Minds On
Notice and wonder
Examine the following set of angles. Add the set together and record your observations.
Add this second following set together and record your observations.
Finally, add this third following set together and record your observations.
Now that you have added the three set of angles, how does one angle relate to another angle in its set? What do you notice about the three sums?
Record your ideas in a notebook or a method of your choice.
Action
Relationships of angles
There are a few ways we can describe relationships between different angles that can help us to determine or solve the measurements of unknown angles, without using a protractor or another measurement tool to measure them. The relationship between angles can be described as supplementary, complementary, or opposite.
Supplementary angles
Supplementary angles describe two angles that make up a straight line. A straight line measures 180°. That means that when there are two angles that make up a straight line, their sum will be 180°. The angle provided indicates that a straight line can be divided with one line into two angles of 120° and 60°. You can confirm they are supplementary by adding 120° + 60° = 180°.
Can you think of any other pairs of angles that could also be described as supplementary angles? How can we use this information to solve an unknown angle?
If the only known angle in the example above was 120°, you could use knowledge of supplementary angles to calculate the other value. We know that a line always equals 180°. That means that 180 minus 120 will equal the measurement of the unknown angle.
Complementary angles
Complementary angles describe two angles that make up a right angle. A right angle measures 90°. That means that when there are two angles that make up a right angle, their sum will always be 90°. The angle provided indicates that a right angle can be divided with one line into two angles of 30° and 60°. You can confirm they are complementary by adding 30° + 60° = 90°.
How can we use this information to solve an unknown angle?
If the only known angle in the example above was 60°, you could use knowledge of complementary angles to calculate the other value. A right angle always equals 90°. That means that 90° − 60° will equal the measurement of the unknown angle.
Opposite angles
When two lines intersect into an “X” shape, they form four angles. Opposite angles describe the relationship between the two angles that are opposite from each other. These angles are congruent, meaning they are equal in measurement.
The example provided identifies the four angles formed by two intersecting lines. A 145° angle is opposite from another 145° angle. The 35° angle is opposite from the other 35° angle. Think of any other angle measurements that could make up the two sets of opposite angles.
Supplementary angles from intersecting lines, divided into 4 angles. Line segment AB crosses line segment CD forming two 145° angles on top and bottom. The angles formed on the right and left sides are both 35°. Adding 145° and 35° (sum = 180°) together with 145° and 35° (again, sum = 180°) totals to 360°
How can we use this information to solve an unknown angle? If the only known angle in the example above was 145° you can figure out its opposite angle easily because it is the same. The other angles could be figured out using one of the other two relationships we just learned, supplementary or complementary. Which could you use to help solve the other angle measurement?
After answering the following questions in your notebook, press ‘Show Answers’ button to access the solutions.
1) Calculate the angle indicated by the question mark.
I know that the angles are supplementary. This means that the total of the two angles must equal to 180°.
This means that 180 − 140 = 40.
The unknown angle is 40°.
2) Calculate the unknown angle indicated by the arc.
I know that the angle inside the circle is 90°. I also know that a full circle measures 360°. This means that the outer angle is 360 − 90 = 270.
The unknown angle is 270°.
3) Calculate the unknown angle indicated by the X mark.
I know that the angles are complementary, which means they must add up to make 90°.
The calculation is 90 − 73 = 17.
The unknown angle is 17°.
4) Given that one angle is 105°, calculate the three unknown angles.
I know that the angles are opposite. The angles must add up to 360°. The angle x is also 105°. That gives me a total of 210.
360 − 210 = 150
I need the total of angles y and z to be 150°. Both angles must be equal.
150 ÷ 2 = 75
x = 105°
y = 75°
z = 75°
Reflect on your learning by answering these two questions.
- What strategies did you use?
- How did your previous knowledge about angles help you to answer these questions?
Record your ideas in a notebook or a method of your choice.
Task 1: Solving for unknown angles
After answering the following questions, press the ‘Show Answers’ button to access the solutions. Use what you know about supplementary, complementary, and opposite angles to solve for the missing measurements of the following angles.
Record your answers in a notebook or a method of your choice.
1) Solve for angles a, b, and c.
I know that these are opposite angles that total 360°. Angle b is also 36° as it has to equal the angle opposite.
36 + 36 = 72
The total of the angles is 360°. This means I need to subtract 72 from 360.
360 − 72 = 288
288° is the sum of angles a and c. I know that they are the same, so I need to divide 288 by 2.
288 ÷ 2 = 144°2) Solve for angles b, c, and d.
I know that these are opposite angles that total 360°. Angle c is 40° as it has be equal to the angle opposite.
40 + 40=80
360 − 80 = 180
180 ÷ 2 = 90
Angles b and d are 140° and angle c is 40°.
I can also solve this problem by using supplementary angles. Angle c is opposite to 40, but it is also supplementary to b and d.
I can use the 40° to find b
180 − 40 = 140°
I can do the same to find d
180 − 40 = 140°
3) Solve for angles a and b.
I know that angle a is 90 degrees, because it has a right-angle symbol (small square). I know the total of all the angles must equal 180°.
90 + 22 = 112
180 − 112 = 68°
Angle b is 68°.
Task 2: Interior angles
Interior angles describe the angles that are inside a shape. All of the angles in a quadrilateral will always add up to 360 degrees. All of the angles in a triangle will always add up to 180 degrees.
You can use the properties of interior angles to solve for an unknown angle in a shape. If all triangles have angles that sum up to 180°, you can subtract the known measurements from 180°. If all quadrilaterals have angles that sum up to 360°, you can subtract the known measurements from 360°.
After answering the following questions, press the ‘Show Answers’ button to access the solutions.
1) There is a triangle with one angle of 70° and one angle of 50°. What is the unknown angle?
I know that the total angles in a triangle equal 180°. I can add the two angles and subtract that total from 180 to find the unknown angle.
70 + 50 = 120
180 − 120 = 60
The unknown angle is 60°.
2) There is a trapezoid with one angle of 100°, 80°, and 75°. What is the unknown angle?
I know that all angles in a quadrilateral add up to 360°. I can add the three known angles, then subtract that total from 360 to find the unknown angle.
80 + 100 + 75 = 255
360 − 255 = 105
The unknown angle is 105°.
I can also solve this problem by using what I know about interior angles. If top and bottom are parallel then the interior angles (in a C pattern, forwards and backwards) equal 180.
180 − 75 = 105
The unknown angle is 105°.
Task 3: Exterior angles
Exterior angles describe an angle that is outside of a shape if you continue a side length along a straight line. Since exterior angles use a straight line, they create two supplementary angles, and we can use supplementary angles to help us solve for the unknown angle.
After answering the following question, press the ‘Show Answers’ button to access the solution.
The following example triangle has an exterior angle. It is a triangle where the bottom line is extended outside to the right of the triangle. Which letter do you think demonstrates the exterior angle?
Angle d is outside the shape, therefore it is the exterior angle.
Task 4: Solving for unknown exterior angles
After answering the following questions, press the ‘Solution’ button to access the solutions.
Practice question 1
For the following isosceles triangle, find the exterior angle m using everything we have learned about the relationships between angles. Press the ‘Hint’ button to learn the definition of an isosceles triangle.
I know that an isosceles triangle has two equal sides and two equal angles. If 70° is the known angle, the unknown angle inside the triangle is also 70°. I know that the exterior angle m uses a straight line, so m is a supplementary angle.
Two supplementary angles must add up to 180°.
180 − 70 = 110
Angle m is 110°.If you would like, you can complete the activity using TVO Mathify. You can also use your notebook or the following fillable and printable document.
Press the ‘TVO Mathify’ button to access this resource and the ‘Activity’ button for your note-taking document.
TVO Mathify (Opens in a new window) Activity (Open PDF in a new window)Practice question 2
Solve for both missing angles of the following trapezoid.
Record your ideas in a notebook or a method of your choice.
I know that a trapezoid is a quadrilateral. Angles in a quadrilateral add up to 360°.
I know that 112° is a supplementary angle, therefore the internal angle must be
180 − 112 = 68
122 + 58 + 68 = 248
x = 360 − 248
x = 112°
Angle a is a supplementary angle. I know that the total of the two angles must add up to 180°.
180 − 58 = 122
a = 122
Angle x is 112° and angle a is 122°.
Consolidation
Task 1: Angles all around the world
Angles are used in every kind of construction, especially bridges and buildings.
Jacques Cartier Bridge, Montreal.
Sydney Harbour Bridge.
Øresund, Öresund Bridge connecting Sweden with Denmark.
Cityscape of Guiyang City, Guizhou Province, China.
Skyscrapers in the city of London, England.
Think about bridges and buildings. Can you find any supplementary, complementary, opposite, interior, or exterior angles?
Record your ideas in a notebook or a method of your choice.
Task 2: Thinking back
Consider and answer the following questions:
- How does knowing the relationship between angles help to solve for unknown angles?
- How would you explain how to determine unknown angles? What are some important facts to understand?
Record your ideas in a notebook or a method of your choice.
Reflection
As you read through these descriptions, which sentence best describes how you are feeling about your understanding of this learning activity? Press the button that is beside this sentence.
I feel...
Now, record your ideas using a voice recorder, speech-to-text, or writing tool.
Connect with a TVO Mathify tutor
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TVO Mathify (Opens in a new window)