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Geometry | Free Practice Problems & Explanations


I. Parallel and Perpendicular Lines

Lesson: Properties of Parallel Lines

Example: Identify each pair of angles as corresponding, alternate interior, alternate exterior, same-side interior, vertical, or adjacent.

Lesson: Proving Lines Parallel

Example: Find the measure of the indicated angle that makes the two lines parallel.

Lesson: Triangle Angle Sum and Exterior Angle Theorem

Example:Find the measure of each angle indicated.
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Lesson: Equations of Lines in the Coordinate Plane

Example: Sketch the graph of the line y=4x-9

Lesson: Writing Equations of Parallel and Perpendicular Lines

Example: Write the slope-intercept form of the equation of a line that passed through (3, 4) and is parallel to y=\dfrac{1}{3}x-5.

II. Congruent Triangles

Lesson: Congruent Triangles

Example: State if the two triangles are congruent. If they are, state how you know.

Triangle Congruence by SSS and SAS

Example: State is the two triangles below are congruent, and if so state how you know.

Triangle Congruence by ASA and AAS

Example: State is the two triangles below are congruent, and if so how?

Lesson: Identifying Corresponding Parts of Congruent Triangles

Example: Give that triangle ABC is congruent to triangle DEF, which pair of angles are congruent?

Lesson: Isosceles and Equilateral Triangles

Example: In  \triangle{XYZ}, \angle{X} \cong \angle{Y}
Which of the following statements is true?

Example: In  \triangle{XYZ}, \angle{X} \cong \angle{Y}. If the measure of \angle{X} is 50 degrees, and the measure of \angle{Z} is 2x-8 degrees, what is the value of x?

Triangle Congruence by HL (Right Triangles)

Example: State is the two triangles below are congruent, and if so how?

Congruence in Overlapping Triangles

Example: Given: \overline{AB} \cong \overline{AC}, what postulate will prove \triangle BAE \cong \triangle CAD?

III. Relationships with Triangles

Midsegments of Triangles

Example: Find the length of VU
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Angle Bisectors in Triangles

Example: Find the missing length indicated.
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Bisectors in Triangles

Example: Find the measure of the indicated length.

Medians and Altitudes

Example: What is the measure of \overline{BD}?

Indirect Proofs

Example:What is the first step to this indirect proof?
\angle A is acute

Lesson: Inequalities in a Triangle

Example:Order the sides of each triangle from shortest to longest.
G021-2-page-001

Inequalities in Two Triangles

Example:
What theorem can be used to prove \overline{AC} > \overline{DF}?

IV. Polygons and Triangles

Lesson: Angle Sum Theorem for Convex Polygons

Example: What is the interior angle sum of a regular hexagon?

Lesson: Properties of Parallelograms

Example: Find the measurement indicated in the parallelogram.
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Proving that a Quadrilateral is a Parallelogram

Example: Given that ABCD is a parallelogram, what is the value of x?

Properties of Rhombuses, Rectangles, and Squares

Example: Which figure below is a rhombus?

Conditions of Rhombuses, Rectangles, and Squares

Example: What value of x in the figure below would make the figure a rectangle?

Trapezoids and Kites

Polygons in the Coordinate Plane

Example: Triangle ABC has vertices at points (2, 4), (5, 7) and (3, 6). Determine what type of triangle \bigtriangleup ABC is.

Applying Coordinate Geometry

Example: Is a quadrilateral with vertices (-A, 0), (0, A), (A,0) and (0, -A) a square?

Proofs Using Coordinate Geometry

Example: Which theorem can be used to proved \bigtriangleup ACD \cong \bigtriangleup CAB?

Solving Proportions

Example: Solve for x
\dfrac{5}{8}=\dfrac{x}{12}

Similar Polygons and Proportions

Example: Solve for x

Proving Triangles Similar

Example:Given \overline{BC} \parallel \overline{DE}, which theorem can be used to prove \triangle ABC \sim \triangle ADE

Similar Right Triangles

Example: Solve for x
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Proportions in Triangles

Example:Find the missing length indicated
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VI. Right Triangles and Trigonometry

Lesson: Pythagorean Theorem

Example: The hypotenuse of a right triangle has a length of 26 inches. One of its legs is 24 inches. What is the length of the other leg?

Special Right Triangles

Example: A right triangle with an angle of measure 45 degrees, has a hypotenuse of length 13. What are the lengths of the other legs?

Lesson: Trigonometry

Example: Right triangle \triangle ABC has the following side lengths: \overline{AB} is 8 ft, \overline{AC} is 3 ft. What is tan(A)?

Elevation and Depression

Example: The height of the Empire State Building is 1,050 feet. At the bottom, there is a basketball 450 feet away. Find the angle of depression from the top of the tower to the basketball.

Law of Sines

Example: In \triangle ABC, the length of \overline{AC} is 12 yd, the length of \overline{AB} is 24 yd, and the measure of angle A is 30 degrees. What is the measure of angle B?

Law of Cosines

Example: In \triangle ABC, the length of \overline{AC} is 11.3 yd, the length of \overline{BC} is 19.9 yd, and the measure of angle C is 112.4 degrees. What is the length of \overline{AB}?

VII. Transformations

Lesson: Translations

Example: \triangle AHF has coordinates A (3, -1), H (5, -1), and F (1,-3). If \triangle AHF is translated 1 unit left and 6 units up, what are the coordinates of its image \triangle A'H'F'?

Reflections

Example: \triangle ABC has coordinates A (2, 4), B (3, 6) and C (-2, -1). If \triangle ABC is reflected across the x-axis, what are the coordinates of its image \triangle A'B'C'?

Lesson: Rotations

Example: \triangle DEF has coordinates D (1, 1), E (3, 1) and F (2, 5). If the triangle is rotated 90 degrees about the origin, what are the coordintates of \triangle D'E'F'?

Compositions of Isometries

Example: What are the coordinates of the point P(5,-2) after (R_{origin} \circ T_{3,-2}) (P)?

Congruent Triangles and Congruence Transformations

Example:

Lesson: Dilations

Example: Triangle X(1,1), Y(3,2), Z(4,1) becomes triangle X'(3,3), Y'(9,6), Z'(12,3) under a dilation. What would the scale factor be for this dilation?

Similarity Transformation

Example: When equilateral triangle \triangle ABC is dilated by a factor of \dfrac{1}{2}, what are the corresponding angle measurements of the image triangle?

VIII. Area

Area of Triangles and Parallelograms

Example: What is the area of a parallelogram with a base of 24 ft and a height of 20 ft?

Area of Trapezoids, Rhombuses, and Kites

Example: What is the area of a trapezoid with bases of 3 ft and 6 ft, with a height of 2.2 ft?

Area of Regular Polygons

Example: What is the area of a regular hexagon that has a side of length 14 inches and an apothem of 12.1 inches?

Perimeter and Area of Similar Figures

Example: The ratio of the perimeters of rectangle I to rectangle II is 1:3. If the area of rectangle I is 12 sq. ft., what is the are of rectangle II?

Trigonometry and the Area of Regular Polygons

Example: What is the area of an equilateral triangle with a side length of 6 m?

Arc Measures and Arc Lengths in Circles

Example: Find the measure of arc JK
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Sector Area

Example: Find the area of each sector.
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Geometric Probability

Example:A circle is inscribed in a square with the following measurements. What is the probability that a random point will lie in the circle? Leave your answers in terms of \pi if necessary.

IX. Surface Area and Volume

Space Figures and Cross Sections

Example: Use Euler’s fourmula to determine the number of faces of a solid with 9 edges and 5 vertices.

Lesson: Surface Areas of Prisms and Cylinders

Example: Sketch the net of the solid below
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Lesson: Surface Areas of Pyramids and Cones

Example:What is the surface area?

Lesson: Volumes of Prisms and Cylinders

Example:Find the volume of each figure. Round your answers to the nearest hundredth, if necessary.
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Lesson: Volumes of Pyramids and Cones

Example: Find the volume of the figure below. Round your answers to the nearest hundreth, if necessary.
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Lesson: Surface Area and Volume of Spheres

Example: What is the volume of a sphere with a diameter of 18 inches?

Areas and Volumes of Similar Solids

Example: The scale factor between two solid figureds is 2:5. If the surface area of the smaller solid is 45 m^2, what is the area of the larger solid?

X. Circles

Lesson: Tangent Lines

Example: Can a radius be drawn to a point of tangency?

Lesson: Congruent Chords

Lesson: Intersecting Chords (Segment Lengths)

Example: Two chords intersect and shown in the diagram below. What is the value of x?
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Lesson: Secants and Tangents w/ Vertex Outside of Circle (Segment Lengths)

Example: Solve for x in the figure below.
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Lesson: Central Angles, Inscribed Angles, and Arcs

Example: Find the measure of the arc or angle indicated.
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Lesson: Intersecting Chords (Arc and Angle Measures)

Example: Find the measure of the indicated angle formed by the intersecting chords.
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Lesson: Tangent and Secant Lines w/ Vertex Outside of Circle (Arcs & Angles)

Example:

Lesson: Chords Perpendicular to a Radius (Segment Measures)

Example: Solve for x.
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Lesson: Equation of a Circle

Example: Use the information provided to write the equation of each circle.

Center: (-4, 8)
Radius: 3