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• 1. Introduction to Functions
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How to Measure Angles in Geometry

Blog Introduction: In geometry, there are three main ways to measure angles. These include using a protractor, measuring with a ruler, and using trigonometric functions. Let's take a closer look at each of these methods so you can be well-prepared for your next geometry test!

Using a Protractor

The most common way to measure angles is by using a protractor. A protractor is a semi-circular device that has degree markings from 0 to 180 degrees. To use a protractor, line the midpoint of the protractor up with the vertex of the angle. Then, line one arm of the angle up with the 0 degree mark and the other arm of the angle up with the 180 degree mark. The angle between the two arms will be the number of degrees of your angle! 

Measuring with a Ruler

Another way to measure angles is by using a ruler. This method is best used for acute angles, or angles less than 90 degrees. To measure an acute angle with a ruler, start by drawing a line segment that is longer than the length of your ruler. Then, place your ruler so that one end is at the vertex of your angle and the other end is on one of the arms of your angle. Without moving your ruler, make a small mark on the line segment at the point where the other arm of your angle intersects the line segment. Finally, measure the distance between the vertex and your mark in millimeters or centimeters and convert this measurement to degrees by dividing by 10 or 100 respectively. 

Using Trigonometric Functions

The last way to measure angles is by using trigonometric functions. This method is best used for obtuse angles, or angles greater than 90 degrees. To measure an obtuse angle using trigonometric functions, start by drawing a line segment that is longer than half the length of your ruler. Then, place your ruler so that one end is at the vertex of your angle and another point on the line segment is at one of the arms of your angle without moving your ruler. Next, make a small mark on the line segment at the point where the other arm of your angle intersects the line segment. Finally, use a calculator to find either sine, cosine, or tangent (whichever is appropriate) of both angles formed by this line segment and use this information to find your desired obtuse angle! 

Conclusion 

There you have it! These are three main methods for measuring angles in geometry. Be sure to practice each method so that you feel confident using them come test time. And remember, if you ever get stuck while working on a problem, ask your teacher for help!

How do you measure angles in geometry?

There are three main ways to measure angles in geometry, which include using a protractor, measuring with a ruler, and using trigonometric functions.

What are measuring angles?

Measuring angles is the process of determining the size or magnitude of an angle. This can be done in a number of ways, depending on the type of angle you are trying to measure.

How do you measure an angle step by step?

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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]

Measuring Angles Worksheets

An angle measure can be defined as the measure of an angle formed by the two rays or arms at a common vertex. Measuring angles worksheets contain examples and problems based on angle measurement in mathematics. Angles are measured in degrees (o).

Angles larger than a straight angle but less than 1 turn (between 180° and 360°) are called reflex angles. An angle equal to 1 turn (360° or 2π radians) is called a full angle.

Benefits of Measuring Angles Worksheets

Angles are used in daily life. There are lots of benefits of solving measuring angles on worksheets. Angles are used in day-to-day activities. Engineers and architects use angles for designs, roads, buildings, and sporting facilities. Athletes use angles to enhance their performance. Carpenters use angles to make chairs, tables, and sofas.

Download Measuring Angles Worksheet PDFs

These math worksheets should be practiced regularly and are free to download in PDF formats.

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Here we will learn about angles, including angle rules, angles in polygons and angles in parallel lines.

There are also angles worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are angles?

Angles measure the amount of turn required to change direction. At GCSE we can measure angles using a protractor using degrees . If the diagram is not drawn to scale we can determine missing angles by using angle facts (also referred to as angle properties or angle rules ).

There can be multiple different approaches to find a missing angle.

Angle types

There are different types of angles.

Step-by-step guide: Types of angles

• Angle rules

We can use angle rules to work out missing angles.

Angle rules are facts that we can apply to calculate missing angles in a diagram.

The five key angle facts that are used widely within the topic are:

• Angles on a straight line

The sum of angles on a straight line is always equal to \bf{180^{o}.}

A straight line would be considered to be half of a full turn; if you were standing on the line facing towards one end, you would have to turn 180 degrees to face the other end of the line.

A straight line can be called a straight angle if there is a vertex on the line and the turn around that vertex is 180^{o}.

• Angles at a point

The sum of angles at a point is always equal to \bf{360^{o}} .

A point would be considered to be a full turn; if you were standing at the point facing in one direction, you would have to turn 360 degrees to return back to your original position.

• Complementary angles

The sum of complementary angles is always equal to \bf{90^{o}} .

Complementary angles therefore make up a right angle.

These angles do not need to be together and form a right angle. If any two angles sum to 90^o they are complementary.

• Supplementary angles

The sum of supplementary angles is always equal to \bf{180^{o}} .

Supplementary angles therefore make up a straight line.

These angles do not need to be together on a straight line. If any two angles sum to 180^o they are supplementary.

• Vertically opposite angles

Vertically opposite angles are equal .

This occurs when two straight lines meet (intersect) at a point known as a vertex , forming an x shape where the opposite pairs of angles are the same size. Also, two adjacent angles are supplementary (they add to equal 180^o ).

Step-by-step guide: Angle rules

• Angles in polygons

We can calculate the interior and exterior angles of any polygon.

• Interior angles

The sum of interior angles in any n -sided shape is determined using the formula,

One interior angle of a regular polygon with n -sides is determined using the formula,

For an irregular polygon, the missing angle is calculated by subtracting all of the known angles from the total sum of the interior angles of the polygon.

• Exterior angles

The sum of exterior angles for any polygon is \bf{360^{o}} .

Whereas the interior angle sum is different for each n -sided shape, the exterior angle sum is always 360^{o}, regardless of how many sides the polygon has.

This is because, as you walk around the perimeter of the shape, the exterior angle is the turn from the direction of one edge to the next edge of the polygon.

For a regular polygon, each exterior angle is equal to 360 divided by the number of sides, n, and so

For an irregular polygon, the unknown exterior angle is calculated by subtracting the known exterior angles from 360^{o}.

The sum of an exterior angle and its adjacent interior angle is 180^{circ} , because they both lie on a straight line.

• Angles in a triangle

The sum of angles in a triangle is \bf{180^{o}} .

Remember that there are four different types of triangles, each with a specific angle property.

• Angles in a quadrilateral

The sum of angles in a quadrilateral is \bf{360^{o}} .

Any quadrilateral can be constructed from two adjacent triangles. This means that as the angle sum of a triangle is equal to 180^{o}, two triangles would have an angle sum of 360^{o}.

There are several different types of quadrilaterals, each with a specific angle property.

Step-by-step guide: Angle in polygons

• Angles in parallel lines

Angles in parallel lines are facts that can be applied to calculate missing angles within a pair of parallel lines. The three key angle facts that are used when looking at angles in parallel lines are,

• Corresponding angles
• Alternate angles
• Co-interior angles

Corresponding angles are equal .

When we intersect a pair of parallel lines with a transversal (another straight line), corresponding angles are the angles that occur on the same side of the transversal line. They are either both obtuse or both acute. 

Alternate angles are equal .

When we intersect a pair of parallel lines with a transversal, alternate angles occur on opposite sides of the transversal line. They are either both obtuse or both acute. 

The sum of two co-interior angles is \bf{180^{o}} .

Co-interior angles occur in between two parallel lines when they are intersected by a transversal . The two angles that occur on the same side of the transversal always add up to 180^{o}.

Step-by-step guide: Angles in parallel lines

How to use angles

We can use angles in lots of different contexts.

We will learn about,

• Types of angles

Explain how to use angles

Angles worksheet

Get your free angles worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Angle rules examples

Example 1: angles on a straight line.

Calculate the value of x.

Add all known angles.

The angle highlighted as a square is a right angle. A right angle measures 90^{o}. Adding the angles together, we can form the expression

2 Subtract the angle sum from \bf{180^{o}} .

As the sum of angles on a straight line total 180^{o},

3 Form and solve the equation.

Here we have no equation to solve as the missing angle is 20^{o}.

Example 2: angles at a point

Subtract the angle sum from \bf{360^{o}} .

Form and solve the equation.

Here there is no equation to solve as we know the angle x=60^{o}.

Example 3: complementary angles

Angle AOB is complementary to BOC. Determine the size of the angle AOB.

Identify which angles are complementary .

Here, the angles AOB and BOC are complementary.

Clearly identify which of the unknown angles the question is asking you to find the value of.

The question would like us to calculate the angle AOB.

Solve the problem and give reasons where applicable .

As the sum of the two angles is 90 degrees, forming an equation, we have

x+10+3x=90,

Solving this for x, we have

\begin{aligned} 4x&=90-10\\\\ 4x&=80\\\\ x&=20. \end{aligned}

Clearly state the answer using angle terminology.

We need to determine the size of angle AOB for the solution and so we need to substitute x=20 into x+10.

Angle AOB = x+10=20+10=30^{o}.

Example 4: supplementary angles

Given that AC is a straight line, determine the size of angle AOB.

Identify which angles are supplementary .

As the line AC is a straight line, the two angles AOB and BOC are supplementary. This means that we can use the angle rule, “the sum of supplementary angles is 180^{o} ”.

We need to determine the size of angle AOB.

Solve the problem and give reasons where applicable.

Adding the two angles 5x+35 and x+25 is equal to 180^{o}, which gives us the equation

5x+35+x+25=180.

Simplifying the left side of the equation, we have

Subtracting 60 from both sides of the equation, we have

6x=180-60=120.

Dividing both sides by 6, we have

x=120\div{6}=20.

We need to determine the size of angle AOB for the solution and so we need to substitute x=20 into angle AOB \ (5x+35).

Angle AOB = 5x+35=(5\times{20})+35=135^{o}.

Example 5: vertically opposite angles

Given that AC and BD are straight intersecting lines at the point O, determine the size of angle COD.

Identify which angles are vertically opposite to one another.

Angle COD is vertically opposite angle AOB and so angle COD = angle AOB.

Angle AOD is vertically opposite angle BOC and so angle AOD = angle BOC.

We need to calculate the size of angle COD \ (2x).

We need to inspect the lines and angles in the diagram to see what other rule(s) we need to apply.

As BD is a straight line, the two angles of 72^o and 2x are supplementary.

This means that if we use the rule “the sum of supplementary angles is 180^{o} ,” we can calculate the size of the angle 2x and hence the missing angle COD.

Forming an equation, we have

Subtracting 72 from both sides of the equation, and then dividing both sides by 2, we have

2x=180-72=108,

x=108\div{2}=54.

As x=54, \ 2x=2\times{x}=2\times{54}=108. As vertically opposite angles are equal,

Angle COD = 108^{o}.

Example 6: interior angles of a regular pentagon

What is the interior angle sum of a regular pentagon?

Identify how many sides the polygon has.

A pentagon has 5 sides.

Identify if the polygon is regular or irregular.

The polygon is regular.

If possible work out how many triangles could be created within the polygon by drawing lines from one vertex to all other vertices.

A regular pentagon contains 3 triangles.

Multiply the number of triangles by \bf{180} to calculate the sum of the interior angles.

State your findings.

The interior angle sum of a regular pentagon is 540^{o}.

Example 7: angles in a triangle

What type of triangle is ABC?

Add up the other angles within the triangle.

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

The remaining angle is 90 degrees and so this is a right angle triangle.

Example 8: angles in a quadrilateral

ABCD is a kite. Calculate the size of angle ADO.

Use angle properties to determine any interior angles.

Angle ABC and ADC are equal as a kite has one pair of equal angles.

This means that ABC = ADC = 2x+10.

Updating the diagram with this information, we have

Add all known interior angles.

The remaining angles must therefore total 260^{o} and so we can form the equation

2x+10+2x+10=260.

Simplifying the left hand side of the equation, we have

Subtract 20 from both sides,

4x=260-20=240.

Divide both sides by 4,

x=240\div{4}=60.

ADO = 60^{o}.

Example 9: exterior angles

Calculate the exterior angle x for the hexagon below.

Identify the number of sides in any polygon given in the question.

This irregular hexagon has 6 sides.

Identify what the question is asking.

We need to find the value of the exterior angle, x.

Solve the problem using the information you have already gathered.

The known exterior angles of this hexagon are: 70^{o}, \ 50^{o}, \ 20^{o}, and 90^{o} (the right angle).

As the interior angle is supplementary to the exterior angle at F, the exterior angle at F is equal to 180-100 = 80^{o}.

We now have the known angles: 70^{o}, \ 50^{o}, \ 20^{o}, \ 90^{o}, \ 80^{o} and the unknown angle x.

Subtracting the known angles from 360, we have

x=360-(70+50+20+90+80)=360-310=50.

The missing exterior angle x=50^{o}.

Example 10: alternate angles

Calculate the size of the angle x.

Highlight the angle(s) that you already know. 

We know angle BGH and we want to find angle GHC.

State the alternate angle, co-interior angle or corresponding angle fact to find a missing angle in the diagram.

BGH is alternate to angle GHC (our unknown value x ), so we can state that x=72^{o}.

Use basic angle facts to calculate the missing angle.

This step isn’t required as the solution has been calculated.

Example 11: corresponding angles

Calculate the value of y.

We do not know the numerical size of any angle in the diagram however we can use angles BIG, \ DJI and EKJ to determine the value of y.

The angle GIB is corresponding to GKF. As EF is a straight line we can calculate the value for x, which will help us to calculate the value of y.

As GIB is corresponding to GKF, angle GKF = 7x.

As EF is a straight line and the sum of angles on a straight line is 180, we can form the equation 2x+7x=180. Solving this equation we have

\begin{aligned} 9x&=180\\\\ x&=180\div{9}\\\\ x&=20 \end{aligned} .

As GKF = 7x, \ 7 \times 20=140 and so GKF = 140^{o}.

State the alternate angle, co-interior angle or corresponding angle fact to find a missing angle in the diagram. 

Angle GKF is corresponding to GJD and so as corresponding angles are equal, GJD = 140^o and so

Example 12: co-interior angles

We know the two angles BGH and GHD.

BGH is co-interior to GHD and so their angle sum is 180^{o}. Forming an equation for the two angles, we have

2x+50+2x+10=180.

Next, we subtract 60 from both sides

4x=180-60=120.

And finally divide by 4.

x=120\div{4}=30

Repeat this process until the required missing angle is calculated.

Common misconceptions

• Measuring angles with a protractor when the diagrams are not drawn accurately

When the question states that the diagram is not drawn accurately we cannot simply measure the missing angles using a protractor. We need to use angle facts to calculate the missing angle.

• Angles on a straight line and vertically opposite angle rules

With the straight line rule that only adjacent angles are considered, and for vertically opposite angles the lines must be straight.

• Exterior angles of a polygon

Exterior angles of a polygon have to travel in the same direction for the sum to be 360^{o}.

• Angles in triangles

Pairing up the incorrect angles in an isosceles triangle and using exterior angles when calculating the interior angle sum of a triangle.

Practice angles questions

1. Calculate the value of x.

The sum of angles on a straight line is 180^{o}.

2. Calculate the value of x.

The sum of angles at a point is 360^{o}.

3. AOC is a right angle. Calculate the value of angle BOC.

The sum of two complementary angles is 90^{o}.

4. AB is a straight line. Calculate the size of angle AOC.

As AB is a straight line, angle AOC is supplementary to angle BOC.

Angle AOC = 3x+x=4x and so

5. The two straight lines AC and BD intersect at the point O. Calculate the value of x.

As AC and BD are straight lines, angle COD is vertically opposite angle AOB.

As vertically opposite angles are equal, we can form the equation

6. The hexagon below has two lines of symmetry. Calculate the size of the interior angle at A.

The interior angle at C = 2x given the vertical line of symmetry.

The interior angle at D,E, and F can be found using the horizontal line of symmetry: D = 2x, \ E = 150^{o}, and F = 2x.

The sum of angles in the hexagon is therefore

As the interior angle at A is equal to 2x,

7. Calculate the value of y.

As the sum of angles in a triangle is 180^{o},

8. Calculate the interior angle at A.

The sum of angles in a quadrilateral is 360^{o}.

9. Calculate the interior angle of a regular octagon.

10. Which diagram only needs us to use alternate angles to determine the value of x?

BJH is alternate to JKL.

JKL is alternate to KLF.

11. Which diagram needs us to use corresponding angles to determine the value of x?

BJH is corresponding to KJF.

The sum of angles in a triangle is 180^o and so angle

Angle JKL is corresponding to HIK.

12. Which diagram uses co-interior angles to calculate the value of x?

DJI is co-interior to BIJ.

EF is a straight line. The sum of angles on a straight line is 180^{o}.

Angles GCSE questions

1. A sock design requires four colours of thread. The pie chart below shows the proportion of each colour.

The ratio of white to yellow is 2 : 1. What fraction of the thread is white?

2. ABC and ACD are two congruent isosceles triangles.

Show that the angle BCD is double the angle BAD.

State any angle rules you use.

BAC = ABC = 20^o and base angles in an isosceles triangle are equal.

ACB = 180-(20 + 20) = 140^o and the sum of angles in a triangle is 180^{o}.

ACD = ACB = 140^o and the two triangles are congruent.

BAD = 20 + 20 = 40^o and the two triangles are congruent.

BCD = 360-(140 + 140) = 80^o and the sum of angles at a point is 360^{o}.

3. Below are the three lines AB, \ CD, and EF, intersected by the line GH.

Show that the lines AB, \ CD, and EF are not parallel.

If AB and CD are parallel, 4x+2x+5=180.

DJK is corresponding to BIJ.

The sum of angles on a straight line total 180^{o}.

x must be the same for both equations.

Alternative Method

If AB and EF are parallel, 4x+3x-15=180.

FKH is corresponding to BIJ.

Learning checklist

You have now learned how to:

• Recognise angles as a property of shape or a description of a turn
• Apply the properties of angles at a point, angles at a point on a straight line, and vertically opposite angles
• Derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon, and to derive properties of regular polygons
• Distinguish between regular and irregular polygons based on reasoning about equal sides and angles
• Understand and use the relationship between parallel lines and alternate and corresponding angles

The next lessons are

• Pythagoras theorem

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Angle Worksheets: Add Insight to Your Preparation

This far-from-exhaustive list of angle worksheets is pivotal in math curriculum. Whether it is basic concepts like naming angles, identifying the parts of an angle, classifying angles, measuring angles using a protractor, or be it advanced like complementary and supplementary angles, angles formed between intersecting lines, or angles formed in 2D shapes we have them all covered for students in grade 4 through high school. Application seals concepts in the minds of children, and hence adding a little challenge into the routine in the form of free worksheets is definitely not a bad idea.

List of Angle Worksheets

Parts of an Angle

Naming Angles

Acute, Right, and Obtuse Angles

Classifying Angles

Reading Protractors

Measuring Angles

Drawing Angles

Estimating Angles

Angles on a Straight Line

Angles Around a Point

• Complementary & Supplementary Angles

Adjacent Angles

Vertical Angles

Linear Pairs of Angles

Pairs of Angles

Angles Formed by a Transversal

Angles in Shapes

Explore the angle worksheets in detail.

How about some practice in identifying the vertex and arms of an angle? Get ahead of the pack with these parts of an angle pdfs and practice identifying and naming the vertex and arms of an angle.

Are you aware of the four ways of naming angles? Buckle up with these printable worksheets, and watch how accurately and effortlessly children name angles using the three points.

Spark interest and encourage children to identify acute, right, and obtuse angles with a bunch of fun-filled exercises like recognizing angles in a clock, angle types in real-life objects, and a lot more!

Become twice as conversant with identifying, classifying, and drawing all six types of angles: acute, right, obtuse, straight, reflex, and complete angles with this collection of pdfs.

Use the protractor tool like a pro to measure and draw angles. Printable protractor templates, a chart illustrating the parts and use of the tool, and protractor reading exercises await students in elementary school.

Reading the correct scale of the protractor to measure angles: the inner or outer scale, measuring and classifying angles, and solving linear equations are the skills grade 4 and grade 5 students acquire with these exercises.

Show your students how to construct angles using a protractor with these drawing angle pdfs. The exercises include constructing angles with 1° increments or 5°, drawing reflex angles, and more.

Expert-level skills aren’t built in a day, to acquire superior skills in estimating angles 4th grade and 5th grade children need to bolster practice with our printable estimating angles worksheets.

Work your way through this compilation of worksheets and examine the angles on a straight line that add up to 180°. Grade 4 and grade 5 students find the measures of the unknown angles by subtracting the given angles from 180°.

Did you know that the angles around a point add up to 360°? Keep this fact in mind as you figure out the measures of the unknown angles by adding the given angles and subtracting the sum from 360°.

Complementary and Supplementary Angles

If it's a pair of angles you see and are trying to figure out if they make a complementary or supplementary pair, the trick is just adding them up and if their sum is 90° they are complementary and if it is 180° they are supplementary. These worksheets are a sure-shot hit with 6th grade and 7th grade learners.

Explore this bunch of printable adjacent angles worksheets to get a vivid picture of the angle addition property exhibited by angles that share the same vertex and are next to each other.

Linked here are exercises on angles formed by intersecting lines! Know the congruent properties of vertical angles or vertically opposite angles and apply them to determine unknown angle measures.

Two angles that are both adjacent and supplementary are a linear pair. The measure of such a pair sum up to 180°. Get to the heart of such angle pairs with these pdf worksheets and solve equations for the unknown angle measures.

Tap your grade 7, and grade 8 student’s potential in identifying the different pairs of angles such as complementary and supplementary angles, linear pair, vertical angles and much more with our engaging set of worksheets.

Construct additional and experiential knowledge with these 8th grade and high school handouts to comprehend the seven types of angle pairs formed by a transversal that include corresponding angles, alternate angles, and consecutive angles.

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Resources tagged with: Angles - points, lines and parallel lines

There are 70 NRICH Mathematical resources connected to Angles - points, lines and parallel lines , you may find related items under Angles, polygons, and geometrical proof .

Angles Inside

Draw some angles inside a rectangle. What do you notice? Can you prove it?

Robotic Rotations

How did the the rotation robot make these patterns?

Polygon Pictures

Can you work out how these polygon pictures were drawn, and use that to figure out their angles?

Triangle in a Trapezium

Can you find and prove the relationship between the area of a trapezium and the area of a triangle constructed within it?

Isosceles Seven

Is it possible to find the angles in this rather special isosceles triangle?

Polygon Rings

Join pentagons together edge to edge. Will they form a ring?

Same Length

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

Olympic Turns

This task looks at the different turns involved in different Olympic sports as a way of exploring the mathematics of turns and angles.

National Flags

This problem explores the shapes and symmetries in some national flags.

Which Solids Can We Make?

Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?

What Shape?

This task develops spatial reasoning skills. By framing and asking questions a member of the team has to find out what mathematical object they have chosen.

The Numbers Give the Design

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

Making Sixty

Why does this fold create an angle of sixty degrees?

Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?

Six Places to Visit

Can you describe the journey to each of the six places on these maps? How would you turn at each junction?

How Safe Are You?

How much do you have to turn these dials by in order to unlock the safes?

Round and Round and Round

Where will the point stop after it has turned through 30 000 degrees? I took out my calculator and typed 30 000 ÷ 360. How did this help?

Semi-regular Tessellations

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

Right Angles

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

Subtended Angles

What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?

Triangles in Circles

Can you find triangles on a 9-point circle? Can you work out their angles?

Octa-flower

Join some regular octahedra, face touching face and one vertex of each meeting at a point. How many octahedra can you fit around this point?

Estimating Angles

How good are you at estimating angles?

Watch the Clock

During the third hour after midnight the hands on a clock point in the same direction (so one hand is over the top of the other). At what time, to the nearest second, does this happen?

Angle Trisection

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

Quad in Quad

Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

Flexi Quads

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

Virtual Geoboard

A virtual geoboard that allows you to create shapes by stretching rubber bands between pegs on the board. Allows a variable number of pegs and variable grid geometry and includes a point labeller.

Pegboard Quads

Make different quadrilaterals on a nine-point pegboard, and work out their angles. What do you notice?

Angle Measurement: an Opportunity for Equity

Suggestions for worthwhile mathematical activity on the subject of angle measurement for all pupils.

Watch Those Wheels

Have you ever noticed the patterns in car wheel trims? These questions will make you look at car wheels in a different way!

Sweeping Hands

Use your knowledge of angles to work out how many degrees the hour and minute hands of a clock travel through in different amounts of time.

Right Angle Challenge

How many right angles can you make using two sticks?

Pythagoras of Samos was a Greek philosopher who lived from about 580 BC to about 500 BC. Find out about the important developments he made in mathematics, astronomy, and the theory of music.

Coordinates and Descartes

Have you ever wondered how maps are made? Or perhaps who first thought of the idea of designing maps? We're here to answer these questions for you.

Maurits Cornelius Escher

Have you ever noticed how mathematical ideas are often used in patterns that we see all around us? This article describes the life of Escher who was a passionate believer that maths and art can be intertwined.

Six circles around a central circle make a flower. Watch the flower as you change the radii in this circle packing. Prove that with the given ratios of the radii the petals touch and fit perfectly.

Lunar Angles

What is the sum of the angles of a triangle whose sides are circular arcs on a flat surface? What if the triangle is on the surface of a sphere?

LOGO Challenge 7 - More Stars and Squares

Can you use LOGO to create a systematic reproduction of a basic design? An introduction to variables in a familiar setting.

LOGO Challenge 8 - Rhombi

Explore patterns based on a rhombus. How can you enlarge the pattern - or explode it?

LOGO Challenge 1 - Star Square

Can you use LOGO to create this star pattern made from squares. Only basic LOGO knowledge needed.

Take the Right Angle

How many times in twelve hours do the hands of a clock form a right angle? Use the interactivity to check your answers.

Parallel Universe

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.

Similarly So

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the minute hand and hour hand had swopped places. What time did the train leave London and how long did the journey take?

On a clock the three hands - the second, minute and hour hands - are on the same axis. How often in a 24 hour day will the second hand be parallel to either of the two other hands?

The points P, Q, R and S are the midpoints of the edges of a non-convex quadrilateral.What do you notice about the quadrilateral PQRS and its area?

A Problem of Time

Consider a watch face which has identical hands and identical marks for the hours. It is opposite to a mirror. When is the time as read direct and in the mirror exactly the same between 6 and 7?

Square World

P is a point inside a square ABCD such that PA= 1, PB = 2 and PC = 3. How big is angle APB ?

Clock Hands

This investigation explores using different shapes as the hands of the clock. What things occur as the the hands move.

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Course: 4th grade   >   Unit 12

Measuring angles: faq.

• Angle measurement & circle arcs
• Measuring angles with a circular protractor
• Angles in circles
• Benchmark angles
• Types of angles by measure
• Angles in circles word problem
• Angles in circles word problems

What is an angle?

• Draw angles
• Measure angles

What's the difference between acute, right, and obtuse angles?

• Estimate angle measures

What are benchmark angles?

What does it mean to decompose an angle.

• Decompose angles

Where do we use angles in the real world?

• In construction, angles are essential for determining the correct shapes and dimensions of structures. For example, carpenters need to measure and cut angles to build frames for doors, windows, and roofs.
• In sports, athletes often use angles to their advantage. A basketball player, for example, will aim the ball at a certain angle to maximize their chances of making a shot. A baseball pitcher might throw a curveball by altering the angle of their wrist.
• When drawing or coloring, we might use angles to make sure the lines and shapes we're creating look the way we want them to.

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