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]
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Unit 6: Lesson 4
- Find measure of vertical angles
- Find measure of angles word problem
- Equation practice with complementary angles
- Equation practice with supplementary angles
- Equation practice with vertical angles
Unknown angle problems (with algebra)
- Your answer should be
- an integer, like 6 6 6 6
- an exact decimal, like 0.75 0.75 0 . 7 5 0, point, 75
- a simplified proper fraction, like 3 / 5 3/5 3 / 5 3, slash, 5
- a simplified improper fraction, like 7 / 4 7/4 7 / 4 7, slash, 4
- a mixed number, like 1 3 / 4 1\ 3/4 1 3 / 4 1, space, 3, slash, 4
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problem solving with angles
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Angle Relationships (Vertical, Complementary, & Supplementary) | Puzzle
Angles of Polygons | Coloring Activity
Classifying Shapes and Angles CSI Math Murder Mystery with Geometry Wars
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Parallel Lines, Transversals, and Angles | Bingo Game
Angle Relationships Task Cards | Parallel Lines, Transversals, & Angles Activity
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Parallel Lines, Transversals, and Angles | Cut and Paste Puzzle
Project Based Learning Math Project Measuring Angles with Protractors
Parallel Lines, Transversals, and Angles | Pyramid Sum Puzzle
Measuring Angles with a Protractor Scavenger Hunt Activity
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Parallel Lines, Transversals, and Angles Puzzle - GOOGLE SLIDES VERSION!
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Protractor Practice - Supplementary & Complementary Angles - Crack the Code
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Unknown Angles Bingo Game PPT with Blank Bingo Card 7.G.B.5
Angle Relationships Puzzle: DIGITAL VERSION (for Google Slides™)
Angle Relationships With Algebra Foldable Notes - Using Equations
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Angle Addition Postulate Task Cards
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Perpendicular and Angle Bisector Word Scramble
4th Grade Find Missing Angles Task Cards, Activities and DIGITAL LEARNING
Also included in: 4th Grade Angles and Measurement Bundle with Distance Learning
COMPLEMENTARY, SUPPLEMENTARY, & VERTICAL ANGLES Scavenger Hunt Task Cards
Relay Race Game BUNDLE (25 Games, 22 with 8 pages of 4 problems)
ANGLES: ANGLE PAIRS - SCAVENGER HUNT WITH RIDDLE AND 20 TASK CARDS
Trig Double Angle and Half Angle Identity Formulas Digital plus Print
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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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Resources tagged with: Angles - points, lines and parallel lines
Filter by: Content type: ALL Problems Articles Games Age range: All 5 to 11 7 to 14 11 to 16 14 to 18 Challenge level:
There are 68 NRICH Mathematical resources connected to Angles - points, lines and parallel lines , you may find related items under Angles, Polygons, and Geometrical Proof .
Angles Inside
Age 11 to 14 challenge level.
Draw some angles inside a rectangle. What do you notice? Can you prove it?
Robotic Rotations
Age 11 to 16 challenge level.
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
Age 14 to 16 challenge level.
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
Age 7 to 11 challenge level.
This task looks at the different turns involved in different Olympic sports as a way of exploring the mathematics of turns and angles.
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?
Making Sixty
Why does this fold create an angle of sixty degrees?
Age 5 to 7 Challenge Level
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
Age 16 to 18 challenge level.
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
Age 7 to 14 challenge level.
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
Age 14 to 18 challenge level.
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?
Orbiting Billiard Balls
What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?
Three Tears
Construct this design using only compasses
Dotty Relationship
Can you draw perpendicular lines without using a protractor? Investigate how this is possible.
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
Age 11 to 16.
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?
Age 7 to 14
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
Age 7 to 16.
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
Age 7 to 16 challenge level.
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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Angles Problem Solving
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The six steps of problem solving involve problem definition, problem analysis, developing possible solutions, selecting a solution, implementing the solution and evaluating the outcome. Problem solving models are used to address issues that...
When multiplying or dividing different bases with the same exponent, combine the bases, and keep the exponent the same. For example, X raised to the third power times Y raised to the third power becomes the product of X times Y raised to th...
Opposite angles, known as vertically opposite angles, are angles that are opposite to each other when two lines intersect. Vertically opposite angles are congruent, meaning they are equal in degrees of measurement.
Solution: Step 1. Read the problem.
Use the math you already know to solve the problem. a. What is the measure of ZFOD? How do you know? b. Name two adjacent angles that together form ZFOD. What
Unknown angle problems (with algebra). CCSS.Math: 7.G.B.5. Problem. Solve for x x xx in the diagram below. Created with Raphaël x ∘ x^\circ x∘ ( 3 x + 10 )
In this lesson you will be given two angles, including one unknown. After being given the total combined angle you will need to determine
This is a pretty tricky problem. Can you solve for the angle? Watch the video for the solution. I thank Barry, and I thank Akshay Dhivare
How to calculate angles in polygons using the interior and exterior angle properties from http://mr-mathematics.com.
Art of Problem Solving's Richard Rusczyk discusses the relationships among angles formed when a transversal intersects parallel lines.
Solving for Angles in Triangles Word Problems With Pictures. 5 work sheets and 5 detailed answer keys. 10 pages total. Rules of parallel
Angles and Parallel Lines : solving problems. Finding missing angles on two parallel lines, using corresponding angles and angles in a triangle.
Resources tagged with: Angles - points, lines and parallel lines. Filter by: Content type: ALL, Problems, Articles, Games. Age range: All
Angles Problem Solving. Grade: All grades. Activity type: Printable. Printable resource - click here to open. ×. To save results or sets tasks for your