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I chose this picture because I think that it clearly shows what vertical angles are

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This source was a good choice because it states all of the information needed for this Learning Target

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Opposite Rays Definition - Math Open Reference

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Definition of a opposite rays - two rays with a common endpoint that form a straight line .

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I chose this source because it clearly stated both of these learning targets

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By Mark Ryan

You're probably already familiar with the Transitive Property and the Substitution Property from algebra. If a = b and b = c, then a = c, right? That's transitivity. And if a = b and b < c, then a < c. That's substitution. Easy enough. Below, you see these theorems in greater detail:

Transitive Property (for three segments or angles): If two segments (or angles) are each congruent to a third segment (or angle), then they're congruent to each other.

The Transitive Property for three things is illustrated in the above figure.

Transitive Property (for four segments or angles): If two segments (or angles) are congruent to congruent segments (or angles), then they're congruent to each other.

The Transitive Property for four things is illustrated in the below figure.

Substitution Property: If two geometric objects (segments, angles, triangles, or whatever) are congruent and you have a statement involving one of them, you can pull the switcheroo and replace the one with the other. (Note that you will not be able to find the term "switcheroo" in your geometry glossary.)

A figure isn't especially helpful for this property, so one isn't included here.

To avoid getting the Transitive and Substitution Properties mixed up, just follow these guidelines:

Check out this TGIF rectangle proof, which deals with angles:

-1 @ -2

No need for a game plan here because the proof is so short - take a look:
Statement 1:
Reason for statement 1: Given.

Reason for statement 2: If two angles form a right angle, then they're complementary (definition of complementary).

Reason for statement 3: Given.

Reason for statement 4: Substitution Property (statements 2 and 3; angle 2 replaces angle 1).

And for the final segment of the program, here's a related proof, OSIM (Oh Shoot, It's Monday):

This is another incredibly short proof that doesn't call for a game plan.
Statement 1:
Reason for statement 1: Given.

Reason for statement 2: A midpoint divides a segment into two congruent segments.

Reason for statement 3: Given.

Reason for statement 4: Transitive Property (for four segments; statements 2 and 3).

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The Transitive and Substitution Properties - dummies

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You're probably already familiar with the Transitive Property and the Substitution Property from algebra. If a = b and b = c, then a = c, right? That's transitivity. And if a = b and b < c, then a < c. That's substitution. Easy enough.

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I chose this because it has all the information stated clearly about this section

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I chose this because it has all the information stated clearly about this lesson

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I chose this website because I think it explains this learning target very well

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By Mark Ryan

Complementary angles are two angles that add up to 90°, or a right angle; two supplementary angles add up to 180°, or a straight angle. These angles aren't the most exciting things in geometry, but you have to be able to spot them in a diagram and know how to use the related theorems in proofs.

You use the theorems listed here for complementary angles:

Complements of the same angle are congruent. If two angles are each complementary to a third angle, then they're congruent to each other. (Note that this theorem involves three total angles.)

Complements of congruent angles are congruent. If two angles are complementary to two other congruent angles, then they're congruent. (This theorem involves four total angles.)

The following examples show how incredibly simple the logic of these two theorems is.
Complements of the Same Angle
Given: Diagram as shown
Complements of Congruent Angles
Given: Diagram as shown

Note: The logic shown in these two figures works the same when you don't know the size of the given angles

And here are the two theorems about supplementary angles that work exactly the same way as the two complementary angle theorems:

*Supplements of the same angle are congruent. If two angles are each supplementary to a third angle, then they're congruent to each other. (This is the three-angle version.)

*Supplements of congruent angles are congruent. If two angles are supplementary to two other congruent angles, then they're congruent. (This is the four-angle version.)

The previous four theorems about complementary and supplementary angles come in pairs: One of the theorems involves three segments or angles, and the other, which is based on the same idea, involves four segments or angles. When doing a proof, note whether the relevant part of the proof diagram contains three or four segments or angles to determine whether to use the three- or four-object version of the appropriate theorem.

Take a look at one of the complementary-angle theorems and one of the supplementary-angle theorems in action:

Before trying to write out a formal, two-column proof, it's often a good idea to think through a seat-of-the-pants argument about why the prove statement has to be true. Think of this argument as a game plan. Game plans are especially helpful for longer proofs, because without a plan, you might get lost in the middle of the proof.

When working through a game plan, you may find it helpful to make up arbitrary sizes for segments and angles in the proof. You can do this for segments and angles in the givens and, sometimes, for unmentioned segments and angles. You should not, however, make up sizes for things that you're trying to show are congruent.

Game plan: In this proof, for example, you might say to yourself, "Let's see. . . . Because of the given perpendicular segments, you have two right angles.

That's it.

Here's the formal proof (each statement is followed by the reason).
Statement 1:
Reason for statement 1: Given. (Why would they tell you this? See reason 2.)

Reason for statement 2: If segments are perpendicular, then they form right angles (definition of perpendicular).

Reason for statement 3: If two angles form a right triangle, then they're complementary (definition of complementary angles).

Reason for statement 4: Given.

Reason for statement 5: If two angles are complementary to two other congruent angles, then they're congruent.

Reason for statement 6: This is assumed from the diagram.

Reason for statement 7: If two angles form a straight angle, then they're supplementary (definition of supplementary angles).

Reason for statement 8: If two angles are supplementary to two other congruent angles, then they're congruent.

Note: Depending on where your geometry teacher falls on the loose-to-rigorous scale, he or she might allow you to omit a step like step 6 in this proof because it's so simple and obvious. Many teachers begin the first semester insisting that every little step be included, but then, as the semester progresses, they loosen up a bit and let you skip some of the simplest steps.

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How to Prove Angles Are Complementary or Supplementary - dummies

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Complementary angles are two angles that add up to 90°, or a right angle; two supplementary angles add up to 180°, or a straight angle. These angles aren't the most exciting things in geometry, but you have to be able to spot them in a diagram and know how to use the related theorems in ...

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I chose this website because i feel like it easily explains how to follow five step procedures

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I chose this website because i feel it explains how to make clear proofs very well

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Modern mathematical proofs that use variables are clear and concise for people who can read them. But often, when the proofs are short, they can be paraphrased and restated in everyday language. There is also a third way to present a proof, which has been used in antiquity: A proof given through a "generic" example.

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I chose this website because it simply defines complementary and supplementary angles

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Complementary or Supplementary Angles

Two angles are complementary if the sum of their angles equals 90 o.
If one angle is known, its complementary angle can be found by subtracting the measure of its angle from 90 o.

Example: What is the complementary angle of 43 o?
Solution: 90 o - 43 o = 47 o

Two angles are supplementary if the sum of their angles equals 180 o.
If one angle is known, its supplementary angle can be found by subtracting the measure of its angle from 180 o.

Example: What is the supplementary angle of 143 o?
Solution: 180 o - 143 o = 37 o

Are the Angles Complementary, Supplementary or Neither?

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Complementary or Supplementary Angles Two angles are complementary if the sum of their angles equals 90 o.If one angle is known, its complementary angle can be found by subtracting the measure of its angle from 90 o.

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I chose this picture for this learning target because it clearly shows the concept of perpindicularity