1. Bell Work: Solve the proportion 9 = 18 3 d

2. Answer: d = 6

3. Lesson 35: Similar and Congruent Polygons

4. Two figures are similar if they have the same shape even though they may vary in size. In the illustration, triangles A, B and C are similar. Triangle D is not similar because its shape is different. A C D B

5. If figures are the same shape and size, they are not only similar, they are also congruent. All three triangles below are similar, but only triangles A and B are congruent. A B C

6. When inspecting polygons to determine whether they are similar or congruent, we compare their corresponding parts.
Corresponding Angles
Corresponding Sides
<A and <D
AB and DE
<B and <E
BC and EF
<C and <F
CA and FD __
___
___
___
___
___ A D C B E F

7. We use tick marks on corresponding side lengths that have equal length and arcs on corresponding angles that have the same measure.

8. The symbol ~ represents similarity
The symbol ~ represents similarity. Since triangle ABC is similar to GHI, we can write triangle ABC ~ triangle GHI G A C H I B

9. The symbol ≅ represents congruence
The symbol ≅ represents congruence. Since triangle ABC and triangle DEF are congruent, we can write triangle ABC ≅ triangle DEF A D B C E F

10. Notice that when we write a statement about the similarity or congruence of two polygons, we name the letters of the corresponding vertices in the same order.

11. Similar Polygons: have corresponding angles which are the same measure and corresponding sides which are proportional in length.

12. Congruent Polygons: have corresponding angles which are the same measure and corresponding sides which are the same length.

13. The following two quadrilaterals are similar
The following two quadrilaterals are similar. The corresponding angles are congruent and the corresponding sides are proportional 12
100°
80° 8
100°
80° 8 8 6 6
80°
100°
80°
100° 12 8

14. The relationship between the sides and angles of a triangle is such that the lengths of the sides determine the size and position of the angles.

15. For example, with three straws of different lengths, we can form one and only one shape of triangle. Three other straws of twice the length would form a similar triangle with angles of the same measure. Knowing that two triangle have proportional corresponding side lengths is enough to determine that they are similar.

16. Side – Side – Side Triangle Similarity:. if two triangles have
Side – Side – Side Triangle Similarity: if two triangles have proportional corresponding side lengths, then the triangles are similar.

17. Triangle ABC ~ Triangle DEF Side – Side – Side4 6 12 8 C 8 B F 16 E
Triangle ABC ~ Triangle DEF
Side – Side – Side

18. All triangles with the same set of angle measures are similar
All triangles with the same set of angle measures are similar. Thus, knowing that two triangles have congruent corresponding angles is sufficient information to conclude that the triangles are similar.

19. Angle – Angle – Angle Similarity: if the angles of one triangle are congruent to the angles of another triangle, then the triangles are similar and their corresponding side lengths are proportional.

20. Triangle ABC ~ Triangle DEF Angle – Angle – Angle
100°
100°
20°
60°
20°
60° A F C D
Triangle ABC ~ Triangle DEF
Angle – Angle – Angle

21. Example: The triangles are similar. Find x. 15 x 5 3 4 12

22. Answer: x = 9
Left Triangle
Right Triangle 3 x 4 12 5 15

23. Example: An architect makes a scale drawing of a building
Example: An architect makes a scale drawing of a building. If one inch on the drawing represents 4 feet, then what is the scale factor form the drawing to the actual building?

24. Answer: If 1 inch represents 4 feet, then 1 inch represents 48 inches
Answer: If 1 inch represents 4 feet, then 1 inch represents 48 inches. Therefore, the scale is 1:48 and the scale factor is 48.

25. Example: An architect makes a scale drawing of a building
Example: An architect makes a scale drawing of a building. What is the length and width of a room that is 5 inches by 4 inches on the drawing?

26. Answer: Each inch represents 4 feet, so 5 inches represents 5 x 4 = 20 feet, and 4 inches represents 4 x 4 = 16 feet. The room is 20 feet long and 16 feet wide.

27. HW: Lesson 35 #1-30 Due Tomorrow