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1. 5-20-13 EOG Review Day 6 - Geometry OBJECTIVE: Draw, construct, and describe geometrical figures and describe the relationships between them AND Solve real-life and mathematical problems involving angle measure, area, surface area and volume. B 12.5

2. Geometry

3. FORMULAS

4. Similar Figures and indirect measurement • In order for two shapes to be considered mathematically similar, the following must be true: • same general shape • congruent corresponding angles • Corresponding side lengths are proportional or share a common scale factor.

5. Similar Figures and Indirect Measurement • For polygons to be similar, three things must be true. Use the following example to figure out the first one. Are these similar figures? How do you know? No, they are not similar because they are not the same shape.

6. Similar Figures and Indirect Measurement Use the following example to figure out the next one. They have the same general shape, but what is different about them? Be specific. Remember, you don’t know side lengths, so don’t focus too much on that aspect. Corresponding angles must be congruent (equal).

7. Similar Figures and Indirect Measurement Now, if we knew side lengths, how could you prove these are not similar mathematically. Corresponding side lengths are not proportional or do not share the same scale factor. 9 9 6 6 10 5

8. To find the lengths of all parts of similar figures when you know the lengths of some corresponding parts, write and solve proportions. When using proportions with scale drawings, use the ratio of the distance in the drawing to the corresponding actual distance. scale = scale actual actual scale factor - the ratio between any two corresponding sides of a similar figures

9. Alan goes to New Mexico from Montana. The distance between New Mexico and Montana, on a map, is 350 cm. If each 10 cm on the map scale drawing equals 20 kilometers, how far apart are New Mexico and Montana? 10 cm = 20 kilometers

10. When the scale factor is less than one, then the resulting image is decreased in size. When the scale factor is greater than one then the object is increased in size.

11. If the parallelogram below is enlarged using a scale factor of 2.2, what will be the perimeter of the new parallelogram?

12. If the parallelogram below is enlarged using a scale factor of 2.2, what will be the area of the new parallelogram?

13. DRAW AND LABEL IN YOUR STUDY GUIDE 3. HINT: When the order of the letters gives you information about corresponding sides.

14. Similar Figures and Indirect Measurement • Are these two figures mathematically similar? They have the same general shape 45° 45° Corresponding angles are congruent 36 27 90° ? 90° 18 24 Corresponding side lengths are proportional. Therefore, the figures are similar.

15. Similar Figures and Indirect Measurement • Are the following figures similar? • No, they are not because the corresponding sides are not proportional. 8 ft. 5 ft. 18 ft. 9 ft.

16. Similar Figures and Indirect Measurement • What happens if you know that rectangles ABCD and PQRS are similar. How could we find x? A B Q R P S D C 8 12 6 X

17. Similar Figures and Indirect Measurement • A Civil War monument in Charlestown, Massachusetts is 221 ft. tall. It casts a shadow 189 feet long at the same time a nearby tree casts a shadow 29 feet long. To the nearest tenth of a foot, how tall is the tree? • The tree is about 33.9 feet tall. 221 a 29 189

18. Similar Figures 1) Find the missing side of the similar triangles below. p n cm cm 12.5 cm 2. Find the length of the bridge in the drawing of the similar figures above. _____________ 5 cm 12 ft

19. Similar Figures and Indirect Measurement Scale Drawing – A drawing that shows a real object with accurate sizes except they have all been reduced or enlarged by a certain amount (called the scale). Example of a scale drawing – Maps Architectural and engineering drawings Charts and pictures in books

20. Find any missing lengths of sides, doors, and Windows using ratios of corresponding sides. 2 .17 .17 Scale = .9 1.5

21. Angle Relationships Important Vocabulary – • Supplementary Angles: • two angles whose measures add to 180° • Complementary Angles: • two angles whose measures add to 90°

22. Angle Relationships Important Vocabulary – • Adjacent Angles: • two angles that share a vertex and a side but no points in their interiors. • Vertical Angles: • angles formed by two intersecting lines, and are opposite each other. Vertical angles are congruent. • Congruent Angles: • angles that have the same measure

23. the opposite angles of intersecting lines must be equal.

24. Symbol for congruence

25. Find the value of x. x 30 x + 30 = 180 - 30 - 30 x = 150

26. Write a variable equation and solve. Find an angle whose supplement is 30 less than twice the angle. x 2x - 30 x + (2x - 30) = 180 3x - 30 = 180 +30 +30 3x = 210 70 x = 70

27. - Angles whose sum is 90 . Complementary Angles b a x+ 40 = 90 - 40 -40 x 40 x = 50

28. - the sum of the interior angles of any triangle is always 180 . 180 Rule for Triangles b 80 x 40 a c 40 + 80 + x = 180 120 + x = 180 x = 60

29. Find each angle below. y + 14 = 63 2y - 10 2y - 10 = 88 y - 20 = 29 y+14 y - 20 180 (y + 14) + (2y - 10) + (y - 20) = 180 4y - 16 = 180 +16 +16 4y = 196 y = 49

30. Angle Relationships supplementary Example 1: • Find the measure of angle 1 if the measure of angle 4 = 135o • Angles 1 and 4 are _______________ angles. • If they are supplementary angles, they must add up to 180 degrees. Angle 1 measures 45 degrees.

31. Angle Relationships supplementary Example 1: • Find the measure of angle 1 if the measure of angle 4 = 135o • Angles 1 and 4 are _______________ angles. • If they are supplementary angles, they must add up to 180 degrees. Angle 1 measures 45 degrees.

32. Angle Relationships supplementary Example 2: • Find the measure of angles 2, 3, and 4 if 1 = 43o • Angles 1 and 2 are _______________ angles. • If they are supplementary angles, they must add up to 180 degrees. Angle 2 measures 137 degrees.

33. Angle Relationships vertical Example 2 continued: • Find the measure of angles 2, 3, and 4 if the measure of angle 1 = 43o • Angles 1 and 3 are _________ angles. • If they are vertical angles, they must be congruent. Angle 3 measures 43 degrees.

34. Angle Relationships supplementary Example 2 continued: • Find the measure of angles 2, 3, and 4 if measure of angle 1 = 43o • Angles 1 and 4 are _______________ angles. • If they are supplementary angles, they must add up to 180 degrees. Angle 4 measures 137 degrees. Is there another way to find the measure of angle 4 without having to set up an equation?

35. Angle Relationships Example 3 – • Find the measure of each angle • Angle HPM – • 108 degrees • Angle JPI – • 38 degrees

36. Angle Relationships Example 3 – • Find the measure of each angle • Angle IPH – • 34 degrees • Angle KPL – • 34 degrees

37. Angles 90 = x – 20 110º 45º 68º 2y + y = 180 60º (w – 50) + (w + 50) = 180 90º 120º

38. http://www.mathwarehouse.com/geometry/triangles/triangle-inequality-theorem-rule-explained.phphttp://www.mathwarehouse.com/geometry/triangles/triangle-inequality-theorem-rule-explained.php

39. Infinite…similar figures can be drawn for any length sides, but the angles would be the same. No because the two shorter sides must add up to be longer than the longest side. No because the two shorter sides must add up to be longer than the longest side. Yes because the two shorter sides do add up to be longer than the longest side. The two shorter sides must add up to be longer than the longest side. 180º

40. IRREGULAR FIGURES To find the area or perimeter of Irregular Figures, follow these steps: 1. Redraw the shapes 2. Name the shapes 3. Write the formulas 4. Substitute 5. Solve 6. Add or Subtract as needed 7. Label units