In the figure show below, the measure of angle x is 150 since the sum of the remote interior angles is

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1 Exterior angles of a triangle An exterior angle of a triangle is equal to the sum of the remote interior angles - in other words, the two interior angles on the opposite side of the triangle. In the figure show below, the measure of angle x is 150 since the sum of the remote interior angles is Find the measure of angle x in the figure to the right. 2. What is the value of x in the figure below? The three exterior angles of any triangle add up to 360. In the figure below a + b + c = 360.

2 3. In the figure below angle a is 120 and angle c is 90. What is the measure of angle b? Similar triangles Similar triangles have the same shape but are different sizes. Corresponding angles are equal and corresponding sides are proportional. In the figure below, triangle ABC is similar to triangle DEF. Similar triangles may be facing different directions which might make it more difficult to see that they are the same shape, such as the figures below. These triangles are similar because they have the same angles. That means that the corresponding sides are proportional. We can use this information to find missing values. To find the missing value of s in the figure above, set up a proportion. Since side bc of the first triangle is proportional to side bc of the second triangle and side ab of the first triangle is proportional to side ab of the second triangle then =. Solve for s by cross multiplying. 4 x 6 = 3s, therefore s = 8

3 4. In the figure below, angle ABC is a right angle and DF is parallel to AC. If AB is 12 inches long, BC is 9 inches long. AD is 6 inches long. What is the area, in square inches, of triangle DBF? 5. The Similar Triangles movie theater company use the Illuminator 100 light bulb in their projectors, but the new manager wants to switch to the Megabulb 100X, a more powerful light bulb that projects movies onto larger screens farther away. The Megabulb 100X projects movies onto screens 108 feet wide and 180 feet from the projector. The original Illuminator 100 projects movies only 81 feet wide. How much farther from the projector, in feet, is the screen for the Megabulb 100X than the screen for the Illuminator 100? 6. A person 5 feet tall casts a shadow 8 feet long. At the same time, a nearby tree casts a shadow 24 feet long. What is the height of the tree?

4 7. In the figure below, angle M is congruent to angle K, angle N and angle L are right angles. Solve for u. Special right triangles. When using the Pythagorean theorem we often get answers with square roots or long decimals. There are a few special right triangles that give integer answers. The most often seen is the right triangle. If one leg is 3 and the other is 4 then the hypotenuse will be 5. When you are able to recognize this kind of triangle, you don't even have to use the Pythagorean theorem, which makes things simpler and easier. Also, try to recognize multiples of the right triangle. For example, all the triangles below are variations of the right triangle. Four times a right triangle makes a triangle. Ten times a makes a right triangle. One half times a right triangle makes a triangle.

5 8. In the right triangle below, one leg is 30 and the hypotenuse is 50. Find b. 9. As shown in the figure below, triangle XYZ is a right triangle. Quadrilaterals ABYX, CDZY, and EFXZ are squares. If the measure of side x = 6 inches, and the measure of hypotenuse z = 10 inches, what is the area of quadrilateral EFXZ? Surface area of a rectangular solid The surface of a rectangular solid - a box, or a cube, for example - is simply adding the area of each of the faces. If the length is l, the width is w, and the height is h, the formula is: Surface area = 2lw + 2wh + 2lh 10. Find the surface area of the rectangular solid below.

6 11. What is the surface area of a rectangular solid with a length of 8 inches, width of 6 inches, and height of 3 inches? 12. The surface area of the rectangular prism below is 300 square inches. The width (side b) is 6 inches and the height (h) is 5 inches. What is the length of the rectangular prism? Triangle inequality theorem: The sum of the lengths of any two sides of a triangle must be greater than the third side. If these inequalities are NOT true, you do not have a triangle. In the example below c < a + b, b < a + c, a < b + c. That means that if a is 4 and b is 3 then side c has to be less than 7. If you think about it, it makes sense because a and b would have to be laid out flat on a straight line to make a 7 (3 + 4 = 7) and it wouldn't be a triangle any more, it would just be a straight line. And the third side couldn't be any longer than that, 8 for example, because couldn't make an 8 even if they were laid out flat.

7 13. One side of a triangle is 3 inches long and another side is 6 inches long. Which of the following could NOT be the length of the other side? A. 10 inches B. 4 inches C. 5 inches D. 6 inches 14. In triangle ABC, side AB is 11 inches long and side BC is 50 inches long. Which of the following CANNOT be the length, in inches of side AC? A. 20 B. 60 C. 40 D If the lengths of all three sides of a triangle are integers, and one side is 9 inches long, what is the smallest possible perimeter of the triangle? Angles and sides of triangles. There is a relationship between an angle and the side opposite that angle. The larger the angle, the larger the side will be. In the triangle below, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. In this figure, angle B is the smallest angle, therefore side b is the shortest side. Angle A is the next smallest angle, so side a is next in length. Angle C is the largest angle, therefore side c is the longest side.

8 16. In the triangle below, list the sides in order from least to greatest. Maximum area of a rectangle with a fixed perimeter. It's always a square. For example, if we were asked to find the largest area of a rectangle with a perimeter of 28 feet, we could use this information to find the answer. Since the largest area of a rectangle with a fixed perimeter is a square, we know that we have a square with a perimeter of 28 feet. That means that the square has sides that are 7 feet long. The area would be 7 x 7 = 49 square feet. 17. What is the maximum possible area, in square inches, of a rectangle with a perimeter of 8 inches? Simplifying square roots Sometimes we need to leave the answer with a radical or square root sign. When we have to do it that way, the calculator isn't very helpful. We need to know how to simplify a square root. This is possible because we can re-write the number as a product of factors and take the square root of the parts that are perfect squares and leave the rest in a square root sign. For example, if I need to take the square root of 12, and I use a calculator I get an answer that looks something like this If I'm asked to give the answer in terms of square roots, then the calculator answer won't help me. I need to try a different approach. I know that 12 isn't a perfect square so it won't have a simple integer answer, so I look for factors of 12 that are perfect squares. 4 is a factor of 12 and 4 is a perfect square so I will re-write the square root like this = = = 2. I can't break it down any farther, so I write it like this 2 which means 2 times the square root of 3. If you multiply that on your calculator you will get They are the same number, they just look different. One advantage to writing answers as square roots is that they are exact, whereas long decimal answers are usually rounded and approximated. 18. Simplify 19. Simplify 20. Simplify 21. Simplify

9 22. In the triangle below angle c is a right angle, and sides AC is equal to side BC. If side AC = 2, what is the length of side AB? Give the answer in terms of a simplified square root. 23. How many units long is one of the sides of a square that has a diagonal 20 units in length? A. 10 B. 10 C. 15 D. 20 E. 15 Volume of a cylinder V=π h (in other words, find the area of the circle and multiply it by the height) 24. The radius of the cylinder below is 3 inches and the height is 4 inches. What is the volume of the cylinder? 25. The volume of the right cylinder below is 100, if the height is 4 what is the radius?

10 Questions that review parallel lines cut by a transversal 26. In the figure below, lines l and m are parallel, which of the following choices list a pair of angles that must be congruent? A. angle 1 and angle 2 B. angle 1 and angle 3 C. angle 2 and angle 3 D. angle 2 and angle 5 E. angle 3 and angle 5 Questions that review triangles (sum of interior angles = 180 ) 27. In triangle ABC the measure of angle A is exactly 37, and the measure of angle B is less than or equal to 63. Which of the following phrases best describes the measure of angle C? A. Exactly 120 B. Exactly 100 C. Exactly 80 D. Greater than or equal to 80 E. Less than or equal to 80 Questions that review Pythagorean theorem 28. What is the height of the trapezoid below?

11 Questions that review area/circumference of a circle 29. A circular pool is being build on a fenced rectangular lot 50 feet wide and 75 feet long. If a border of 10 feet is needed between the outside edge of the pool and the fence, what is the largest diameter of a pool that can be built? Questions that review area of a trapezoid. 30. The area of a trapezoid is found by multiplying the height by the average of the bases: A = h( ). An isosceles trapezoid has bases of length 4 inches and 8 inches. The area of the trapezoid is 30 square inches. What is the height of the trapezoid, in inches?