1-1. Calculate the values of the expressions below. Show all steps in your process.

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1 1-1. Calculate the values of the expressions below. Show all steps in your process. a. 2 (3(5 + 2) 1) b. 6 2(4 + 5) + 6 c d ( ) 1-2. Simplify the expressions below as much as possible. Show all work. a. 2a + 4(7 + 5a) b. 4(3x + 2) 5(7x + 5) c. x(x + 5) d. 2x + x(x + 6) 1-3. Examine the graph below. Then, in a sentence or two, suggest reasons why the graph rises at 11:00 a.m. and then drops at 1:15 p.m Describe how to find the area and perimeter of a shape. Then examine the rectangle above. If the perimeter of this shape is 120 cm, which equation below represents this fact? Once you have selected the appropriate equation, solve for x. a. 2x x 1 = 120 b. 4(6x 1) = 120 c. 2(6x 1) + 2(2x + 5) = 120 d. (2x + 5)(6x 1) = Delilah drew 3 points on her paper. When she connects these points, must they form a triangle? Why or why not? Draw an example on your paper to support your reasoning.

2 1-6. Rebecca placed a transparent grid of square units over each of the shapes she was measuring below. Using her grid, determine the area of each shape. Explain the process you used One goal of this course will be to review and enhance your algebra skills, solve for x in each equation below, show all steps leading to your solution, and check your answer. a. 34x 18 = 10x 9 b. 4x 5 = 4x + 10 c. 3(x 5) + 2(3x + 1) = 45 d. 2(x + 4) + 6 = The day before Gerardo returned from a two-week trip, he wondered if he left his plants inside his apartment or outside on his deck. He knows these facts: If his plants are indoors, he must water them at least once a week or they will die. If he leaves his plants outdoors and it rains, then he does not have to water them. Otherwise, he must water them at least once a week or they will die. It has not rained in his town for 2 weeks. When Gerardo returns, will his plants be dead? Explain your reasoning in a sentence For each of the equations below, solve for y in terms of x. a. 2x 3y = 12 b. 5x + 2y = Draw four different rectangles that each have an area of 24 square units (use whole numbers) Then find the perimeter of each one. Which rectangle has the greatest perimeter?

3 1-11. The distance along a straight road is measured as shown in the diagram below. If the distance between towns A and C is 67 miles, find the distance between towns A and B Examine the rectangle below. a. What is the perimeter in terms of x? In other words, find the perimeter. b. If the perimeter is 78 cm, find the dimensions of the rectangle. Show all your work. c. Verify that the area of this rectangle is 360 sq. cm. Explain how you know this Rosalinda examined the angles below and wrote the equation below (2x + 1º) + (x 10º) = 90º Does her equation make sense? If so, explain why her equation must be true. If it is not correct, determine what is incorrect and write the equation. a. If you have not already done so, solve her equation, clearly showing all your steps. What are the measures of the two angles? b. Verify that your answer is correct For each equation below, solve for x. Show all work. 1. 5x 2x + x = x 2 x = 7 x 3. 3(x l) = 2x 3 + 3x 4. 3(2 x) = 5(2x 7) + 2

4 1-15. Angela had a rectangular piece of paper and then cut a rectangle out of a corner as shown below. Find the area and perimeter of the resulting shape. Show all your work The perimeter of the triangle below is 52 units. Write and solve an equation based on the information in the diagram. Use your solution for x to find the measures of each side of the triangle. Be sure to confirm that your answer is correct. Show all your work Bertie placed a transparent grid made up of unit squares over each of the shapes she was measuring below. Using her grid, approximate the area of each region For each equation below, solve for the given variable. Show the steps leading to your solution and check your answer. a. 75 = 14y + 5 b. 7r + 13 = 71 c. 3a + 11 = 7a 13 d. 2m + m 8 = 7

5 1-19. Examine the diagrams below. What is the geometric relationship between the labeled angles? What is the relationship of their measures? Then, use the relationship to write an equation and solve for x In the problems below, write and solve an equation for the variable, then use that variable value to answer the question. Show all your work. a. Find m MNP. b. Find m FGH. c. Find m DBC. d. Find m LPQ and m LPN Simplify the following expressions. a (2x 4) 4x b. (x 3)(3x + 4) c. 5x(2x + 7) + x (3x 5)

6 1-22. Multiply the expressions below. Then simplify the result, if possible. a. (x + 2)(x + 3) b. (3x + 5)(x 2) c. (2x + 1)(5x 4) Jacob wrote the conjecture: Vertical angles have equal measure. (Remember that a conjecture is an educated guess that has not yet been proven.) Do you think Jacob s vertical angle conjecture holds for any pair of vertical angles? Jacob s explanation included the diagram showing intersecting lines. He then wrote that a + b = 180º and a + c = 180º. Are these statements true? Why? a. How can you use Jacob s statements in part (b) to prove that vertical angles always have equal measure? Examine the diagrams below. For each pair of angles marked on the diagram, quickly decide what relationship their measures have. Your responses should be limited to one of three relationships: same (equal measures), complementary (have a sum of 90 ), and supplementary (have a sum of 180 ). c. d.

7 1-25. Marcos was walking home after school thinking about special angle relationships when he happened to notice a pattern of parallelogram tiles on the wall of a building. Marcos saw lots of special angle relationships in this pattern, so he decided to copy the pattern into his notebook. The beginning of Marcos s diagram is shown below. This type of pattern is sometimes called a tiling. In this tiling, a parallelogram is copied and translated to fill an entire page without gaps or overlaps. a. Since each parallelogram is a translation of another, what can be stated about the angles in the rest of Marcos tiling? Copy the tiling diagram. Color all angles that must be equal the same color. b. Consider the angles inside a single parallelogram. Which angles must have equal measure? c. What about the relationship between lines? Can you identify any lines that must be parallel? Mark all of the lines on your diagram with the same number of arrows to show which lines are parallel For each diagram below, solve for x. Explain what relationship(s) used for each problem.

8 1-27. Julia wants to learn more about the angles in Marcos s diagram and has decided to focus on just a part of his tiling. An enlarged view of that section is shown in the image below, with some points and angles labeled. A line that crosses two or more other lines is called a transversal. In Julia s diagram, which line is the transversal? Which lines are parallel? a. Name all the pairs of corresponding angles you can find in Julia s diagram. b. Name all the pairs of alternate interior angles you can find in Julia s diagram. c. Name all the pairs of same side interior angles you can find in Julia s diagram. Suppose b = 60º and x = 60º Use what you know about vertical angles and supplementary angles relationships to find the measures of all the other angles in Julia s diagram Identify the geometric angle relationship(s) in each diagram. Use what you know about those relationships to write an equation and solve for x. Show all your work.

9 1-29. Frank wonders whether corresponding angles always have equal measure. For parts (a) through (d) below, decide if the corresponding angles have the same measure. Then determine if you have enough information to find the measures of x and y. If you do, find the angle measures and state the relationship. a. b. c. d. e. Answer Frank s question: Do corresponding angles always have equal measure? If not, when are their measures equal? Copy each diagram below and find the value of x, if possible. If it is not possible, explain how you know. State the relationships you use. Justify every measurement you find. c.

10 1-31. Use your knowledge of angle relationships to solve for x in the diagrams below. Justify your solutions by naming the geometric relationship Copy the diagrams on your paper and find all missing angles in the diagrams below Copy the diagram below. Then use the information provided in the diagram to find the measures of angles a, b, c, and d. For each angle, name the geometric relationship to justify your conclusion.

11 1-34. Marcos decided to change his tiling problem by drawing diagonals in each of the parallelograms. Copy his pattern, shown below, on your paper. a. Use a colored pencil to shade one of the triangle s angles. Then use the same color to shade every angle on the resource page that is equal to the shaded angle. b. Repeat this process for the other two angles of the triangle, using a different color for each angle in the triangle. When you are done, every angle in your tiling should be shaded with one of the three colors. c. Now examine your colored tiling. What relationship can you find between the three different-colored angles? You may want to focus on the angles that form a straight angle. What does this tell you about the angles in a triangle? d. How can you convince yourself that your conjecture is true for all triangles? That is, given parallel lines (since the tiling was generated by translating parallelograms), why does a = d and c = e in the diagram at below? Use your theorem from problem 1-34 about the angles in a triangle to find x in each diagram below. Show all work.

12 1-36. Examine the triangle below. Show all your work. a. If m D = 48 and m F = 117, then what is m E? b. Solve for x if m D = 4x + 2, m F = 7x 8, and m E = 4x + 6. Then find m D. c. If m D = m F = m E, what type of triangle is ΔFED? How much can you figure out about the figure at right using your knowledge of angle relationships? Find the measures of all the labeled angles in the diagram below. Justify your solutions with the name of the angle relationship you used. Carefully record your work as you go Solve for x in each diagram below. c. d.

13 1-39. Use the angle relationships in each of the diagrams below to solve for the given variables. Show all work. c. d The exterior angles of a quadrilateral are labeled a, b, c, and d in the diagram below. Find the measures of a, b, c, and d and then find the sum of the exterior angles Answer the questions below. Show all your work. a. If you know the number of sides of a regular polygon, how can you find the measure of an interior angle directly? Find the measurements of an interior angle of a 15-gon. b. If you know the number of sides of a regular polygon, how can you find the measure of an exterior angle directly? Find the measurements of an exterior angle of a 10-gon. c. What if you know that the measure of an interior angle of a regular polygon is 162? How many sides must the polygon have? Show all work. d. If the measure of an exterior angle of a regular polygon is 15, how many sides does it have? What is the measure of an interior angle? Show how you know Solve for x. Explain what relationship(s) you used for each problem.

14 1-43. Use your knowledge of polygons to answer the questions below, if possible. a. How many sides does a polygon have if the sum of the measures of the interior angles is 1980? b. If the exterior angle of a regular polygon is 90, how many sides does it have? What is another name for this shape? c. Each interior angle of a regular pentagon has measure 2x + 4. What is x? Explain how you found your answer. d. The measures of four of the exterior angles of a pentagon are 57, 74, 56, and 66. What is the measure of the remaining angle? e. Find the sum of the interior angles of an 11-gon. Does it matter if it is regular or not? Solve for x in each of the diagrams below. c What relationships do you see between the angles in the diagram? What relationships can you find between the lines and/or line segments? List as many relationships as you can based on the information in the diagram.

15 1-46. What can the Triangle Angle Sum Theorem help you learn about special triangles? a. Find the measure of each angle in an equilateral triangle. Justify your conclusion. b. Consider the isosceles right triangle (also sometimes referred to as a half-square ) below. Find the measures of all the angles in a half-square Find the measures of x, y, and z below. Justify each conclusion with the name of a geometric relationship Use the relationships in the diagrams below to solve for x, if possible. If it is not possible, state how you know. Justify your solution by stating which geometric relationships you use. c.