# Geometry | Free Practice Problems & Explanations

### 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 $y=4x-9$

### 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 $y=\dfrac{1}{3}x-5$.

### 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 $\triangle BAE \cong \triangle CAD$?

### 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 $\overline{BD}$?

### 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.

### Inequalities in Two Triangles

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

### 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 $x$?

### Lesson: 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?

### 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 $x$

### Proportions in Triangles

Example:Find the missing length indicated

### 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?

### 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 $\triangle ABC$ 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 $\overline{AB}$?

### 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 $\triangle A'H'F'$?

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

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 $(R_{origin} \circ T_{3,-2}) (P)$?

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?

### 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 $\pi$ if necessary.

### 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 $m^2$, what is the area of the larger solid?

### Lesson: Tangent Lines

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

### Lesson: Intersecting Chords (Segment Lengths)

Example: Two chords intersect and shown in the diagram below. What is the value of $x$?

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

Example: Solve for $x$ 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.

Example:

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

Example: Solve for $x$.

### Lesson: Equation of a Circle

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

Center: (-4, 8)