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JRLeon Geometry Chapter 11.1 HGSH Lesson 11.1 (581)

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1 JRLeon Geometry Chapter 11.1 HGSH Lesson 11.1 (581)

2 JRLeon Geometry Chapter 11.1 HGSH Lesson 11.1 (581)

3 JRLeon Geometry Chapter 11.1 HGSH Lesson 11.1 The statement CORN ~ PEAS says that quadrilateral CORN is similar to quadrilateral PEAS. Just as in statements of congruence, the order of the letters tells you which segments and which angles in the two polygons correspond. Notice that the ratio of the lengths of any two segments in one polygon is equal to the ratio of the corresponding two segments in the similar polygon. Observe que la relación de las longitudes de dos segmentos en un polígono es igual a la relación de los dos segmentos correspondientes en el polígono similar.

4 JRLeon Geometry Chapter 11.1 HGSH Lesson 11.1 Do you need both conditions— congruent angles and proportional sides—to guarantee that the two polygons are similar? If only the corresponding angles of two polygons are congruent, are the polygons similar?

5 JRLeon Geometry Chapter 11.1 HGSH Lesson 11.1 Do you need both conditions— congruent angles and proportional sides—to guarantee that the two polygons are similar? If corresponding sides of two polygons are proportional, are the polygons necessarily similar?

6 JRLeon Geometry Chapter 11.1 HGSH Lesson 11.1 You can use the definition of similar polygons to find missing measures in similar polygons.

7 JRLeon Geometry Chapter 11.1 HGSH Lesson 11.1 A transformation in which a polygon is enlarged or reduced by a given factor around a given center point. Dilation - of a polygon Una transformación en el que un polígono se amplía o reduce por un factor dado alrededor de un punto central dado.

8 JRLeon Geometry Chapter 11.1 HGSH Lesson 11.2 (589) In Lesson 11.1, you concluded that you must know about both the angles and the sides of two quadrilaterals in order to make a valid conclusion about their similarity. However, triangles are unique. Recall from Chapter 4 that you found four shortcuts for triangle congruence: SSS, SAS, ASA, and SAA. Are there shortcuts for triangle similarity as well? ( SSS, SAS, ASA, SAA or AAA)? Let’s first look for shortcuts using only angles Let’s first look for shortcuts using only angles. The figures below illustrate that you cannot conclude that two triangles are similar given that only one set of corresponding angles are congruent. How about two sets of congruent angles?

9 JRLeon Geometry Chapter 11.1 HGSH Lesson 11.2 The AA Similarity Postulate The AA (angle angle) similarity postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar Triangle Sum Conjecture You know from the Triangle Sum Conjecture that:  mA +  mB +  mC = 180°, and  mD +  mE +  mF =180°. transitive property By the transitive property,  mA +  mB +  mC =  mD +  mE +  mF. You also know that  mA =  mD, and  mB =  mE. You can substitute for  mD and  mE in the longer equation to get:  mA +  mB +  mC =  mA +  mB +  mF. Subtracting equal terms from both sides, you are left with  mC =  mF. As you may have guessed, there is no need to investigate the AAA, ASA, or SAA Similarity Conjectures. Thanks to the Triangle Sum Conjecture, or more specifically the Third Angle Conjecture, the AA Similarity Conjecture is all you need.

10 JRLeon Geometry Chapter 11.1 HGSH Lesson 11.2 Now let’s look for shortcuts for similarity that use only sides. The figures below illustrate that you cannot conclude that two triangles are similar given that two sets of corresponding sides are proportional. How about all three sets of corresponding sides?

11 JRLeon Geometry Chapter 11.1 HGSH Lesson 11.2 SSS Similarity Side-side-side similarity. When two triangles have corresponding sides with identical ratios as shown, the triangles are similar. Three sides in proportion (SSS) So SSS, AAA, ASA, and SAA are shortcuts for triangle similarity. That leaves SAS and SSA as possible shortcuts to consider.

12 JRLeon Geometry Chapter 11.1 HGSH Lesson 11.2 SAS Similarity Side-angle-side similarity. When two triangles have corresponding sides with identical ratios and the included angles are congruent as shown, the triangles are similar. Two sides and included angle (SAS)

13 JRLeon Geometry Chapter 11.1 HGSH Lesson 11.2 One question remains: Is SSA a shortcut for similarity? Recall from Chapter 4 that SSA did not work for congruence because you could create two different triangles. Those two different triangles were neither congruent nor similar. So, no, SSA is not a shortcut for similarity.

14 JRLeon Geometry Chapter HGSH Lessons 11.1 – 11.2 Class work: Pg. 585 – Problems 2 through 16 EVEN Pg. 591 – Problems 2 through 16 EVEN Homework: Pg. 585 – Problems 1 through 15 ODD Pg. 591 – Problems 1 through 15 ODD


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