6th Grade - Complementary And Supplementary Angles

 
     
 
     
 
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6th
Geometry
Complementary and Supplementary Angles
Use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle.
Understand that complementary angles are two or more angles that equal 90 degrees. Supplementary angles are two or more angles that measure exactly 180 degrees. Use the definition of complementary and supplementary angles to find the measure of a missing angle. The sum of the measure of all three angles in every triangle will always be 180 degrees. Use this information to find the measure of a missing angle in a triangle.
 

Sample Problems

(1)

Angles m and n are complementary. If angle m = 60°, what is the measure of angle n?

(30)

(2)

Angles s and t are supplementary. If angle t = 120°, what is the measure of angle s?

(60)

(3)

A triangle has angles labeled a, b, and c, where a = 50° and b = 60°. What is the measure of angle c?

(70)

(4)

In the figure below, angle VNL is 150°. What is the measure of angle LNB?


(30)

(5)

If angle CMB has the measure of 110°, what are the angle measures of CMP, PMQ and BMQ?


(CMP = 70, BMQ = 70, PMQ 110)

Learning Tips

(1)

Complementary angles come together to make one 90°, right angle. A right angle is easy to spot because it looks like a capital L. Right, 90° angles can be found everywhere. Where the floor meets the wall is one example of a 90° angle. The square shape at the bottom left of the figure is showing that the figure truly is a right angle. This is due to the fact that an 88° angle, for example, can look very close to a capital L. Before determining if two or more angles are complementary, you must be certain that you are truly looking at a right angle. A square in the corner where the vertex, such as in the figure below, always indicates that you are looking at a right angle. It is important to remember that the right angle (L shape) can be turned in any direction. In other words, it may be turned upside down, to the left, etc…

Once we know that a figure is a right angle, we can calculate a missing angle by subtracting the known angle from 90°. Since the sum of two complementary angles is 90°, we can use the inverse of addition, subtraction to find a missing angle. Let’s say we have two complimentary angles. If the first angle measures 60° and the second angle is equal to m, then we know that 60 + m = 90°. To solve this equation, we would use the inverse operation, subtract to isolate and solve for m. So, 90 – 60 = m and m = 30°. Now that you know why it works, you can just take a short cut and subtract the angle you know from 90 every time you see complementary angles.

(2)

Supplementary angles can be identified by two or more angles joining together to make a 180° angle. A 180° angle is also known as a straight angle. This type of angle is easy to identify because the two angles are connected to a straight line. In the figure below, we can see the straight angle as the common line underneath each angle. It is important to remember that the straight angle can be turned in a number of directions. It can be sideways, diagonal, etc… The direction or placement of the line does not matter, if it is a straight line, it is worth 180°.

We can use this information to find a missing angle of supplementary angles. Once again, we will use the inverse of addition, subtraction, to find our answer. However, this time since the sum of the measure of the angle is 180°, we will subtract from this number. So, if we have two supplementary angles and know that the first is worth 40°, we will subtract 40 from 180 to find our missing angle. In this case, our missing angle is 140°. We can prove it because 140 + 40 = 180 and the sum of supplementary angles is 180°.

(3)

Memorization of each geometric term is very important. There are a variety of ways to try to memorize these angle terms. Flash cards with angles drawn on one side and the academic term on the other side is one option. You may also want to come up with mnemonic devices to aid in memorization. For example, complimentary angles measure 90 degrees together. A 90 degree angle can be shown by making an L shape with your thumb and pointer finger. Many people use this hand signal as a symbol for loser. You can use the phrase “compliment the loser” to help relate complementary angles to a right angle. Also, supplementary starts with an “s”, as does straight angle. It will help the kinesthetic learner to say the word and use hand motions to show the angle. For example, hold up your thumb and pointer finger to make an L shape and say, “complimentary” or keep your arm straight in front of you and say “supplementary”.

(4)

The use of rulers helps many children break a difficult problem into smaller parts so that they can see the angles.

Two rulers can be used to find complimentary angles. However, in order for this to work properly, you must place the rulers together so that they make an L shape with out any gaps or overlapping. So, for example when looking at a problem such as,


You would place the first rule along line UX and the second ruler along XS. This will allow you to isolate this part of the problem and only think about 90° angles.

One ruler can be used to find supplementary angles and isolate them. For example, in the figure above, if we needed to find the measure of angle SXQ we could place a ruler along the straight line YQ. Then we could use the measure of angle YXS to find SXQ.



(5)

The sum of the three angles of a triangle is 180°, the same as a straight angle. It is easy to understand why if can imagine cutting the top point of the triangle. If you were to do this, both sides would fall in opposite directions and create a straight line. You can show this using a long strip of paper. Take the strip of paper and fold two sides up into a triangular shape. Leave the base flat on the table. Now, let the two side you folded up drop back down onto the table, you will have a straight line, which is equal to 180°, just like all 3 angles of a triangle. So, an equilateral triangle would have 3 angles worth 60° each.


Since we know the sum of all three angles of a triangle measure, 180°, we use this to find the measure of a missing angle. For example, if we have a triangle angles measuring: 90°, 25° and x. We would be able to find the value of x in two simple steps.

  1. Add up the angles you know. 90 + 25 = 115

  2. Subtract your sum from step 1 from 180° (the total of all the angles together. 180 – 115 = 65

The value of the missing angle, x is 65°. We can prove this because 90 + 25 + 65 = 180°.

(6)

It is quick and easy to check your angle measures when working with complementary, supplementary and triangle measures. After your find your missing angle, you simple add it to the other angles and make sure you get a sum of 90 for complementary angles and 180 for supplementary angles and all three angles of a triangle. Checking your work in math is crucial. For, you may understand the concept perfectly and make a minor computation error and go from an A answer to an F in an instant. Here are some examples of how to check your work to ensure those good grades:

Check your work for complementary angles:

EXAMPLE 1: If angle A = 70°, then B = 20°

Check it, 70 + 20 = 90°. CORRECT, the sum of complementary angles is always 90°!

EXAMPLE 2: If angle A = 45°, then B = 55°

Check it, 45 + 55 = 100. INCORRECT, the sum needs to be 90!!! Fix it before moving on.

Check your work for supplementary angles:

EXAMPLE 1: If angle A = 130°, then B = 50°

Check it, 130 + 50 = 180. CORRECT, the sum of supplementary angles is always 180°!

EXAMPLE 2: If angle A = 162°, then B = 22°

Check it, 162 + 22 = 184. INCORRECT, the sum should be 180. A common mistake was made, forgetting to borrow with subtraction, but it’s not to let to fix it.

Check your work for the missing angle of a triangle:

EXAMPLE 1: If angle A = 90°, B = 50°, then C = 40°

Check it, 90 + 50 + 40 = 180°. CORRECT, all three angles of a triangle have the sum of 180°!

EXAMPLE 2: If angle A = 53°, B = 33°, then C = 9

Check it, 53 + 33 + 9 = 98. INCORRECT, the sum should be 180, not 98. Forgetting to borrow with subtraction is a common error that affects the math grades of many 6th graders. This can be avoided by simply using addition to check the final sum.


Extra Help Problems

(1)

Angles x and y are complimentary. If angle x = 74°, what is the measure of angle y?

(16)

(2)

Angles m and n are supplementary. If angle n = 162°, what is the measure of angle m?

(18)

(3)

The angles of a triangle are w, x, and y, where y = 45° and x = 90°. What is the measure of angle w?

(45)

(4)

Angle 1 compliments angle 2. If angle 1 measures 89°, what is the measure of angle 2?

(1)

(5)

Angle Z supplements angle A. If angle A measures 136°, what is the measure of angle Z?

(44)

(6)

A right triangle has an angle measuring 43°. What is the measure of the third angle?

(47)

(7)

An equilateral triangle has 3 equivalent angle measures. What is the measure of each angle? Explain.

(60)

(8)

If two angles measure 72° and 18°, what kind of angle pair are they? Explain.

(complementary, together they measure 90)

(9)

An obtuse, isosceles triangle has angle 1 measuring 120° and angle 2 measuring 37°. What is the measure of the third angle?

(23)

(10)

An acute isosceles triangle has angle X measuring 75° and angle y measuring 30°. What is the measure of the missing angle?

(75)

(11)

Find the measure of the angle that complements 23°.

(67)

(12)

Find the measure of the angle that supplements 117°.

(63)

(13)

Looking at the figure, if x = 60 and y = 60, what is z equal to?

(60)

(14)

Looking at the figure, if z = 60°, what is the measure of w?

(120)

(15)

In the figure below, angle BEC is equal to 65°. Use this information to find the measure of angle AEB.

(115)

(16)

If BEC is equal to 65° and angle AED is vertical to it, what is the measure of AED? Use the property of supplementary and vertical angles to find all four angle measures.

(65)

(17)

In the figure below, if angle C = 120°, what is the measure of angle D?


(60)

(18)

If the angle above A in the figure is 120°, what are the measures of angle A and B together?

(180)

(19)

Use the properties of supplementary and vertical angles to find the measure of all angles in the figure below, if angle SBV is 48°.

(ABG is 48, GBV is 132, SBA is 132)

(20)

In the figure, angle YXS is 45°. Use the properties of vertical supplementary and complementary angles to find the measures of angles YXS, SXQ, JXQ, and UXJ?

(YXS-45, SXQ-135, JXQ-45, UXJ-90)

 

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