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1. 1. SIMILARITY AND CONGRUENCE
2. 2. Insisivi Eka S Mutiara Aura K Sri Ayu Pujiati
3. 3. SIMILARITY CONGRUENCE
4. 4. SIMILARITY ツ
5. 5. SIMILAR FIGURES SIMILAR FIGURES
6. 6. SIMILAR FIGURES Similar figures are two figures that are the same shape and whose sides are proportional
7. 7. ~ This is the symbol that means “similar.” These figures are the same shape but different sizes.
8. 8. Example : A 25 cm x 15 cm rectangle and a 10 cm x 6 cm rectangle are given. Are the rectangles similar? 15 cm 6cm 10 cm 25 cm
9. 9. ANSWER (≧∇≦)/ The two rectangles have equal corresponding angles each of which is a right angle. Ratio of the length = 25 cm : 10 cm = 5 : 2 Ratio of the width = 15 cm : 6 cm = 5 : 2 Thus, Two rectangles are similar . Because the corresponding angles are equal and the corresponding sides are proportional.
10. 10. SIMILAR TRIANGLES Similar triangles are two triangles that have the same shape but not
11. 11. TWO TRIANGLE ARE SIMILAR IF : The Corresponding sides are in proportion Corresponding pairs of sides are in proportion
12. 12. SIMILAR TRIANGLE Angle A ~ Angle D Angle B ~ Angle E Angle C ~ Angle F AB = BC = AC DE EF DF
13. 13. Proving Similarity (AAA) - Angle, Angle, Angle If three angles of one triangle are congruent, respectively, to three angles of a second triangle, then the triangles are similar. AAA AA
14. 14. (`▽´)-σ Example I : In ABC and DEF, AB = 9 cm, BC = 6 cm , CA = 12 cm, DE = 15 cm, EF = 10 cm, FD = 20 cm. Explain why the two triangles are considered similar. Name all the pairs of equal angles ! C 12 A 6 B F 20 D 10 15 E ANSWER
15. 15. ᾈňšὠὲ ŕ (•"̮•) In △ABC : AB = 9 cm BC = 6 cm CA = 12 cm In △ DEF : DE = 15 cm EF = 10 cm FD = 20 cm AB : DE = 9 cm : 15 cm =3:5 BC : EF = 6 cm : 10 cm =3:5 CA : FD = 12 cm : 20 cm =3:5 Thus, △ABC and △FED are similar since all the corresponding sides are proportional • The Pairs of equal angles are : A=D,B=E, C=F
16. 16.
17. 17. CONGRUENT FIGURES CONGRUENCE CONGRUENCE CONGRUENCE CONGRUENCE CONGRUENT TRIANGLES
18. 18. CONGRUENT FIGURES Two figures are congruent if they have same size and same shape.
19. 19. The Properties of Two Congruent Figures Has same shape and same size All corresponding pairs of angles are congruent Corresponding pairs of sides are congruent.
20. 20. D C H G E A B ‘ F
21. 21. Since parallelogram ABCD and EFGH are congruent : EH = AB, thus AB = 7 cm AD = GH , thus AD = 12 cm
22. 22. When we talk about congruent triangles, we mean everything about them Is congruent. All 3 pairs of corresponding angles are equal…. And all 3 pairs of corresponding sides are eq
23. 23. Proving Triangles Congruent • To prove that two triangles are congruent it is only necessary to B show that some corresponding parts are congruent. • For example, suppose that in AB DE and in that and AC DF and A D C A E • Then intuition tells us that the remaining sides must be congruent, and… • The triangles themselves must be congruent. F D
24. 24. The properties of congruent triangle
25. 25. If we can show all 3 pairs of corr. sides are congruent, the triangles have to be congruent.
26. 26. Show 2 pairs of sides and the included angles are congruent and the triangles have to be congruent Included angle Non-included angles
27. 27. AAA PROPERTY (ANGLE,ANGLE, ANGLE) THIS MEANS WE ARE GIVEN ALL THREE ANGLES OF A TRIANGLE, BUT NO SIDES.
28. 28. ASA PROPERTY (ANGLE,SIDE, ANGLE) C A F D IN TWO TRIANGLES, IF ONE PAIR OF ANGLES ARE CONGRUENT, ANOTHER PAIR OF ANGLES ARE CONGRUENT, AND THE PAIR OF SIDES IN BETWEEN THE PAIRS OF CONGRUENT ANGLES ARE CONGRUENT, THEN THE TRIANGLES ARE CONGRUENT. B FOR EXAMPLE, IN THE FIGURE, IF THE CORRESPONDING PARTS ARE CONGRUENT AS MARKED, THEN WE CITE “ANGLE-SIDE-ANGLE (ASA)” AS THE E REASON THE TRIANGLES ARE CONGRUENT.
29. 29. AAS PROPERTY (ANGLE,ANGLE, SIDE) C B A F D E In two triangles, if one pair of angles are congruent, another pair of angles are congruent, and a pair of sides not between the two angles are congruent, then the triangles are congruent. For example, in the figure, if the corresponding parts are congruent as marked, then
30. 30. THE END