1.6 Angle Relationships

We have seen that angles are formed by the intersection of lines, rays, or segments, Now we need to understand the relations between the angles formed by these intersections.

Complimentary Angles:

Two angles are complementary angles if their combined measures total 90°.

Example A

Problem:

Given the following figures, identify any and all complimentary angles

Solution:
.
When we look at the figures we see that

AFC is a right angle, so it is 90^o. We can now find the measure of

BFC.

Now that we know

BFC = 40^o and we already know that

BFA = 50^o we can say that

BFC and

BFA are complimentary angles.

We also see that we have another angle

MNP and its measure is also 50^o . Since

MNP and

BFC. would also add up to be 90^owe can also say that

BFC and

MNP are complimentary angles.

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Supplementary Angles:

Two angles are supplementary angles if their combined measures total 180°.

Example B

Problem

Given the following figures, identify any and all supplementary angles

Solution:
.
From the last example we found that

AFB = 90^oand that

BFC = 40^o.

Now we need to know what the measures of

DFC,

DFE, and

EFA are.

So we now have:

There are even more angle that we need to find before we answer the question, and they are

DFB, and

EFA.

So now we have identified all of the angles in the figure.

Now that we know all of the angles we can identify the pairs of angle that are supplementary, add up to 180^o.

The supplementary angles that are next to each other are:

DFB and

BFA

BFC and

BFE

AFE and

EFD

EFD and

DFC

DFC and

CFA

The supplementary angles that are NOT next to each other are:

DFB and

MNP <== Don't forget that angle

DFC and

AFE

DFE and

CFA

So, in these figures there are 8 different pairs of supplementary angles.

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Adjacent Angles:

Adjacent angles have the same vertex and share a common side

Example C

Problem:

Given the following figure, identify any and all adjacent angles

Solution:

All we need to do is identify the pair of angles that have a common side.

Here we have

DFC and

CFB as adjacent angles.

Here we have

CFB and

BFA as adjacent angles.

Here we have

BFA and

AFE as adjacent angles.

Here we have

EFD and

DFC as adjacent angles.

Also, we have

CFB and

BFE as adjacent angles.

And we have

DFB and

BFA as adjacent angles.

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Linear Pair:

A linear pair is formed by two adjacent angles whose non-common sides are opposite rays. The sum of the measures of a linear pair is 180°.

Example D

Problem:

Given the following figure, identify any and linear pairs

Solution:

Linear pairs are similar to supplementary angles, except that they need to be adjacent angle as well.

Here we have

DFC and

CFA as a linear pair.

Here we have

CFA and

AFE as a linear pair.

Here we have

AFE and

EFD as a linear pair.

Here we have

EFD and

DFC as a linear pair.

Here we have

CFB and

BFE as a linear pair.

Here we have

DFB and

BFA as a linear pair.

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Vertical Angles:

Vertical angles are nonadjacent angles formed by two intersecting lines whose angle measures are equal.

Example E

Problem:

Given the following figure, identify any and vertical angles.

Solution:

To find vertical angles we need to find angles that are opposite of each other.

Here we have

DFE and

CFA as vertical angles and each angle measure is 90^o.

Here we have

DFC and

AFE as vertical angles and each angle measure is 90^o.

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Back to Example A, B, C, D, E

Practice Problems

Use the following figure to answer the practice problems.
Unless otherwise stated, figures that appear to have lines are lines.

A. Identify any and all complimentary angles with their angle measures.

B Identify any and all supplementary angles with their angle measures.

C. Identify any and all adjacent angles

D. Identify any and all linear pairs with their angle measures.

E. Identify any and all vertical angles with their angle measures.

Assignment:

Find the measure of each of the following angles.
Unless otherwise stated, figures that appear to have lines are lines.

1. Complement of RST 2. Supplement of RST
3. Complement of EDF 4. Supplement of EDF
Tell whether the pair of angles are adjacent angles, form a linear pair, or are vertical angles. 5. JTY and YTH 6. YTZ and PTZ 7. JTP and YTZ 8. PTZ and HTZ
Find the measure of each of the following angles. 9. ABD 10. EBF 11. ABE 12. EBC
13. LMQ 14. QMP 15. NMO 16. NMP
Given the figure on the right: 17. Solve for X. Find the measure of each of the following angles. 18. KLG 19. HLG
Given the figure on the right: 20. Solve for X. 21. Solve for Y Find the measure of each of the following angles. 22. EFG 23. GFH 24. HFI 25. EFI