs(3:4 = 7:8)a proportion?If it were true, cross products would match:This is not true, so it isnota proportion.Example 3:Means–Extremes Switching PropertyIfis a proportion, then switching the means or the extremes still gives a true proportion.For example, if

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Study Guidethen we can switch:Both new statements are still proportions.Example 4:SolveIffind the ratio (x/y).Switch the means (5 and(y)):So the ratio of(x)to(y)is5 to 4.1.2Upside-Down PropertyIfthen you can flip both fractions:If a proportion is true, its reciprocal is also true.

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

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Study GuideExample 6:SolveIffind (13/8).Since(13 = 5 + 8), use Property 4:Example 7:Geometry ApplicationFigure 1 Using theSegment Addition Postulate.Suppose in a figure:We want to find (AC/BC).

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Study GuideUsing theSegment Addition Postulate:Now apply Property 4:Example 8:Real-World ApplicationA map scale says:3 cm on the map = 5 actual milesIf 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 are16⅔ miles apart.

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Study GuideSummaryProportions are powerful tools. They help us:•Solve equations•Compare ratios•Work with geometry•Solve real-world problems like maps and scaling2.Similar Polygons2.1What Are Similar Polygons?Figure 1Similar quadrilaterals.Two polygons that have thesame shapeare calledsimilar polygons.We use the symbol~to show similarity.For example:Notice this is different from the congruence symbol (≅).•Congruentfigures are the same shapeandsize.•Similarfigures are the same shape, but not necessarily the same size.

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Study Guide2.2What Must Be True?When two polygons are similar,bothof 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 arenot similar.Example 1:Similar QuadrilateralsFigure 2Quadrilaterals 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 3Quadrilaterals 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 =∠HThe order of the letters matters! It tells you which angles match.•Corresponding Sides Are ProportionalEach side in the first quadrilateral matches a side in the second one.2.3When Polygons Are NOT SimilarIt’s possible for one condition to be true without the other.But remember—both must be truefor similarity.Case 1: Side Ratios Are Equal, but Angles Are NotEven 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 arenot similar.Case 2: Angles Are Equal, but Side Ratios Are NotSometimes all corresponding angles are equal, but the side lengths are not proportional.

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

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