# 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.

### Lesson: Equations of Lines in the Coordinate Plane

*Example:* Sketch the graph of the line

### 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 .

#### 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 ,

Which of the following statements is true?

*Example:* In , . If the measure of is 50 degrees, and the measure of is degrees, what is the value of ?

### 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: , what postulate will prove ?

#### III. Relationships with Triangles

### Midsegments of Triangles

*Example:* Find the length of *VU*

### Angle Bisectors in Triangles

*Example:* Find the missing length indicated.

### Bisectors in Triangles

*Example:* Find the measure of the indicated length.

### Medians and Altitudes

*Example:* What is the measure of ?

### Indirect Proofs

*Example:*What is the first step to this indirect proof?

is acute

### Lesson: Inequalities in a Triangle

*Example:*Order the sides of each triangle from shortest to longest.

### Inequalities in Two Triangles

*Example:*

What theorem can be used to prove ?

#### 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.

### Proving that a Quadrilateral is a Parallelogram

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

### Properties of Rhombuses, Rectangles, and Squares

### Lesson: Rhombuses

### Lesson: Rectangles

### Lesson: Squares

*Example:* Which figure below is a rhombus?

### Conditions of Rhombuses, Rectangles, and Squares

*Example:* What value of 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 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 ?

### Solving Proportions

*Example:* Solve for

### Similar Polygons and Proportions

*Example:* Solve for

### Proving Triangles Similar

*Example:*Given , which theorem can be used to prove

### Similar Right Triangles

*Example:* Solve for

### Proportions in Triangles

*Example:*Find the missing length indicated

#### 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

### Lesson: 30-60-90

### Lesson: 45-45-90

### Lesson: Pythagorean Triples

*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 has the following side lengths: is 8 ft, 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 , the length of is 12 yd, the length of is 24 yd, and the measure of angle A is 30 degrees. What is the measure of angle B?

### Law of Cosines

*Example:* In , the length of is 11.3 yd, the length of is 19.9 yd, and the measure of angle C is 112.4 degrees. What is the length of ?

#### VII. Transformations

### Lesson: Translations

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

### Reflections

### Lesson: Reflection Over an Axis

### Lesson: Reflection Over a Horizontal or Vertical Line

### Lesson: Reflection Over the Line *y=x*

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

### Lesson: Rotations

*Example:* 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 ?

### Compositions of Isometries

*Example:* What are the coordinates of the point after ?

### Congruent Triangles and Congruence Transformations

*Example:*

### Lesson: Dilations

*Example:* Triangle , , becomes triangle , , under a dilation. What would the scale factor be for this dilation?

### Similarity Transformation

*Example:* When equilateral triangle is dilated by a factor of , 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*

### Sector Area

*Example:* Find the area of each sector.

### 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 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

### 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.

### Lesson: Volumes of Pyramids and Cones

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

### 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 , 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 ?

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

*Example:* Solve for in the figure below.

### Lesson: Central Angles, Inscribed Angles, and Arcs

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

### Lesson: Intersecting Chords (Arc and Angle Measures)

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

### Lesson: Tangent and Secant Lines w/ Vertex Outside of Circle (Arcs & Angles)

*Example:*

### Lesson: Chords Perpendicular to a Radius (Segment Measures)

*Example:* Solve for .

### Lesson: Equation of a Circle

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

Center: (-4, 8)

Radius: 3

## Caddell Prep offers online test prep, web-based math assignments, and in-person tutoring and test prep.