math geometry angles problem solving

Angle Properties - Find the Angle Problems

These lessons give a summary of the different angle properties and how they can be used to find missing angles.

Related Pages Pairs Of Angles Corresponding Angles Alternate Interior Angles & Alternate External Angles More Geometry Lessons

“Find the angle” problems are very common in tests like the SAT, GRE or the GCSE. In such problems, you will be given some lines and angles and you will be required to find a particular angle or angles.

In order to answer this type of questions,

• you would need to know some commonly used angle properties.
• you would need to practice lots of such problems. The more you practice, the easier it becomes to “see” which properties need to be applied.

Some Common Angle Properties

The sum of angles at a point is 360˚.

Vertical angles are equal.

The sum of complementary angles is 90˚.

The sum of angles on a straight line is 180˚.

Alternate Angles (Angles found in a Z -shaped figure)

Corresponding Angles (Angles found in a F -shaped figure)

Interior Angles (Angles found in a C -shaped or U -shaped figure) Interior angles are supplementary. Supplementary angles are angles that add up to 180˚.

The sum of angles in a triangle is 180˚.

An exterior angle of a triangle is equal to the sum of the two opposite interior angles.

The sum of interior angles of a quadrilateral is 360˚.

How to use the above angle properties to solve some “find the angle” problems?

Find the Measure of the Missing Angle

Angles and Parallel Lines : solving problems

Finding missing angles on two parallel lines, using corresponding angles and angles in a triangle.

Angles formed by Parallel Lines and Transversals

How to use Properties of Vertical Angles, Corresponding Angles, Interior Angles of a Triangle, and Supplementary Angles to find all the angles in a diagram. Other Properties discussed include Alternate Interior Angles, Alternate Exterior Angles, Complementary Angles, and the Exterior and Opposite Interior Angles of a triangle.

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Angles of a triangle

Here you will learn about angles of a triangle including what the sum of both interior and exterior angles of a triangle are, how to find missing angles, and how to use this alongside other angle facts to form and solve equations.

Students will first learn about angles of a triangle as a part of geometry in 8 th grade and will extend their knowledge throughout high school geometry.

What are angles of a triangle?

Angles of a triangle refer to the sum of the angles of a triangle, found at each vertex in a triangle.

A triangle has both interior angles and exterior angles.

[FREE] Angles Worksheet (Grade 4)

Use this quiz to check your grade 4 students’ understanding of angles. 10+ questions with answers covering a range of 4th grade angles topics to identify areas of strength and support!

Interior angles of a triangle

Interior angles of a triangle are angles that are formed inside a triangle by its three sides. Each interior angle is formed by two adjacent sides of the triangle.

The sum of the interior angles of a triangle is \bf{180}^{\circ}.

For example,

Exterior angles of a triangle

The exterior angles of a triangle are angles that are formed on the outside of the triangle when its sides are extended.

The sum of exterior angles of a triangle is \bf360^{\circ}.

The exterior angle is also equal to the sum of the two opposite interior angles. Each exterior angle is supplementary to its adjacent interior angle.

Other angle facts

Sometimes the problem will involve using other angle facts. Let’s recap some of the other important angle facts:

Common Core State Standards

How does this relate to 8 th grade math and high school math?

• Grade 8: Geometry (8.G.A.5) Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
• High school: Geometry (HS.G.CO.C.10) Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180^{\circ}; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

How to find a missing angle of a triangle

In order to find the measure of a missing angle of a triangle:

Add up the angles that are given within the triangle.

Subtract this total from \bf{180}^{\circ}.

Angles of a triangle examples

Example 1: scalene triangle.

Find the measure of the unknown angle labeled a in the following triangle:

The angles 57^{\circ} and 79^{\circ} are given. Add these together:

2 Subtract this total from \bf{180}^{\circ}.

Subtract 136^{\circ} from 180^{\circ} \text{:}

Example 2: right triangle

Find the measure of the unknown angle labeled b in the following triangle:

The angles 90^{\circ} and 19^{\circ} are given. Add these together:

90+19=109^{\circ}

Subtract 109^{\circ} from 180^{\circ} \text{:}

\begin{aligned}& 180-109=71^{\circ} \\\\ & b=71^{\circ}\end{aligned}

Example 3: isosceles triangle

Find the measure of the unknown angle labeled c in the following triangle:

When two sides of a triangle are equal, the angles at the ends of those sides will also be equal.

The angle 64^{\circ} is given. This is an isosceles triangle (two equal length sides and two equal angles), so the other angle at the bottom will also be 64^{\circ}.

64+64=128^{\circ}

Subtract 128^{\circ} from 180^{\circ} \text{:}

\begin{aligned}& 180-128=52^{\circ} \\\\ & c=52^{\circ}\end{aligned}

How to find a missing angle of a triangle using other angle facts

In order to find the missing angle of a triangle using other angle facts:

Use angle facts to identify all possible angles.

Calculate the missing angles in the triangle.

Example 4: using angles at a point

Find the measure of the angle labeled e \text{:}

Angles at a point add up to 360^{\circ}, so you can subtract the known exterior angle from 360^{\circ}.

360-310=50^{\circ}

With the measure of two angles within the triangle, you can find the missing angle:

\begin{aligned}& 100+50=150 \\\\ & 180-150=30^{\circ} \\\\ & e=30^{\circ}\end{aligned}

Example 5: using opposite angles

Find the measure of the angle labeled f \text{:}

You are given two of the angles in the triangle, so start by finding the third angle:

\begin{aligned}& 90+61=151 \\\\ & 180-151=29^{\circ}\end{aligned}

Use the fact that opposite angles are equal to find f.

f=29^{\circ}

Example 6: two different triangles

Find the measure of the angle labeled g \text{:}

You know two of the angles in the triangle on the right, so now calculate the third.

\begin{aligned}& 48+18=66 \\\\ & 180-66=114^{\circ}\end{aligned}

You can use the fact that angles on a straight line add up to 180^{\circ}.

180-114=66^{\circ}

Since the sides of the triangle are equal, the triangle on the left is an isosceles triangle and the two angles at the bottom of the triangle are equal. Therefore you can find the third angle.

Teaching tips for angles of a triangle

• Visual aids, such as diagrams, and hands-on activities allow students to explore the properties of triangles in a different way.
• Allow students to work in pairs or small groups while working through practice problems on worksheets. This encourages discussions about math between peers, which allows for students to share their personal insights and strategies used.
• Provide real-time feedback to students as they are working through practice problems to clarify any misunderstandings as they arise. The use of a triangle calculator can support this, without having to take any additional time from you.

Easy mistakes to make

• Mixing up the sum of interior angles and exterior angles Students may use 360^{\circ} instead of 180^{\circ} for the sum of the interior angles of the triangle and vice versa.

Related angles in polygons lessons

• Interior and exterior angles of polygons
• Quadrilateral angles
• Interior angles of a polygon
• Sum of exterior angles of a polygon
• Angles of a hexagon

Angles in a triangle practice questions

1. Find the measure of the angle b in the following triangle:

2. Find the measure of the angle c \text{:}

This is an isosceles triangle and the two angles at the bottom of the triangle are equal.

3. Find the measure of angle x in the following triangle:

This is an isosceles triangle and the two angles on the right are equal.

4. What is the size of each angle in an equilateral triangle?

An equilateral triangle has three equal sides so

5. Find the measure of the angle labeled w in the following triangle:

The angle opposite 24^{\circ} is also 24^{\circ} since vertically opposite angles are equal. The triangle is an isosceles triangle and the two angles on the left are the same size.

6. Find the measure of the angle labeled v \text{:}

Let’s find the missing angle in the triangle on the left first,

Then use the fact that angles on a straight line add up to 180^{\circ} to find the unlabeled angle in the right hand triangle.

Now, find angle v \text{:}

Angles of a triangle FAQs

No, the interior angles within a triangle can vary depending on the triangle. However, all of the measures of angles within a triangle will always equal 180 degrees.

Each exterior angle of a triangle is supplementary to its adjacent interior angle. The sum of each exterior angle and its adjacent interior angle will equal 180 degrees.

The Law of Sines is a principle in trigonometry that relates the side lengths of a triangle to the sines of its angles. It states that in any triangle, the ration of the length of a side to the sine of its opposite angle is constant for all three sides. It can be expressed as: \cfrac{a}{\sin (A)}=\cfrac{b}{\sin (B)}=\cfrac{c}{\sin (C)}. See also: Law of Sines

The next lessons are

• Congruence and similarity
• Transformations
• Mathematical proofs
• Trigonometry

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Appendix B: Geometry

Using properties of angles to solve problems, learning outcomes.

• Find the supplement of an angle
• Find the complement of an angle

Are you familiar with the phrase ‘do a [latex]180[/latex]?’ It means to make a full turn so that you face the opposite direction. It comes from the fact that the measure of an angle that makes a straight line is [latex]180[/latex] degrees. See the image below.

[latex]\angle A[/latex] is the angle with vertex at [latex]\text{point }A[/latex].

We measure angles in degrees, and use the symbol [latex]^ \circ[/latex] to represent degrees. We use the abbreviation [latex]m[/latex] to for the measure of an angle. So if [latex]\angle A[/latex] is [latex]\text{27}^ \circ [/latex], we would write [latex]m\angle A=27[/latex].

If the sum of the measures of two angles is [latex]\text{180}^ \circ[/latex], then they are called supplementary angles. In the images below, each pair of angles is supplementary because their measures add to [latex]\text{180}^ \circ [/latex]. Each angle is the supplement of the other.

The sum of the measures of supplementary angles is [latex]\text{180}^ \circ [/latex].

The sum of the measures of complementary angles is [latex]\text{90}^ \circ[/latex].

Supplementary and Complementary Angles

If the sum of the measures of two angles is [latex]\text{180}^\circ [/latex], then the angles are supplementary .

If angle [latex]A[/latex] and angle [latex]B[/latex] are supplementary, then [latex]m\angle{A}+m\angle{B}=180^\circ[/latex].

If the sum of the measures of two angles is [latex]\text{90}^\circ[/latex], then the angles are complementary .

If angle [latex]A[/latex] and angle [latex]B[/latex] are complementary, then [latex]m\angle{A}+m\angle{B}=90^\circ[/latex].

In this section and the next, you will be introduced to some common geometry formulas. We will adapt our Problem Solving Strategy for Geometry Applications. The geometry formula will name the variables and give us the equation to solve.

In addition, since these applications will all involve geometric shapes, it will be helpful to draw a figure and then label it with the information from the problem. We will include this step in the Problem Solving Strategy for Geometry Applications.

Use a Problem Solving Strategy for Geometry Applications.

• Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
• Identify what you are looking for.
• Name what you are looking for and choose a variable to represent it.
• Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
• Solve the equation using good algebra techniques.
• Check the answer in the problem and make sure it makes sense.
• Answer the question with a complete sentence.

The next example will show how you can use the Problem Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles.

An angle measures [latex]\text{40}^ \circ[/latex].

1. Find its supplement

2. Find its complement

Write the appropriate formula for the situation and substitute in the given information. [latex]m\angle A+m\angle B=90[/latex] Step 5. Solve the equation. [latex]c+40=90[/latex]

[latex]c=50[/latex] Step 6. Check:

[latex]50+40\stackrel{?}{=}90[/latex]

In the following video we show more examples of how to find the supplement and complement of an angle.

Did you notice that the words complementary and supplementary are in alphabetical order just like [latex]90[/latex] and [latex]180[/latex] are in numerical order?

Two angles are supplementary. The larger angle is [latex]\text{30}^ \circ[/latex] more than the smaller angle. Find the measure of both angles.

• Question ID 146497, 146496, 146495. Authored by : Lumen Learning. License : CC BY: Attribution
• Determine the Complement and Supplement of a Given Angle. Authored by : James Sousa (mathispower4u.com). Located at : https://youtu.be/ZQ_L3yJOfqM . License : CC BY: Attribution
• Prealgebra. Provided by : OpenStax. License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

Geometry Worksheets

Welcome to the geometry worksheets page at Math-Drills.com where we believe that there is nothing wrong with being square! This page includes Geometry Worksheets on angles, coordinate geometry, triangles, quadrilaterals, transformations and three-dimensional geometry worksheets.

Get out those rulers, protractors and compasses because we've got some great worksheets for geometry! The quadrilaterals are meant to be cut out, measured, folded, compared, and even written upon. They can be quite useful in teaching all sorts of concepts related to quadrilaterals. Just below them, you'll find worksheets meant for angle geometry. Also see the measurement page for more angle worksheets. The bulk of this page is devoted to transformations. Transformational geometry is one of those topics that can be really interesting for students and we've got enough worksheets for that geometry topic to keep your students busy for hours.

Don't miss the challenging, but interesting world of connecting cubes at the bottom of this page. You might encounter a few future artists when you use these worksheets with students.

Most Popular Geometry Worksheets this Week

Lines and Angles

In this section, there are worksheets for two of the basic concepts of geometry: lines and angles.

Lines (or straight lines to be precise) in geometry are continuous and extend in both directions to infinity. They have no width, depth or curvature. In math activities, they are often represented by a drawn straight path with some width. To show that they are lines, arrows are drawn on each end to show they extend to infinity. A line segment is a finite section of a line. Line segments are often represented with points at each end of a drawn straight path. Rays start at a point and extend in a straight line to infinity. This is shown with a point at one end of a drawn straight path and an arrow at the other end.

• Identifying Lines, Line Segments and Rays Identify Lines, Segments and Rays

Angles can be classified into six different types. Acute angles are greater than 0 degrees but less than 90 degrees. Right angles are exactly 90 degrees. Obtuse angles are greater than 90 degrees but less than 180 degrees. Straight angles are exactly 180 degrees. Reflex angles are greater than 180 degrees but less than 360 degrees. Complete/Full angles are exactly 360 degrees.

• Identifying Angle Types Worksheets Identifying Acute and Obtuse Angles Identifying Acute and Obtuse Angles (No Angle Marks) Identifying Acute, Obtuse and Right Angles Identifying Acute, Obtuse and Right Angles (No Angle Marks) Identifying Acute, Obtuse, Right and Straight Angles Identifying Acute, Obtuse, Right and Straight Angles (No Angle Marks) Identifying Acute, Obtuse, Right, Straight and Reflex Angles Identifying Acute, Obtuse, Right, Straight, Reflex and Complete/Full Angles

There are several angle relationships of which students should be aware. Complementary angles are two angles that together form a 90 degree angle; supplementary angles are two angles that together form a 180 degree angle; and explementary angles are two angles that together form a 360 degree angle. Vertical angles are found at line intersections; angles opposite each other are equal. Students can practice determining and/or calculating the unknown angle(s) in the following angle relationships worksheets.

• Angle Relationships Worksheets Complementary Angles Complementary Angles (Diagrams Rotated) Supplementary Angles Supplementary Angles (Diagrams Rotated) Mixed Complementary and Supplementary Angles Questions (Diagrams Rotated) Explementary Angles Explementary Angles (Diagrams Rotated) Mixed Adjacent Angles Questions (Diagrams Rotated) Vertical/Opposite Angles Vertical/Opposite Angles (Diagrams Rotated) Mixed Angle Relationships Questions(Diagrams Rotated)
• Angles of Transversals Intersecting Parallel Lines Interior Alternate Angles Exterior Alternate Angles Alternate Angles Corresponding Angles Co-Interior Angles Transversals

Measuring angles worksheets, can be found on the Measurement Page

Triangles, Quadrilaterals and Other Shapes

The quadrilaterals set can be used for a number of activities that involve classifying and recognizing quadrilaterals or for finding the properties of quadrilaterals (e.g. that the interior angles add up to 360 degrees). The tangram printables are useful in tangram activities. There are several options available for the tangram printables depending on your printer, and each option includes a large version and smaller versions. If you know someone with a suitable saw, you can use the tangram printable as a template on material such as quarter inch plywood; then simply sand and paint the pieces.

• Shape Sets Quadrilaterals Set Tangrams
• Identifying Regular Polygons Identifying Regular Shapes from Triangles to Octagons

Worksheets for classifying triangles by side and angle properties and for working with Pythagorean theorem.

If you are interested in students measuring angles and sides for themselves, it is best to use the versions with no marks. The marked versions will indicate the right and obtuse angles and the equal sides.

• Classifying Triangles Worksheets Classifying Triangles by Side Properties Classifying Triangles by Angle Properties Classifying Triangles by Side and Angle Properties Classifying Triangles by Side Properties (No Marks) Classifying Triangles by Angle Properties (No Marks) Classifying Triangles by Side and Angle Properties (No Marks)

A cathetus (plural catheti) refers to a side of a right-angle triangle other than the hypotenuse.

• Calculating Triangle Dimensions Using Pythagorean Theorem Calculate the Hypotenuse Using Pythagorean Theorem (No Rotation) Calculate the Hypotenuse Using Pythagorean Theorem Calculate a Cathetus Using Pythagorean Theorem (No Rotation) Calculate a Cathetus Using Pythagorean Theorem Calculate any Side Using Pythagorean Theorem (No Rotation) Calculate any Side Using Pythagorean Theorem

Trigonometric ratios are useful in determining the dimensions of right-angled triangles. The three basic ratios are summarized by the acronym SOHCAHTOA. The SOH part refers to the ratio: sin(α) = O/H where α is an angle measurement; O refers the length of the side (O)pposite the angle measurement and H refers to the length of the (H)ypotenuse of the right-angled triangle. The CAH part refers to the ratio: cos(α) = A/H where A refers to the length of the (A)djacent side to the angle. The TOA refers to the ratio: tan(α) = O/A.

• Calculating Angles and Sides Using Trigonometric Ratios Calculating Angles Using the Sine Ratio Calculating Sides Using the Sine Ratio Calculating Angles and Sides Using the Sine Ratio Calculating Angles Using the Cosine Ratio Calculating Sides Using the Cosine Ratio Calculating Angles and Sides Using the Cosine Ratio Calculating Angles Using the Tangent Ratio Calculating Sides Using the Tangent Ratio Calculating Angles and Sides Using the Tangent Ratio Calculating Angles Using Trigonometric Ratios Calculating Sides Using Trigonometric Ratios Calculating Angles and Sides Using Trigonometric Ratios

Quadrilaterals are interesting shapes to classify. Their classification relies on a few attributes and most quadrilaterals can be classified as more than one shape. A square, for example, is also a parallelogram, rhombus, rectangle and kite. A quick summary of all quadrilaterals is as follows: quadrilaterals have four sides. A square has 90 degree corners and equal length sides. A rectangle has 90 degree corners, but the side lengths don't have to be equal. A rhombus has equal length sides, but the angles don't have to be 90 degrees. A parallelogram has both pairs of opposite sides equal and parallel and both pairs of opposite angles are equal. A trapezoid only needs to have one pair of opposite sides parallel. A kite has two pairs of equal length sides where each pair is joined/adjacent rather than opposite to one other. A bowtie is sometimes included which is a complex quadrilateral with two sides that crossover one another, but they are readily recognizable. Any other four-sided polygon can safely be called a quadrilateral if it doesn't meet any of the criteria for a more specific classification.

• Classifying Quadrilaterals Classifying Simple Quadrilaterals Classifying All Quadrilaterals Classifying All Quadrilaterals (+ Rotation)

Coordinate Plane Worksheets

Coordinate point geometry worksheets to help students learn about the Cartesian plane.

• Plotting Random Coordinate Points Plotting Coordinate Points in All Quadrants Plotting Coordinate Points in Positive x Quadrants Plotting Coordinate Points in Positive y Quadrants

There are many other Cartesian Art plots scattered around the Math-Drills website as many of them are associated with a holiday. To find them quickly, use the search box.

• Cartesian Art Cartesian Art Maple Leaf
• Coordinate Plane Distance and Area Calculating Pythagorean Distances of Coordinate Points Calculating Perimeter and Area of Triangles on Coordinate Planes Calculating Perimeter and Area of Quadrilaterals on Coordinate Planes Calculating Perimeter and Area of Triangles and Quadrilaterals on Coordinate Planes

Transformations Worksheets

Transformations worksheets for translations, reflections, rotations and dilations practice.

Here are two quick and easy ways to check students' answers on the transformational geometry worksheets below. First, you can line up the student's page and the answer page and hold it up to the light. Moving/sliding the pages slightly will show you if the student's answers are correct. Keep the student's page on top and mark it or give feedback as necessary. The second way is to photocopy the answer page onto an overhead transparency. Overlay the transparency on the student's page and flip it up as necessary to mark or give feedback.

Also known as sliding, translations are a way to mathematically describe how something moves on a Cartesian plane. In translations, every vertex and line segment moves the same, so the resulting shape is congruent to the original.

• Translations Worksheets Translation of 3 vertices by up to 3 units. Translation of 3 vertices by up to 6 units. Translation of 3 vertices by up to 25 units. Translation of 4 vertices by up to 6 units. Translation of 5 vertices by up to 6 units.
• Translations Worksheets (Multi-Step) Two-Step Translation of 3 vertices by up to 6 units. Two-Step Translation of 4 vertices by up to 6 units. Three-Step Translation of 3 vertices by up to 6 units. Three-Step Translation of 4 vertices by up to 6 units.

Reflect on this: reflecting shapes over horizontal or vertical lines is actually quite straight-forward, especially if there is a grid involved. Start at one of the original points/vertices and measure the distance to the reflecting line. Note that you should measure perpendicularly or 90 degrees toward the line which is why it is easier with vertical or horizontal reflecting lines than with diagonal lines. Measure out 90 degrees on the other side of the reflecting line, the same distance of course, and make a point to represent the reflected vertex. Once you've done this for all of the vertices, you simply draw in the line segments and your reflected shape will be finished.

Reflecting can also be as simple as paper-folding. Fold the paper on the reflecting line and hold the paper up to the light. On a window is best because you will also have a surface on which to write. Only mark the vertices, don't try to draw the entire shape. Unfold the paper and use a pencil and ruler to draw the line segments between the vertices.

• Reflections Worksheets Reflection of 3 Vertices Over x = 0 and y = 0 Reflection of 4 Vertices Over x = 0 and y = 0 Reflection of 5 Vertices Over x = 0 and y = 0 Reflection of 3 Vertices Over Various Lines Reflection of 4 Vertices Over Various Lines Reflection of 5 Vertices Over Various Lines
• Reflections Worksheets (Multi-Step) Two-Step Reflection of 3 Vertices Over Various Lines Two-Step Reflection of 4 Vertices Over Various Lines Three-Step Reflection of 3 Vertices Over Various Lines Three-Step Reflection of 4 Vertices Over Various Lines

Here's an idea on how to complete rotations without measuring. It works best on a grid and with 90 or 180 degree rotations. You will need a blank overhead projector sheet or other suitable clear plastic sheet and a pen that will work on the page. Non-permanent pens are best because the plastic sheet can be washed and reused. Place the sheet over top of the coordinate axes with the figure to be rotated. With the pen, make a small cross to show the x and y axes being as precise as possible. Also mark the vertices of the shape to be rotated. Using the plastic sheet, perform the rotation, lining up the cross again with the axes. Choose one vertex and mark it on the paper by holding the plastic sheet in place, but flipping it up enough to get a mark on the paper. Do this for the other vertices, then remove the plastic sheet and join the vertices with line segments using a ruler.

• Rotations Worksheets Rotation of 3 Vertices around the Origin Starting in Quadrant I Rotation of 4 Vertices around the Origin Starting in Quadrant I Rotation of 5 Vertices around the Origin Starting in Quadrant I Rotation of 3 Vertices around the Origin Rotation of 4 Vertices around the Origin Rotation of 5 Vertices around the Origin Rotation of 3 Vertices around Any Point Rotation of 4 Vertices around Any Point Rotation of 5 Vertices around Any Point
• Rotations Worksheets (Multi-Step) Two-Step Rotations of 3 Vertices around Any Point Two-Step Rotations of 4 Vertices around Any Point Two-Step Rotations of 5 Vertices around Any Point Three-Step Rotations of 3 Vertices around Any Point Three-Step Rotations of 4 Vertices around Any Point Three-Step Rotations of 5 Vertices around Any Point
• Dilations Worksheets Dilations Using Center (0, 0) Dilations Using Various Centers
• Determining Scale Factors Worksheets Determine Scale Factors of Rectangles (Whole Numbers) Determine Scale Factors of Rectangles (0.5 Intervals) Determine Scale Factors of Rectangles (0.1 Intervals) Determine Scale Factors of Triangles (Whole Numbers) Determine Scale Factors of Triangles (0.5 Intervals) Determine Scale Factors of Triangles (0.1 Intervals) Determine Scale Factors of Rectangles and Triangles (Whole Numbers) Determine Scale Factors of Rectangles and Triangles (0.5 Intervals) Determine Scale Factors of Rectangles Triangles (0.1 Intervals)
• Mixed Transformations Worksheets (Multi-Step) Two-Step Transformations Three-Step Transformations

Constructions Worksheets

Constructions worksheets for constructing bisectors, perpendicular lines and triangle centers.

It is amazing what one can accomplish with a compass, a straight-edge and a pencil. In this section, students will do math like Euclid did over 2000 years ago. Not only will this be a lesson in history, but students will gain valuable skills that they can use in later math studies.

• Constructing Midpoints And Bisectors On Line Segments And Angles Midpoints on Horizontal Line Segments Perpendicular Bisectors on Horizontal Line Segments Perpendicular Bisectors on Rotated Line Segments Angle Bisectors (Angles not Rotated) Angle Bisectors (Angles Randomly Rotated)
• Constructing Perpendicular Lines Construct Perpendicular Lines Through Points on a Line Segment Construct Perpendicular Lines Through Points Not on Line Segment Construct Perpendicular Lines Through Points on Line Segment (Segments are randomly rotated) Construct Perpendicular Lines Through Points Not on Line Segment (Segments are randomly rotated)
• Constructing Triangle Centers Centroids for Acute Triangles Centroids for Mixed Acute and Obtuse Triangles Orthocenters for Acute Triangles Orthocenters for Mixed Acute and Obtuse Triangles Incenters for Acute Triangles Incenters for Mixed Acute and Obtuse Triangles Circumcenters for Acute Triangles Circumcenters for Mixed Acute and Obtuse Triangles All Centers for Acute Triangles All Centers for Mixed Acute and Obtuse Triangles

Three-Dimensional Geometry

Three-dimensional geometry worksheets that are based on connecting cubes and worksheets for classifying three-dimensional figures.

Connecting cubes can be a powerful tool for developing spatial sense in students. The first two worksheets below are difficult to do even for adults, but with a little practice, students will be creating structures much more complex than the ones below. Use isometric grid paper and square graph paper or dot paper to help students create three-dimensional sketches of connecting cubes and side views of structures.

• Connecting Cube Structures Side Views of Connecting Cube Structures Build Connecting Cube Structures
• Classifying Three-Dimensional Figures Classify Prisms Classify Pyramids Classify Prisms and Pyramids

This section includes a number of nets that students can use to build the associated 3D solids. All of the Platonic solids and many of the Archimedean solids are included. A pair of scissors, a little tape and some dexterity are all that are needed. For something a little more substantial, copy or print the nets onto cardstock first. You may also want to check your print settings to make sure you print in "actual size" rather than fitting to the page, so there is no distortion.

• Nets of Three-Dimensional Figures Nets of Platonic and Archimedean Solids Nets of All Platonic Solids Nets of Some Archimedean Solids Net of a Tetrahedron Net of a Cube Net of an Octahedron Net of a Dodecahedron (Version 1) Net of a Dodecahedron (Version 2) Net of an Icosahedron Net of a Truncated Tetrahedron Net of a Cuboctahedron Net of a Truncated Cube Net of a Truncated Octahedron Net of a Rhombicuboctahedron Net of a Truncated Cuboctahedron Net of a Snub Cube Net of an Icosidodecahedron

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