Yoga okta nugraha Adam mulya giffari Rifky hadiansyah congruence IX H Yoga okta nugraha Adam mulya giffari Rifky hadiansyah

1.1 conditions for similarity of two plane figures 1.1.1 condition for congruence of two figures Two plane figures which area perfectly coincident are called two congruent figures

1.1.2 scaled pictures and models A scaled picture or model has the same shape as the real object. The size of a scaled picture or model is enlarged or reduced from the original by the same ratio The corresponding part size are the length of the original and the length of model the width of the original and the width of the model length of the model = width of the model Length of the original width of the original Height of the model Height of the original

1.1.3 condition for similarity of two figures We have seen that two figures a scale model and the original have the same shape but different in size and measurements, such figures are called similar figures Length EF = 3 x length AB or EF : AB = 3:1 Length EH= 3 x length AD or EH : AD = 3:1 Thus, the ratios of the corresponding side are equal: EF : AB = EH : AD = 3:1 The corresponding angles have the same measure, i.e.: <A = <E = 90⁰ <B = <F = 90⁰ <C = <G = 90⁰ <D = <H = 90⁰ 1. all the corresponding sides are proportional 2. all the corresponding angles are equal in measure

1.2 congruent triangles 1.2.1 conditions for congruence of two triangles The image formed by an object placed in front of a flat mirror is congruent to the object , such is the case with a triangle and its image ∆A’B’C is a reflection of ∆ABC against the line XY. ∆ABC and ∆A’B’C are congruent Next ∆A’B’C is translated to the right such that it coincides with ∆DEF . ∆A’B’C and ∆Def are congruent Sand then ∆ABC and ∆Def are congruent <A = <E AB=ED <B = <D BC=DF <C = <F AC=EF From above description we can draw the following conclusion two triangles are congruent ifa: 1. all the corresponding sides are proportional 2. all the corresponding angles are equal in measure

1.2.2 the properties of two congruent triangles The determine the congruence of the two triangles, we can use the parts in a triangle, i.e. the length of its sides and the measure of its internal angles A. SSS property (side,side,side) The three corresponding side are equal B. AAA property (angle,angle,angle) The three corresponding angles are equal SAS property (side,angle,side) the two sides and the included angles are equal If the three sides of one triangle are equal to the three sides of the other triangle, then the 2 triangles are congruent If the three angles of 1 triangle are equal to the three angles of the other triangle, then two triangles aren’t necessarily congruent

1.2.3 finding the sides and angles in congruent triangles To find the side angles in congruent triangles first identify those angles which are equal in measure and those sides which are equal in length if the congruence of the triangles is not known, we should examine for their congruence first

1.3.1 Conditions for Similarity of Two Triangles Similarity of Triangles Based of Two Triangles If the corresponding angles of two triangles are equal, then the corrseponding sides are proportional If the corresponding angles of two triangles are equal, then the triangles must be similar If two triangles have two pairs of equal angles then the other pair are equal. Equal angles are opposite to the corresponding sides.

1.3.1 Conditions for Similarity of Two Triangles Similarity of Triangles Based on Corresponding Sides If all the corresponding sides of two triangles are proportional, then all the corresponding angles are equal Thus, if all the corresponding sides of two triangles are proportional, then the two triangles must be similar To identify corresponding sides when the lengths of all sides are known, pair the longest line with the longest line, the medium length line with the medium length line, and the shortest line with the shortest line

1.3.1 Conditions for Similarity of Two Triangles Similarity of Triangles Based on Two Sides and the Included Angles If two Triangles have a pair of equal angles and the sides which include the equal angles of both triangles are proportional, the the two triangles are

1.3.2 Finding the Lengths of the Sides in Similar Triangles We have disscused that if two triangles have three pairs of equal angles then the two triangles are similar so both of them have corresponding sides which are in proportion. Thus, if two triangles have three pairs of equal angles, we can find the lengths of the sides using proportions of corresponding sides

1.3.3 Telling the Differences between Similar Triangles and Congruent Triangles Between two triangles there must be one of the following possible relationship. Being congruent Being similar Being neither congruent nor similar We will only discuss the differences between two congruent triangles and two similar triangles. Look at the following table. Two Congruent Triangles Two Similar Triangles The corresponding sides are equal in length The triangles are of the same size The corresponding sides are in proportion The triangles are different in size

1.3.3 Telling the Differences between Similar Triangles and Congruent Triangles Two congruent triangles have corrseponding angles equal in measure and corresponding sides equal in length. Two similar triangles have corrseponding angles equal in measure but corresponding sides unequal in legth (only proportional).

Applying The Notion of Similarity Drawing sketch can help us in solving different word problems involving the idea of similarity

The shadows a pole and a tree make are 5 m and 20 m long , respectively. If the height of the pole is 4m, find the height of the tree ??? 20 cm 4 cm

answer 4/t = 5/20 5 x t = 4 x 20 5t = 80 t = 80/5 t = 16

A picture I put on a 40 x 55 cm cardboard A picture I put on a 40 x 55 cm cardboard. On the right and left side of the picture is a space 4 cm wide each. If the picture and the cardboard are similar, what is the width of the space below the picture 40 cm 5 cm 55 cm 4 cm 4 cm 7 cm

answer Height of the cardboard = 55 cm Width of the cardboard = 40 cm Width of picture = 40 cm – ( 4 + 4 ) cm = 32 cm Height of picture = x cm x/50 = 32/40 40x = 55 x 32 x = 55 x 32 4 x = 55 11 x 4 40 5 5 1 v x = 44 The width of the space below the pictures is : l = [55 – ( 5 + 44 )] cm = ( 55 – 49 ) cm = 6 cm