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Geometry Help | Free Geometry Lessons & Practice Problems

Go through the lessons and practice problems below to help you learn Geometry and excel in school. We’ll track your progress and help you identify your strengths and weaknesses. Geometry help is available to everyone, but you need to create an account in order to access the practice questions and track your progress.

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.
G005_8

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.

Lesson: Triangle Congruence by SSS and SAS

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

Lesson: 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?

Lesson: Triangle Congruence by HL (Right Triangles)

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

Lesson: 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
G016-3-1

Angle Bisectors in Triangles

Example: Find the missing length indicated.
G017_4

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.
G024_1

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
G035_1

Proportions in Triangles

Example:Find the missing length indicated
G036_1

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
G055_3

Sector Area

Example: Find the area of each sector.
G056-4_1

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
G059_1

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.
G061_1

Lesson: Volumes of Pyramids and Cones

Example: Find the volume of the figure below. Round your answers to the nearest hundreth, if necessary.
G062_2

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?
G067_6

Lesson: Secants and Tangents w/ Vertex Outside of Circle (Segment Lengths)

Example: Solve for x in the figure below.
G068_3

Lesson: Central Angles, Inscribed Angles, and Arcs

Example: Find the measure of the arc or angle indicated.
G069_1

Lesson: Intersecting Chords (Arc and Angle Measures)

Example: Find the measure of the indicated angle formed by the intersecting chords.
G070_1

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.
G072_7

Lesson: Equation of a Circle

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

Center: (-4, 8)
Radius: 3