## 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 $180$?’ 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 $180$ degrees. See the image below.

$\angle A$ is the angle with vertex at $\text{point }A$.

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

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

The sum of the measures of supplementary angles is $\text{180}^ \circ$.

The sum of the measures of complementary angles is $\text{90}^ \circ$.

## Supplementary and Complementary Angles

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

If angle $A$ and angle $B$ are supplementary, then $m\angle{A}+m\angle{B}=180^\circ$.

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

If angle $A$ and angle $B$ are complementary, then $m\angle{A}+m\angle{B}=90^\circ$.

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.

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 $\text{40}^ \circ$.

1. Find its supplement

2. Find its complement

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

$c=50$ Step 6. Check:

$50+40\stackrel{?}{=}90$

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 $90$ and $180$ are in numerical order?

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

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## Angle Relationships Task Cards | Parallel Lines, Transversals, & Angles Activity

Also included in:  Angle Relationships Activity Bundle | Parallel Lines, Transversals, & Angles

## Measuring Angles with a Protractor Scavenger Hunt Activity

Also included in:  Middle School Math Scavenger Hunt Mega Bundle - 45 Fun Low-Prep Math Activities

## Parallel Lines, Transversals, and Angles Puzzle - GOOGLE SLIDES VERSION!

Also included in:  All Things Algebra® - Digital Activities Bundle for Google Slides™ - VOLUME 1

## Protractor Practice - Supplementary & Complementary Angles - Crack the Code

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## Angle Relationships With Algebra Foldable Notes - Using Equations

Also included in:  Angle Relationships Activity Bundle - 5 Fun Low-Prep Activities - 7.G.B.5

Also included in:  Geometry Basics Activity Bundle

## 4th Grade Find Missing Angles Task Cards, Activities and DIGITAL LEARNING

Also included in:  4th Grade Angles and Measurement Bundle with Distance Learning

## 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,

## 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.

## 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.

Age 14 to 18 challenge level.

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.

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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#### IMAGES

1. Find The Missing Angles Worksheet Grade 5

2. Angles Problem Solving

3. Question Video: Solving Problems Involving Adjacent Angles

4. Multistep Angle Problems

5. TIP BELAJAR MATEMATIK (TIPS FOR LEARNING MATHEMATICS): PROBLEM SOLVING INVOLVING ANGLES IN CIRCLE

6. Angles Problem Solving

#### VIDEO

1. Finding the Missing Angle Geometry Problem

2. 4.3.1 Multiple Angle Equations

3. 4.4 Solving Problems Using Obtuse Angles.wmv

4. Mathematics from a different angle #mathematics #maths #shorts

5. Geometry Problem: Finding the Missing Angle

6. Math rule| angles of a triangle math problem #shorts #math #viral #education #mathhack

1. What Are the Six Steps of Problem Solving?

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...

2. How Do You Solve a Problem When You Have Different Bases With the Same Exponents?

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...

3. What Is an Opposite Angle?

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.

4. Using Properties of Angles to Solve Problems

Solution: Step 1. Read the problem.

5. Lesson 18 Problem Solving with Angles

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

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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 )

7. Solve Problems with Angles 14-7

In this lesson you will be given two angles, including one unknown. After being given the total combined angle you will need to determine

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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

9. Problem Solving with Angles of Polygons

How to calculate angles in polygons using the interior and exterior angle properties from http://mr-mathematics.com.

10. Art of Problem Solving: Angles and Parallel Lines

Art of Problem Solving's Richard Rusczyk discusses the relationships among angles formed when a transversal intersects parallel lines.

11. Problem Solving With Angles Teaching Resources

Solving for Angles in Triangles Word Problems With Pictures. 5 work sheets and 5 detailed answer keys. 10 pages total. Rules of parallel

12. Angle Properties

Angles and Parallel Lines : solving problems. Finding missing angles on two parallel lines, using corresponding angles and angles in a triangle.

13. Angles

Resources tagged with: Angles - points, lines and parallel lines. Filter by: Content type: ALL, Problems, Articles, Games. Age range: All

14. Angles Problem Solving