Geometry: 1,001 Practice Problems For Dummies (+ Free Online Practice)
Overview
Practice makes perfect! Get perfect with a thousand and one practice problems!
1,001 Geometry Practice Problems For Dummies gives you 1,001 opportunities to practice solving problems that deal with core geometry topics, such as points, lines, angles, and planes, as well as area and volume of shapes. You'll also find practice problems on more advanced topics, such as proofs, theorems, and postulates. The companion website gives you free online access to 500 practice problems and solutions. You can track your progress and ID where you should focus your study time. The online component works in conjunction with the book to help you polish your skills
and build confidence.
As the perfect companion to Geometry For Dummies or a stand-alone practice tool for students, this book & website will help you put your geometry skills into practice, encouraging deeper understanding and retention. The companion website includes:
- Hundreds of practice problems
- Customizable practice sets for self-directed study
- Problems ranked as easy, medium, and hard
- Free one-year access to the online questions bank
With 1,001 Geometry Practice Problems For Dummies, you'll get the practice you need to master geometry and gain confidence in the classroom.
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About The Author
Allen Ma and Amber Kuang are math teachers at John F. Kennedy High School in Bellmore, New York. Allen, who has taught geometry for 20 years, is the math team coach and a former honors math research coordinator. Amber has taught all levels of mathematics, from algebra to calculus, for the past 14 years.
Sample Chapters
geometry: 1,001 practice problems for dummies (+ free online practice)
CHEAT SHEET
Geometry is full of formulas, properties, and theorems. You can become successful in geometry by remembering the most important ones and learning how to apply them. Use this reference sheet as you practice various geometry problems to grow your knowledge and skills.Geometry practice problems with triangles and polygonsA polygon is a geometric figure that has at least three sides.
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Additional Book Resources
Articles from
the book
Geometry
When two parallel lines are intersected by a third line (called a transversal), congruent pairs of angles are formed, including alternate interior angles and alternate exterior angles.
The following practice questions ask you to use this information to find a missing angle, and then to apply some algebra to calculate a missing variable.
Geometry
Angles that form a linear pair combine to form a straight angle. (A straight angle measures 180 degrees.) The following practice questions ask you to solve problems based on linear pairs.
Practice questions
In the following figure,
at E. In the following questions, fill in the blank to make the statement true.
Geometry
If you are asked to find the area of a regular polygon, you can do so by using a formula that includes the perimeter of the polygon and a measurement called the apothem.
The apothem is the line segment from the center of the polygon to the midpoint of one of the sides, and is perpendicular to that side. The perimeter is the total distance around the polygon.
Geometry
In geometry, the centroid of a triangle is the point where the medians intersect. The following practice questions ask you to find the coordinates of a centroid in a triangle and to find the distance from one of the vertices to the centroid, given the median length.
Practice questions
Use the given information to solve the practice questions.
To solve geometry problems about circles, you will need to know the following circle theorems involving tangents, secants, and chords. These theorems can be used to find information about angles, intercepted arcs, and length of segments of a circle. In addition, you find the standard and general form of a circle, the formulas for area and circumference, and the area of a sector of a circle.
Geometry
You can use your knowledge of geometric constructions (as well as a compass and straight edge) to create congruent angles. The following practice questions test your construction skills.
If you're drawing two arcs for a construction, make sure you keep the width of the compass (or radii of the circles) consistent.
Geometry
When you deal with geometry problems where you have to construct 30- and 45-degree angles, you may need to do more than one construction to create what the problem is asking for.
The following practice questions ask you to apply your knowledge of constructions to some creative problems.
Practice questions
Construct a 30-degree angle.
Coordinate geometry is the study of geometric figures graphed on a coordinate plane. The slope formula can be used to determine whether lines are parallel or perpendicular. The midpoint can be used to determine if segments are bisected and also can be used to find the center of a circle. The distance formula can be used to determine the lengths of sides of geometric figures.
Geometry
Coordinate geometry proofs require an understanding of the properties of shapes such as triangles, quadrilaterals, and other polygons. The following practice questions ask you to apply the midpoint and slope formulas to prove different facts about two different quadrilaterals.
Practice questions
The points R (4, 1), H (8, 3), O (10, 7), and M (6, 5) are the vertices of a rhombus.
Geometry
When two parallel lines are intersected by a third line — a transversal — corresponding pairs of angles are formed. You can use the properties of these angles to find missing angles.
Practice questions
Use the figures and the given information to solve for the missing angles in the following questions.
as shown in the following figure.
Geometry
In geometry, a dilation is a transformation that changes only the size of a geometric shape, while leaving its shape and orientation unchanged. In the following practice questions, you're asked to calculate the constant of dilation, and then find the dilated image of given coordinates.
The center of dilation is the origin.
Geometry
In geometry, you can use the exterior angle of a triangle to find a missing interior angle. The following practice questions ask you to do just that, and then to apply some algebra, along with the properties of an exterior angle, to find a missing variable.
Practice questions
Use the figure and the given information to answer the following questions.
Many formulas are associated with the study of three-dimensional shapes in geometry. Here, you find formulas for calculating the volume, surface area, and lateral area of cylinders, cones, spheres, pyramids, cube, and rectangular prisms.
Cylinders
The lateral area of a cylinder equals
Cones
The later
Geometry
In geometry, you may be asked to formulate a proof with overlapping triangles. In order to prove parts of a triangle are congruent, you first need to prove that the triangles are congruent to each other.
The following example asks you to do just that.
Practice questions
Use the following figure to answer the questions regarding overlapping triangles.
Geometry
A polygon is a geometric figure that has at least three sides. The triangle is the most basic polygon. You will find the following formulas and properties useful when answering questions involving triangle inequalities, right triangles, relationships between the angles and sides of triangles, and interior and exterior angles of polygons.
Geometry
Geometry is full of formulas, properties, and theorems. You can become successful in geometry by remembering the most important ones and learning how to apply them. Use this reference sheet as you practice various geometry problems to grow your knowledge and skills.Geometry practice problems with triangles and polygonsA polygon is a geometric figure that has at least three sides.
Geometry
In an indirect geometric proof, you assume the opposite of what needs to be proven is true. Therefore, when the proof contradicts itself, it proves that the opposite must be true.
Practice questions
Use the following figure to answer the questions regarding this indirect proof.
Given:
are straight lines.
Prove:
do not bisect each other.
Geometry
In geometry, when you have an inscribed angle on a circle, the measure of the inscribed angle and the length of the intercepted arc are related.
An inscribed angle is equal to half of the intercepted arc.
The following practice questions ask you to find the measure of an inscribed arc and an inscribed angle.
Practice questions
Equilateral triangle
is inscribed in Circle O.
Geometry
In geometry, you can find the sum of the interior or exterior angles of a polygon based on the number of sides the polygon has. You can then apply this information to find individual interior or exterior angles.
The sum of the exterior angles of any polygon is 360 degrees. The formula
tells you the sum of the interior angles of a polygon, where n represents the number of sides.
Geometry
If you are given two side lengths of a triangle, you can use geometry to determine possible values for the missing third side. In the case of an isosceles triangle, you can find the exact value.
The following practice questions ask you to find the missing third side in both situations.
Practice questions
If two sides of an isosceles triangle are 6 and 12, what must the third side be?
Geometry
To find the orthocenter of a triangle, you need to find the point where the three altitudes of the triangle intersect. In the following practice questions, you apply the point-slope and altitude formulas to do so.
Practice questions
Use your knowledge of the orthocenter of a triangle to solve the following problems.
Geometry
You can apply your knowledge of geometric constructions to solve a variety of problems. The following practice questions test your ability to construct parallel and perpendicular lines.
If you're drawing two arcs for a construction, make sure you keep the width of the compass (or radii of the circles) consistent, and make your arcs large enough so that they intersect.
Geometry
If you want to know whether lines are parallel or perpendicular to each other (or neither), you first need to write their equations in slope-intercept form: y = mx + b.
The following practice geometry questions ask you to rewrite pairs of line equations, and then compare their slopes.
Practice questions
State whether the following two lines are parallel, perpendicular, or neither: 2y + 3 = 4x and 4y + 2x = 12.
Geometry
In geometry, the triangle inequality theorem states that when you add the lengths of any two sides of a triangle, their sum will be greater that the length of the third side.
By using the triangle inequality theorem and the exterior angle theorem, you should have no trouble completing the inequality proof in the following practice question.
Geometry
In geometry, you may be given specific information about a triangle and in turn be asked to prove something specific about it. The following example requires that you use the SAS property to prove that a triangle is congruent.
Practice questions
Use the following figure to answer each question.
Given
bisect each other at B.
Geometry
Say that you have two triangles and you need to prove that the sides of the triangles are in proportion to each other. How do you do it? Elementary! You just need to prove the triangles are similar by AA (angle-angle).
If two triangles are similar, this means the corresponding sides are in proportion.
The following practice problem asks you to finish a proof showing the sides of two triangles are in proportion.
Geometry
In two-dimensional geometry, a parallelogram is a quadrilateral (a four-sided figure) with two pairs of congruent sides and two pairs of congruent angles. The following practice questions ask you to use the properties of parallelograms to find missing angles and variables.
Practice questions
In parallelogram MATH, diagonals
intersect at E.
Geometry
When you draw an altitude to the hypotenuse of a right triangle, you create two new triangles with some interesting properties: first, they are also right triangles, and second, they are similar to each other and to the original right triangle.
The following practice questions ask you to use 'mean proportionals' to get to the solutions—no mean feat!
Geometry
In geometry, a transformation can change the size, location, or appearance of a geometric figure. Rigid motion refers to a limited transformation: only an object's location is changed, not its shape or size.
The following practice questions ask you to determine the rigid motion that will map one triangle onto another.
Geometry
In geometry, when you rotate an image, the sign of the degree of rotation tells you the direction in which the image is rotating. A positive degree measurement means you're rotating counterclockwise, whereas a negative degree measurement means you're rotating clockwise.
The following practice questions test your knowledge of rotations by asking you to rotate an octagon, and then to rotate coordinates of an image about the origin.
Geometry
When two triangles are similar, this means that their sides are in proportion. The following word problems ask you to use the properties of similar triangles—and a little bit of algebra—to find the solutions.
Similar triangles are different only in size. The corresponding angles still have the same measure.
Practice questions
The lengths of the sides of a triangle are 16, 23, and 31.
Geometry
In geometry, special right triangles are great to work with because the ratio of their sides will always be the same, making calculations easier. The two special triangles you need to know are the isosceles (or 45-45-90) and 30-60-90 right triangles.
You can use your knowledge of special right triangles to answer the following questions.
Geometry
A rectangle is a parallelogram with four right angles, which allows you to apply the Pythagorean theorem when trying to find missing sides or angles. The following practice geometry questions ask you to find the diagonal and side length for two different rectangles.
Practice questions
Find the length of the diagonal of rectangle RSTW.
Geometry
A rhombus is a parallelogram with some interesting and useful properties. For example, all of its sides are congruent, and it contains diagonals that are perpendicular bisectors and that bisect the angles of the rhombus.
You can use these properties in the following practice geometry questions, first, to solve for a missing variable x, and second, to find the perimeter of a rhombus.
In geometry, the Euler line is a serious multi-tasker: it contains the centroid, circumcenter, and orthocenter of a triangle. If you know any two of these points, you can determine the Euler line.
The following practice questions ask you, first, to use the equation for the Euler line to find the y-coordinate for the circumcenter of a triangle, and second, to find the equation for the Euler line given the triangle's centroid and orthocenter.
Geometry
When you transform geometric shapes, you may be changing their size, location, appearance, or all three. The following practice questions ask you to perform transformations that include reflection, rotation, translation, and dilation.
Practice questions
In a composition of transformations, two or more transformations are combined to form a new transformation.
Geometry
You can identify triangles by their angle measurements. Using the fact that all angles in a triangle add up to 180 degrees, you apply some algebra to determine whether a triangle is equiangular, isosceles, scalene, right, acute, or obtuse.
Practice questions
Triangle WXY is drawn with
extended to
Use the diagram and the given information to classify each triangle as acute, obtuse, equiangular, or right.
Geometry
You can classify equilateral, isosceles, scalene, and right triangles based on their side measurements. The following practice questions ask you to identify a triangle using algebra.
Practice questions
Refer to the triangle PER. Use the following information to calculate the length of each side of the triangle and classify the triangle as isosceles, equilateral, scalene, and/or right.
Geometry
In geometry, the point in a triangle where the angle bisectors of the triangle intersect is called the incenter. The following practice questions test your skills at finding the incenter of a given triangle.
Practice questions
Point I is the incenter of triangle CEN. Use the following figure and the given information to solve the problems.
Geometry
In geometry, if you’re given a right triangle with missing angles or sides, you can use trigonometric ratios—sine, cosine, or tangent—to find them.
To help you decide which of the three trigonometric ratios to use, you can label the sides of the triangle as adjacent or opposite. This labeling is dependent on the given angle in the right triangle.
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