Answer each of the following problems. Make sure to show your work. Points D, E, and F are collinear because they lie on the same line in the plane.
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1 Answer each of the following problems. Make sure to show your work. Notation 1. Given the plane DGF in the diagram, which points are collinear? Points D, E, and F are collinear because they lie on the same line in the plane. 2. Which point is coplanar with A, B, and C in the diagram below? Point D lies in the same plane as points A, B, and C and so is coplanar with them. 3. In the diagram below, which two rays are opposite one another? Points M, X, and N are collinear. Point X lies in between M and N so the rays XM and XN are opposites. Also, the points P, X, and Q are collinear. Point X lies in between P and Q, so rays XP and XQ are opposites. 4. Why is the straightedge of a ruler not the same as a line? In geometry, a line has an infinite length while a ruler has a finite length. Patterns and Conjectures 5. Predict the next number in the sequence: 1, 4, 16, 64 The pattern is that each number is multiplied by four. So 1 x 4 = 4; 4 x 4 = 16; 16 x 4 = 64; 64 x 4 = 256.
2 6. Make a conjecture about the sum of primes based on the following pattern: 4 = = = = = and so on. Based on the pattern, the sum of the prime numbers (2, 3, 5, 7 ) is equal to an even number (4, 6, 8, 10, 12). 7. The next number in the sequence -5, -2, 4, 13 is 25. Is this conjecture true or false? The conjecture is true because for this pattern, you add multiples of 3 to get to the next number. In other words, = -2. Then, = 4. Then, = 13. Then, = What role do conjectures and counterexamples play in the process of finding a pattern? First, you look for a pattern, and you make a conjecture about the rules for the pattern. Finally, you verify that the conjecture is true. If it is not true, you show a counterexample to demonstrate this fact. Definitions and Postulates 9. If EG = 17 and FG = 12, use the segment addition postulate to find the length of segment EF. EG = EF + FG EF = EG FG EF = = 5
3 10. What is a postulate? A postulate is a statement that is somewhat obvious in nature but very useful in geometry because it can be used to help prove theorems. 11. What types of statements can be used as reasons in proofs? Definitions, axioms, postulates, and given statements are all acceptable reasons for proofs. 12. Using the segment addition postulate, find the value of x if AB = 3x + 8 BC = 4x + 6 AC = 28 Compute using betweenness of points. AB + BC = AC 3x x + 6 = 28, now use algebra to find x = 2.
4 Basic Geometric Shapes 13. Name the types of quadrilaterals and polygons in the picture. The 1st shape is a rectangle (opposite sides equal, right angles); the 2nd shape is a square (all sides/angles equal); the 3rd shape is an octagon (8 sides); the 4th shape is a square (all sides/angles equal); and the 5th shape is a rectangle (opposite sides equal, right angles). 14. What is another name for an equilateral quadrilateral? A 4-sided shape that has equal sides is also called a rhombus. 15. The planet Mercury has a diameter of 4,878 km, whereas the Moon has a radius of km. Which object is larger and why? The planet Mercury has a diameter of 4,878 km which is double the radius. So the radius is 4,878/2 = 2,439 km. The radius of the planet Mercury is slightly larger than that of the Moon. 16. When is a rhombus a rectangle? When is a rectangle not a rhombus? A rhombus is a 4-sided shape with all equal sides. The rhombus could have right angles and with the opposite sides being equal, it would be a rectangle. On the other hand, a rectangle only has equal opposite sides, which is not all of the sides like a rhombus. A rectangle also needs to have all right angles, which a rhombus does not.
5 Congruent Objects and Shapes 17. How many pairs of angles are congruent in the figures below? In the diagram, there is a single angle marking for 1 pair of angles (V and M) and a double angle marking for angles(x and L). Hence, 2 pairs of angles are marked congruent. The third angle pair would also congurent, beacuse the triangles are congruent by ASA. 18. What is the geometric definition for similar objects? Similar objects have congruent angles but proportional sides. 19. If two polygons are similar, must they also be congruent? Explain your answer. According the geometry definition of similarity, similar polygons have congruent angles, but sides that are proportional are not necessarily congruent. So similar polygons would not necessarily be congruent unless the sides of the polygon are proportional by a factor of Are the sides and angles of a rectangle congruent to each other? Explain. By definition, a rectangle has right angles. Therefore, the opposite sides of a rectangle must be congruent to each other while all of the angles are congruent to one another.
6 Perimeter, Circumference and Area 21. This pizza s border is stuffed with cheese. How many inches around is the border if the diameter of the pizza is 8 inches? The circumference of the pizza is c = πd which is (3.14)(8) = What is the area of the grey matting in this 16 by 20 picture frame if the picture to be framed is 8 x 10? A 16x20 inch picture frame has an area of (16)(20)=320 in 2. An 8x10 inch picture to be framed has (8)(10)=80 in 2. The remaining area is = 240 in If the carpet in this den is 4 by 6, what is its perimeter? The carpet s perimeter is 2L + 2W = 2(4)+2(6)= = 20.
7 24. How is the area of a rectangle related to the area of a triangle? A rectangle can be divided into two right triangles. So the area of a rectangle is (width x length) whereas for a triangle the width becomes the base and the length becomes the height. This derives to the area formula for a triangle which is (½ base height). Introduction to Transformations 25. How was the purple triangle transformed into the orange triangle? The purple triangle has been shifted 1 unit right and 5 units up.
8 26. What type of transformation is illustrated in the picture below? A rotation will produce this image. 27. What type of transformation does the image below depict? Rotation 28. How is a reflection different than a rotation? A reflection produces a mirror image of an object along a line of reflection whereas a rotation turns the objects about a fixed point. Geometric Constructions 29. A divides a segment or angle in half. bisector
9 30. A line that intersects a segment at right angles while dividing the segment in half is called a. perpendicular bisector 31. How does a geometric construction differ from a drawing? A geometric construction is a set of steps used to create geometric figures using just a straightedge and a compass without using formal measurements; whereas a drawing can use any tool. 32. Is it possible to have a bisector that is not perpendicular to the segment that it bisects? Explain. A bisector cuts a line in half by a point. Therefore, the bisecting line or segment itself can be slanted in any direction. Congruent Constructions 33. The picture represents a final step used in... The picture illustrates the final step in duplicating a line segment. 34. lines intersect at a right angle. Perpendicular 35. Explain how protractors are used in geometric constructions. You do not use a protractor to make a construction. You use a compass and a straightedge, but you can use a protractor to check your work when you are finished.
10 36. What is a segment bisector? Explain. A bisector cuts a line in half by a point. The point can be just a point or from a line, segment, or ray. Reasoning through Logic Puzzles 37. Which symbol represents "or" in a truth table? V is the symbol that represents or in a truth table. 38. If the tire is flat, then I will have to change it. Using the statement above, first state your variables in terms of p and q, identifying which variable represents which statement. Then write this statement in symbolic form. p: The tire is flat. q: I will have to change it. p -> q 39. If the puppy is tired, then he will sleep. Using the statement above, first state your variables in terms of p and q and which variable represents which statement. Then write this statement in symbolic form. p: The puppy is tired. q: The puppy will sleep. p -> q 40. Given: p: x is prime. q: x is even. What is the truth value of p^q when x is replaced by 2? If both p and q are true, the statement p^q is true. If both p and q are false, the statement p^q is still true. All other options yield a false result.
Student Name: Tools of Geometry Module Review. Answer each of the following problems. Make sure to show your work. Notation
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