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Contents: Similarity Similar figures Properties of similar figures Figures which are always similar Theorems of similar triangles Area and perimeter of similar triangles Congruency Congruent figures Theorems of congruent triangles SIMILARITY If two objects have the same shape, but one is the enlargement of the other then; they are called "similar." This is symbol that means “similar” Real life examples of similar figures: Enlargement of a photograph, when a photograph is enlarged its size differ from each other but their shape is similar. Shadow of a cardboard shape, if the shade is held in front if a light source, parallel to a screen then a shadow of the shape will appear on the screen. The shape will be known as a similar figure as it will have the same shape on the screen but a different size. Painting car, often car painters use a similar small model of their car to paint on the model before painting the original car; to be more accurate. Example: a real life solving problem QUESTION: Suppose you wanted to find the height of the tree. Unfortunately all that you have is a tape measure, and you are too short to reach the top of the tree. How will you find the height of the tree? SOLUTION: You can measure the length of the tree’s shadow. Then, measure the length of your shadow. If you know how tall you are, then you can determine how tall the tree is. If your height is 6feet and your shadows height is 2feet than you can find the height of tree. If you are 6ft and the shadow is 2ft, it means that the shadow is 3 times smaller than you. Therefore if the tree’s shadow is 10ft than the tree will be 3 times it.So, the tree will be 30ft. Thus, the example above shows that the shadow of both the tree and you is a similar figure; as only their size differs not their shape. Similar figures Are figures for which all corresponding angles are equal and all corresponding sides are proportional. Corresponding sides are two sides of two similar figures which are relatively positioned. The two triangles which are similar must be equi-angular.Also,when two figures are similar; the ratios of the lengths of their corresponding sides are equal . Example: which sides corresponds to which? Properties of two figures to be Similar 1.) Perimeters of similar triangles: Perimeters of similar triangles are in the same ratio as their corresponding sides and this ratio is called the scale factor. There are two similar triangles are . This ratio is called the scale factor. Thus, the perimeters of two similar triangles are in the ratio of their scale factor. 2.) Areas of similar triangles: The ratio of the areas of two similar triangles is equal to the ratio of the squares of the corresponding sides, i.e. the square of the scale factor. To prove that, draw perpendicular from A and P to meet seg.BC and seg.QR at D and S respectively. Since Thus the areas of two similar triangles are in the same ratio as the square of their scale factors. 3.)Corresponding angles are the same So in the figure above, ∠P =∠L, ∠Q =∠M, and ∠R =∠N. From this, it follows that the corresponding exterior angles will also be the same. 4.)Corresponding sides are all in the same proportion By definition each pair of corresponding sides are in the same proportion, or ratio. Formally, in two similar triangles PQR and LMN : So, for example, if in two similar polygons one side is twice the length of the corresponding side in the other, Then all the other sides will be twice the length of their corresponding side also. 5.)Corresponding diagonals are in the same proportion In each polygon the corresponding diagonals are in the same proportion. Their ratio is the same as the ratio of the sides. FIGURES WHICH ARE ALWAYS SIMILAR 1.)Circles are always similar because its angle is always 360◦ although its size vary from big to small. 2.)All equilateral triangles are similar as all three angles of an equilateral triangle are equal. Thus they add up to 180◦. Even if they vary in different size all the three angles will be equal. 3.) All squares are similar as all their four angles add up to 360◦,and each angle is 90◦. Thus, the four sides of a square are proportionally corresponding. All squares are similar Theorems of similar figures 1.)a segment joining the midpoints of two sides of a triangle is parallel to the third side, and its measure is one-half the measure of the third side. Example: R Question: find the measure of each angle for the figure given. The figure represents letters S and T as midpoints; and R is 25◦, STR is 115◦. a.) Q b.) RST C.) P d.)PST e.) QTS T Solution: a.)Q=115◦ {as corresponds . Therefore; Q STR } b.)RST=40◦ {180◦-(115◦+25◦)=40◦ c.)P=40◦ {PRST} d.)PST=140◦ {180◦-40◦=140◦} e.)QTS=65◦ {180◦-115◦=65◦} S Q P PPP P 2.) In similar triangles, corresponding altitudes are proportional to corresponding sides. Example: D Question: find the value of x and y. Solution: ACD EGH X A = set up a proportion 9x = 84 cross products are equal x=9 8 B = 12y = 72 y=6 H 12 22 2 9 7 Y C E F G 3.) If a segment is parallel to one side of a triangle and intersects the other sides in two points, then the tr iangle formed is similar to the original triangle. 4.) If a segment is parallel to one side of a triangle and intersects the other sides in two points, then the segment divides those two sides proportionally. Example: P Question: find PT and PR 4 T S Solution: = X 12 7 7x = 48; therefore x = 6 the answer will be, PT = 6 and PR = 12 + 6 Q R =18 5.) If three parallel lines intersects two transversals, then they divide the transversals proportionally. 6.) In similar triangles, corresponding medians are proportional to corresponding sides. Example: V Question: are the triangles UVW and KLM similar? Solution: find the ratios of corresponding sides. = = = = The sides that include V and L are proportional. V L Given from the figure the answer will be,UVW KLM 15 9 U W L 12 2 K 20 M Area of similar triangles The ratio of the areas of similar triangles is equal to the square of the ratio of corresponding sides. In the figure below, if triangles ABC and XYZ are similar then, A X = = B C Y find the area of triangle XYZ given that the area of triangle ABC is 12 . D = W Z = = K being the scale factor. Example: Question: X In triangle XYZ, Z = 60◦ and X A =50◦;and In triangle ABC,B = 70◦and C=60◦. Hence B C Y Z =5cm and =10cm. Solution: In triangle XYZ, Y = 70◦;and In triangle ABC,A = 50◦. Hence the two triangles are similar because they are equiangular ,BC and YZ correspond, therefore K= = = =4 =2 =4 Area of XYZ = 4 12 = 48 Exercise: 1.) In the following figure, the triangle ABC and EFG are similar. If the area of ABC is E the area of EFG. A , calculate 6cm 2cm M B C F G 2.) In the triangle LPR, MN is parallel to PR. The area of triangle LMN= LPR= . A.)Calculate the area of MNRP L M B.)Calculate the ratio fraction in its lowest terms. N 8cm P and the area of triangle C.)If MN= 8cm, find the length of PR. R as a Perimeters of similar triangles In similar triangles, that is corresponding angles are the same and corresponding sides are proportional then, ratio of corresponding sides = ratio of their perimeter. Triangle DEF is bigger than triangle ABC. Congruency It is when two objects are of the same shape and size. Then, this objects are known as ‘congruent’. Real life examples of congruent figures: Human’s eyes, every human’s both eyes are of the same shape and size. Congruent figures Two figures are congruent if they have the same shape and size. When two triangles are congruent, you can fit one on top of the other so that the two figures match exactly. The sides and angles that match are corresponding parts. The vertices that math are corresponding vertices. B A C CORRESPONDING VETRICES CORRESPONDING PARTS A corresponds to D B corresponds toE C corresponds toF A corresponds to D B corresponds to E C corresponds to F corresponds to corresponds to corresponds to When writing a congruement statement, list the corresponding vertices in the same order. Below are, all the ways for the triangles above. Note that all these congruency statements imply the same correspondence. ABCDEF BACEDF CABFDE ACB DFE BCA EFD CBAFED Congruent triangles are triangles whose vertices can be made to correspond in such a way that the corresponding parts of the triangles are congruent. Every triangle has six parts; three sides and three angles. If two triangles are congruent, then six pairs of corresponding parts are congruent. Also, if the vertices can be matched so that all six pairs of corresponding parts are congruent, then the triangles are congruent, Theorems of congruent figures 1.) If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle,then the two triangles are congruent . 2.) If the legs of one right triangle are congruent to the legs of another right triangle, then the two right triangles are congruent. 3.) If a leg in an acute angle of one right triangle are congruent to the corresponding parts of another right triangle,then the two right triangles are congruent. 4.) If the hypothenese and an acute angle of a rihght triangle are congruent to the hypothense and an acute angle of another right triangle, then the right triangles are congruent. 5.) The base angles of an isosceles triangle are congruent. 6.) If a triangle is equilateral , then it is equiangular. 7.) The measure of each angle of any equilateral triangle is 60◦. 8.) If two angles of a triangle are congruent, then they are the base angles of an issoscels triangle. 9.) Every equiangular triangle is equilateral. 3-D shapes The conditions that the two cubes above are similar are: Corresponding angles are congruent. Corresponding sides are in proportional to each other. The ratios of their measures are equal of the two corresponding sides. The shapes above are called ‘similar’ as their four angles are corresponding to each other. Also, their sides are proportionally corresponding.they are called similar because all four angles are equal 90◦; thus all cubes have the same angle.the main concept is that all cubes add upto 360◦. GRADE X - ASSESSMENT - SIMILARITY AND CONGRUENCY Writing skill 3 creativit y flow neatness Drawing skill 2 pictures labeling 0 1 Formatting skill 3 page paragraphs math equations 1 Polygon 2 alwayssimilar polygon s appln 2 Triangle 2 theorem s applns 3D shapes 3 perimete r vol, SA 1 1 Questions 3 3D shapes congruenc y variety 0 Example s 2 ex+ans 0 Total 20 6 Hiral, you have not taken this work seriously… most of the work is copy pasted from some other source which is not the intension of giving such assignment.. you wouldn’t have learnt anything from this work… be serious!