Adjacent angles are supplementary. Perpendicular lines form all right angles. You named angles and determined their congruence or incongruence when two or more lines crossed. In this chapter, you will actually measure angles using an instru- ment called the protractor. Protractors have two scales—choose the scale that starts with 0 on the side you have chosen. Where the second arm of your angle crosses the scale on the protractor is your measurement. Position the protractor as if you were measuring an angle. Choose your scale and make a mark on the page at the desired measurement. Note: Because adjacent angles share a single vertex point, adjacent angles can be added together to make larger angles.
This technique will be partic- ularly useful when working with complementary and supplementary angles in Chapter 5. Set 15 Using the diagram below, measure each angle. Complete the figure with question Set 17 Choose the best answer. Set 18 A bisector is any ray or line segment that divides an angle or another line segment into two congruent and equal parts. What is the measure between Avenue Y and Avenue Z? What is the special name for this angle?
A new courthouse opened on Avenue Y. An alley connects the courthouse to Avenue C perpendicularly. However, it cannot be determined that they do not share any interior points, that they form a straight line, that they form a right angle, or that they are the same shape and size. The answer must be choice e. When angles are added together to make larger angles, the vertex always remains the same. Choice b does not name the vertex at all, so it is also incorrect. Choice e is incorrect because we are given that the angles are adjacent; we know they share side XZ; and we know they do not share sides XR and XA.
B Ave. Z Ave. Y CH alley Ave. A T Ave. Bisect means cuts in half or divides in half. Add the alley to your drawing. Good job! Excellent work! You have mastered the use of protractors. You can now move into an entire chapter dedicated to comple- ments and supplements. Perhaps the three most useful angle pairs to know in geometry are complementary, supplementary, and vertical angle pairs. Set 19 Choose the best answer for questions 95 through 99 based on the fig- ure below. Set 20 Choose the best answer. Set 21 Fill in the blanks based on your knowledge of angles and the figure below.
Set 22 State the relationship or sum of the angles given based on the figure below. Set 20 Choice b assumes both angles are also congruent; however, that information is not given. Choices c and d are incorrect. Unlike the question above, where every complementary angle must also be acute, supplementary angles can be acute, right, or obtuse. If an angle is obtuse, its supplement is acute.
If an angle is right, its supplement is also right. Two obtuse angles can never be a supplementary pair, and two acute angles can never be a supplementary pair. Without more information, this question cannot be determined. Thus if one angle is obtuse, the other angle is acute. Adjacent complementary angles.
Thus if one angle is acute, the other angle is obtuse. When two angles are supplementary to the same angle or angles that measure the same, then they are congruent. Set 22 Consequently, the angles are congruent and their measurements are equal. A determination cannot be made. Vertical angle pairs are formed when lines intersect. Adjacent supplementary angles. How so? Because they are always open!
The two rays of an angle extend out in different directions and continue on forever. On the other hand, poly- gons are the introverts in mathematics. If you connect three or more line segments end-to-end, what do you have? It has the fewest sides and angles that a polygon can have. Set 24 Fill in the blanks based on your knowledge of triangles and angles. Set 25 Choose the best answer. Which of the following sets of interior angle measures would describe an acute isosceles triangle?
Which of the following sets of interior angle measures would describe an obtuse isosceles triangle? Which of the following angle measurements would not describe an interior angle of a right angle? Set 26 Using the obtuse triangle diagram below, determine which of the pair of angles given has a greater measure. Isosceles acute triangle BDE. Base angles D and B are congruent. Not a triangle. Any triangle can have one right angle or one obtuse angle, not both. Acute scalene triangle PQR. All three angles are acute, and all three angles are different.
Isosceles right triangle ABD. Acute equilateral triangle DEZ. An equiangular triangle is an equilateral triangle, and both are always acute. Scalene right triangle CHI. Add the measure of each angle together. Acute equilateral triangle KLM. Set 25 Choice a is not an acute triangle because it has one right angle. Though choice c describes an equilateral triangle; it also describes an isosceles triangle. Choice a is not an obtuse triangle; it is a right triangle.
A right triangle has a right angle and two acute angles; it does not have any obtuse angles. Angles and sides are measured in different units. The supplement to an obtuse angle is always acute. They are congruent and equal. Same shape, same size. That is also you, but much smaller. Look at the people around you. Some triangles are exactly alike; some are very alike, and some are not alike at all. The next chapter will look at proving similar triangles. Set 27 Choose the best answer. If congruency cannot be determined, choose choice d. SSS b. SAS c. ASA d. It cannot be determined.
The girls decide to use an arm length to separate each girl from her two other squad mates. Which postulate proves that their triangles are congruent? Two sets of the same book are stacked triangularly against opposite walls. Both sets must look exactly alike. They are twelve books high against the wall, and twelve books from the wall. Which postulate proves that the two stacks are congruent? Set 28 Use the figure below to answer questions through Name each of the triangles in order of corresponding vertices. Name corresponding line segments. Set 29 Use the figure below to answer questions through Set 30 Use the figure below to answer questions through Name each set of congruent triangles in order of corresponding vertices.
Set 31 Use the figure below to answer questions through Name a set of congruent triangles in order of corresponding vertices. What postulate proves it? Congruency cannot be determined. In later chapters you will learn more about similar triangles; but in this chapter you need to know that congruent angles are not enough to prove triangles are congruent. As long as the arm lengths are consistent, there will be only one way to form those cheering triangles. The legs of each stack measure 12 books.
Both stacks are right triangles with leg lengths of 12 and Set 28 Always coordinate corresponding vertices. Remember to align corresponding vertices. Set 30 There are two sets of congruent triangles in this question. Set 32 Similar triangles share congruent angles and congru- ent shapes. Only their sizes differ.
So, when does size matter? Ratios and Proportions A ratio is a statement comparing any two quantities. If I have 10 bikes and you have 20 cars, then the ratio of my bikes to your cars is 10 to Ratios are commonly written with a colon between the sets of objects being compared. The ratio of my blue pens to my black pens is ; I add four more black pens to my collec- tion.
The answer: 14 blue pens. Caution: When writing a proportion, always line up like ratios. The ratio is not equal to the ratio ! Set 33 Choose the best answer. Angle-Angle b. Side-Side-Side c. Side-Angle-Side d. Angle-Side-Angle State the ratio of side AB to side EF. Name each of the triangles in order of their corresponding vertices. State the postulate that proves similarity. Name a pair of similar triangles in order of corresponding vertices. Prove that WX YB are parallel. Set 36 Use the figure below to answer questions through Find AE Set 37 Fill in the blanks with a letter from a corresponding figure in the box below.
The angles of a right isosceles triangle always measure 45 — 45 — Since at least two corresponding angles are congruent, right isosceles triangles are similar. A ratio is a comparison. If one side of a triangle measures 16 inches, and a corresponding side in another triangle measures 24 inches, then the ratio is The comparison now reads, or 2 to 3. Choices a, c, and d simplify into the same incorrect ratio of or When writing a proportion, corresponding parts must parallel each other.
The proportions in choices b and c are misaligned. Choice a looks for the line segment 20 — x, not x. First, state the ratio between similar triangles; that ratio is or The ratio means that a line segment in the larger triangle is always 5 times more than the corresponding line segment in a similar triangle. If the line segment measures 30 inches, it is 5 times more than the corresponding line segment.
Set 34 Always coordinate corresponding endpoints. The sides of similar triangles are not congruent; they are proportional. That ratio means that a line segment in the smaller triangle is half the size of the corresponding line segment in the larger triangle. If that line segment measures 20 inches, it is half the size of the corresponding line segment. Set 35 Angle-Angle postulate. Since there are no side measurements to compare, only an all-angular postulate can prove triangle similarity.
When alternate interior angles are congruent, then lines are parallel. Set 36 Though it is easy to overlook, vertex C applies to both triangles. This is a little tricky.
501 Measurement Conversion Questions (Skill Builder in Focus)
The ratio is 6x:1x, or If the side of the smaller triangle measures 7, then the corresponding side of the larger triangle will measure 6 times 7, or According to the Angle-Angle postulate, at least two congruent angles prove similarity. To be congruent, an included side must also be congruent. They are congruent. They are not congruent; they are only similar. According to the Angle-Angle postulate, at least two congruent angles prove sim- ilarity. Choice l has a set of corresponding and congruent angles, which proves similarity; but choice l also has an included congruent side, which proves congruency.
Any triangle with congruent sides and congruent angles is an equilateral, equiangular triangle. A number multiplied by itself is raised to the second power. How- ever, if all the sides of any given triangle are known, but none of the angles are known, the Pythagorean theorem can tell you whether that triangle is obtuse or acute. Set 38 Choose the best answer. If the sides of a triangle measure 3, 4, and 5, then the triangle is a. If the sides of a triangle measure 12, 16, and 20, then the triangle is a.
If the sides of a triangle measure 15, 17, and 22, then the triangle is a. If the sides of a triangle measure 6, 16, and 26, then the triangle is a. If the sides of a triangle measure 12, 12, and 15, then the triangle is a. If the sides of a triangle measure 2, 3, and 16, then the triangle is a. Eva and Carr meet at a corner. The legs of a table measure 3 feet long and the top measures 4 feet long.
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If the legs are connected to the table at a right angle, then what is the distance between the bottom of each leg and the end of the tabletop? Dorothy is standing directly meters under a plane. It is meters away from her, and she has not moved. How far apart are the planes from each other? Timmy arranges the walls of his shed on the ground. The base of the second side measures 15 feet. If the walls are at a right angle from each other, the measure from the end of one side to the end of the second side equals a.
Find the value of x. Find the value of y. Set 41 Use the figure below to answer questions through Find the value of a. Find AC. Set 43 Use the figure below to answer questions through Find the value of w. Find the value of Z. This is a popular triangle, so know it well. A triangle is a right triangle. This is also a 3—4—5 triangle. When the sum of the smaller sides squared is greater than the square of the largest side, then the triangle is acute. When the sum of the smaller sides squared is less than the square of the largest side, then the triangle is obtuse. The Pythagorean theorem does not include any angles.
The corner forms the right angle of this triangle; Eva and Carr walk the distance of each leg, and the question wants to know the hypotenuse. The connection between the leg and the tabletop forms the right angle of this triangle. If you chose answer d, you forgot to take the square root of the The distance between Dorothy and the second plane is the hypotenuse. Notice that if you divided each side by , this is another triangle. The hypotenuse is unknown. Plug 6, 6, and y into the Pythagorean theorem.
The third side measures 2. Plug the given measures into the Pythagorean theorem. Set 42 Both triangles are isosceles, and they share a common vertex point. Ultimately, all their angles are congruent. The ratio between corresponding line segments A Set 43 As a rule, there is a ver- tex for every side of a polygon. Vertices of a convex polygon all point outwards all regular polygons are also convex polygons.
Envision each of these objects as simply as possible, otherwise there will always be exceptions. Name the polygon. Is it convex or concave? How many diagonals can be drawn from vertex O? How many sides does the polygon have? Set 47 Use the diagram below to answer questions through Set 48 Use the diagram below to answer questions through Set 49 Use your knowledge of polygons to fill in the blank. How many triangles can be drawn in the accompanying polygon at one time?
Not a polygon. A grid is not a polygon because its lines intersect at points that are not endpoints. A classic television screen is rectangular; it has four sides and four vertices. The human face is very complex, but primarily it has few if any straight line segments. An ergonomic chair is a chair designed to contour to your body.
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It is usually curved to support the natural curves of the hip and spine. Like the human face, lace is very intricate. Unlike the human face, lace has lots of line segments that meet at lots of different points. Set 45 Set 46 Set 47 If you list every vertex in consecutive order, then your name for the polygon given is correct. OV, OW Set 48 List every vertex in consecutive order and your answer is correct. Consecutive sides. When a line segment connects nonconsecutive end- points in a polygon, it is a diagonal. Consecutive vertices. Set 50 For solutions to and , refer to image below.
Remember when drawing your triangles that a diagonal must go from endpoint to endpoint. For solutions to and , refer to the image below. Neither are parallelo- grams, rectangles, or rhombuses. But squares are rhombuses, rectangles, and parallelograms. How can this be? Parallelograms, rectangles, rhombuses, and squares are all members of a four-sided polygon family called the quadrilaterals. Each member has a unique property that makes it distinctive from its fellow members. A square shares all those unique properties, making it the most unique quadrilateral.
Below are those particular characteristics that make each quadrilateral an individual. Four line segments connected end-to-end will always form a. A square whose vertices are the midpoints of another square is a. The sides of a square measure 2. A rhombus, a rectangle, and an isosceles trapezoid all have a. Set 52 Fill in the blanks based on your knowledge of quadrilaterals. More than one answer may be correct. Set 53 Choose the best answer.
Set 54 Use the figure below to answer questions through Using your knowledge of triangles and quadrilaterals, show that diagonals AC and BD intersect perpendicularly. Using your knowledge of triangles and quadrilaterals, what is the length of imaginary side BP? Using your knowledge of triangles and quadrilaterals, what is the length of diagonal DB? All parallelograms have opposite congruent sides including rectangles, rhombuses and squares.
Find the point along a line segment that would divide that line segment into two equal pieces. Connect the midpoint of a square together and you have another square that is half the existing square. Three squares in a row will have three times the length of one square, or 2.
However, the width will remain the length of just one square, or 2. Rhombuses and squares have congruent sides and diagonals that are perpendicular. Because their sides are not congruent, rectangles and trapezoids do not have diagonals that cross perpendicularly. Diagonals of a trapezoid are not congruent unless the trapezoid is an isosceles trapezoid. Diagonals of any trapezoid do not bisect each other. Set 52 A parallelogram, a rectangle, a rhombus, or a square.
When a transversal crosses a pair of parallel lines, alternate interior angles are congruent, while same side interior angles are supplementary. Draw a parallelogram, a rectangle, a rhombus, and a square; extend each of their sides. Again, look at the drawing you made above to see why consecutive angles are supplementary. The angle opposite the given angle must be congruent.
Choice b does not align the angles in consecutive order; choice c mistakenly subtracts 21 from 90 when consecutive angles are supplementary, not complementary. Opposite angles in an isosceles trapezoid are supplementary. Choice a describes a consecutive angle along the same parallel line. Set 54 First, opposite sides of a rhombus are parallel, which means alternate interior angles are congruent.
BP Set 55 Choose the best answer. A regular octagonal gazebo is added to a Victorian lawn garden. Each side of the octagon measures 5 ft. Timmy randomly walks ten steps to the left. He does this nine more times. His path never crosses itself, and he returns to his starting point. The perimeter of Periwinkle High is 1, ft. It has four sides of equal length. Each side measures a. Roberta draws two similar pentagons. The perimeter of the larger pentagon is 93 ft. If the perimeter of the smaller pentagon equals 31 ft. Isadora wants to know the perimeter of the face of a building; however, she does not have a ladder.
The mailbox is 4. Which perimeter is not the same? Which choice below has a different perimeter than the others? Regular Figure A with sides that measure 4. Regular Figure B with sides that measure 1. Regular Figure C with sides that measure 5. Regular Figure D with sides that measure 6. Set 57 Find the perimeter of the following figures. Set 59 Use the figure below to answer questions through Timmy walked ten ten-step sets.
Choices b and d use the wrong increment, feet. Polygon CRXZ is a rectangle whose sides measure Set 56 In choice d, the perimeter of the Jacuzzi measures 4 feet by 10 sides, or It is obvious that the Jacuzzi has a different perimeter. Plug each choice into this formula.
Set 57 Add the measure of each exterior side together. Set 59 Opposite sides of a rectangle are congruent. Perimeter is always expressed in linear units. Area is always expressed in square units. Set 62 Choose the best answer. Area is a. If two triangles are similar, the ratio of their areas is a. An apothem a. Set 63 Circle whether the statements below are true or false.
A rhombus with opposite sides that measure 5 feet has the same area as a square with opposite sides that measure 5 feet. True or False. A rectangle with opposite sides that measure 5 feet and 10 feet has the same area as a parallelogram with opposite sides that measure 5 feet and 10 feet. A rectangle with opposite sides that measure 5 feet and 10 feet has twice the area of a square with opposite sides that measure 5 feet.
A parallelogram with opposite sides that measure 5 feet and 10 feet has twice the area of a rhombus whose height is equal to the height of the parallelogram and whose opposite sides measure 5 feet. A triangle with a base of 10 and a height of 5 has a third the area of a trapezoid with base lengths of 10 and 20 and a height of 5. Set 64 Find the shaded area of each figure below. Find the shaded area of quadrilateral ABCD.
D C L 10 ft. O N Find the shaded area of Figure X. Find the shaded area of Figure Y. Find the shaded area of Figure Z. Find the area of quadrilateral ABCD. V 15 ft. U I 10 ft. H Find the length of CH Q P Find the measure of side x. Find the measure of side y. Find the measure of side z. All areas are positive numbers.
501 Maths Questions
Choice a is incorrect because if an area represented negative space, then it would be a negative number, which it cannot be. An apothem extends from the center of a polygon to a side of the polygon. All apothems are perpendicular bisectors and only span half the length of a polygon. A radius to be discussed in a later chapter extends from the center point of a polygon to any vertex.
Two consecutive radii form a central angle. Apothems are not radii. Set 63 If the rhombus is not a square, it is a tilted square which makes its height less than 5 feet. Consequently, the area of the square is 25 square feet, but the area of the rhombus is less than 25 square feet. If the parallelogram is not a rectangle, it is a tilted rectangle which makes its height less than 5 feet. Conseqently, the area of the rectangle is 50 square feet, but the area of the parallelogram is less than 50 square feet. One triangle has an area of 25 square feet.
The trapezoid has an area that measures 75 square feet. If one side of the square measures 8 feet, the other three sides of the square each measure 8 feet. If one side of a regular pentagon measures 10 feet, the other sides of a pentagon measure 10 feet. Since there are four conjoined regular hexagons, each with an area of square feet, you multiply square feet by 4. The height of right triangle ABO is ft. The area of the void must be subtracted from square feet. Subtract 7. Parallelogram ABCE: 16 ft. Extend TW Solve the area of parallelogram VUTW: 2 ft.
Their areas each equal 2.
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The area of square LMPQ equals the product of two sides: 3 ft. The sum of all the areas equal 9 sq. Subtract the area of rectangle CFED: 5 ft. Set 66 The area of a regular polygon equals half the product of its perimeter by its apothem: 45 sq. The perimeter of a regular polygon equals the length of each side multiplied by the number of sides: 30 ft.
You know the lengths of two corresponding sides, and you know the area of the smaller triangle. Apply the rule 6 sq. Cross-multiply: 6 sq. Set 67 The congruent height of each trapezoid is known, and one congruent base length is known. Imagine a perpendicular line from vertex N to the Surface Area of a Prism A prism has six faces; each face is a planar rectangle. Set 68 Choose the best answer. A rectangular prism has a. How many faces of a cube have equal areas?
Mark plays a joke on Tom. He removes the bottom from a box of bookmarks. When Tom lifts the box, all the bookmarks fall out. What is the surface area of the empty box Tom is holding if the box measures 5. Crafty Tara decides to make each of her friends a light box. To let the light out, she removes a right triangle from each side of the box such that the area of each face of the box is the same. What is the remaining surface area of the box if each edge of the box measures 3. Jimmy gives his father the measurements of a table he wants built.
The 25th Annual Go-Cart Race is just around the corner, and Dave still needs to build a platform for the winner. Sarah cuts three identical blocks of wood and joins them end-to- end. How much exposed surface area remains? Block3 k2 1. Block1 8. When the faces of a rectangular prism are laid side-by-side, you always have three pairs of congruent faces. That means every face of the prism and there are six faces has one other face that shares its shape, size, and area.
A cube, like a rectangular prism, has six faces. If you have a small box nearby, pick it up and count its faces. It has six. In fact, if it is a cube, it has six congruent faces. Set 69 Tara removes six triangular pieces, one from each face of the cube. It is given that each triangular cutout removes 6. These next few problems are tricky: Carefully look at the diagram.
Notice that the top of each cubed leg is not an exposed surface area, nor is the space they occupy under the large rectangular prism. It is reasonable to assume that where the cubes meet the rectangular prism, an equal amount of area from the prism is also not exposed. Like the question above, there are concealed surface areas in this question.
Find the surface area for the base rectangular prism. Do not worry about any concealed parts; imagine the top plane rising with each step. Of the next two prisms, only their sides are considered exposed surfaces the lip of their top surfaces have already been accounted for. Subtracting a foot from each side of the base prism, the second prism measures 13 feet by 5 feet by 1 foot. The last prism measures 11 feet by 3 feet by 1 foot. Add all the exposed surface areas together: sq. Subtract the concealed surface area from the total surface area: Set 70 Plug the variables into the formula for the Sa of a prism: sq.
In geometry, it is neither half empty, nor half full; it is half the volume. Volume is what is inside the shapes you and I see. The sides of a right prism perpendicularly meet the base. Again, that base can be any polygon. The most common oblique prism is the Pyramid. What is the name of a right sided prism? Which measurement uses the largest increment? Find the volume of a right heptagonal prism with base sides that measure 13 cm, an apothem that measures 6 cm, and a height that measures 2 cm.
Find the volume of a pyramid with four congruent base sides. Find the volume of a pyramid with an eight-sided base that measures sq. Set 73 Find each unknown element using the information below. Find the height of a right rectangular prism with a Find the base area of a right nonagon prism with an 8, cubic ft. The base of the pyramid forms an equilateral triangle. What is the perimeter of one face side? What is the surface area? What is the volume? Set 75 Use the solid figure below to answer questions through What is the width and length? What is the height? Choice a is a hexagonal pyramid; none of its six sides perpendicularly meets its base.
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The sides of choice b only perpendicularly join one base side, and choice c is an oblique quadrilateral; its base is facing away from you. Choice d is the correct answer; it is a triangular right prism. You will be tempted to answer rectangle. Remember all right prisms have rectangles. Do as you did above: subtract two base sides—the prism has ten sides, one for each edge of a decagon.
A hexagonal prism must have a hexagon as one of its sides. A right hexagonal prism has two hexagons. Choice a is a pentagonal right prism; choice c is a decagonal right prism; and choice d is not a prism at all. Choices a and b are eliminated because they are not pyramids. Choice d is also eliminated because its base polygon is not equivalent to the given base polygon, an equilateral triangle. Again, you are looking for a pyramid with the same base measurements of the given cube. Area and Surface area use square measurements, an inch times an inch, to describe two-dimensional space.
Volume uses the largest measurement; it uses the cubic measurement, an inch times an inch times an inch. Volume is three-dimensional; its measurement must account for each dimension. Set 72 Unlike the example above, this pyramid has an octagonal base. Set 73 If the volume of a right rectangular prism measures If the volume of a right nonagon prism measures 8, cubic feet and its height is 8. Set 74 A cube has six congruent faces; each face has four congruent sides. Set 75 This is the area of one base side.
Plug the measures you found in the previous question into this formula. To properly review circles, we start with a point. All the points that lie on the circle are equidistant from the center point. A radius is a line segment that extends from the center of the circle and meets exactly one point on the circle. Circles with the same center point but different radii are concentric circles. A central angle is an angle formed by two radii. A diameter is a chord that joins two points on a circle and passes through the center point.
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