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My weakness for this chapter is that there are a lot of theorems to remember. It is difficult to remember the acceptable wording for the theorems and when to use them. It is also difficult to know what I need to include in the proof and what is pointless information to add. Doing homework that pulls multiple theorems into one problem has been very helpful. It has been helpful having the answers on hand so if I am stuck I can work through the problem by using the answer key.

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LT 10: I can apply the substitution property. 

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LT 10: I can apply the substitution property. 

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Substitution is used to plug in for values.
In geometry, the substitution property is used when no variables are present, when sums or equations are referenced (even not explicitly).

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LT 5: I can prove angles congruent by using complementary and supplementary angle theorems. 

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LT 5: I can prove angles congruent by using complementary and supplementary angle theorems. 

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Theorem: If two angles are comp (or supp) to the same angle, then they are congruent.
Must have:
~ 2 pairs of comp (or supp) angles
~3 angles involved/mentioned
Theorem: If two angles are comp (or supp) to congruent angles, then they are congruent
Must have:
~2 pairs of comp (or supp angles
~4 mentioned angles
~1 pair of congruent angles

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LT 9: I can apply the transitive properties of angles and segments. 

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LT 9: I can apply the transitive properties of angles and segments. 

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Transitive Property: If two angles are congruent to the same angle then they are congruent

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LT 8: I can apply the multiplication and division properties of segments and angles. 

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LT 8: I can apply the multiplication and division properties of segments and angles. 

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Multiplication Property: If angles or segments are congruent, then their like doubles (or triples) are congruent.
Small to Big
Division Property: If angles or segments are congruent, then their like halves/ thirds are congruent.
Big to Small

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LT 6: I can apply the addition properties of segments and angles. 

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LT 6: I can apply the addition properties of segments and angles. 

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If congruent angles (or segments) are added to congruent angles (or segments) then the sums are congruent.

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LT 7: I can apply the the subtraction properties of segments and angles. 

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LT 7: I can apply the the subtraction properties of segments and angles. 

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If congruent angles (or segments), subtracted from congruent angles (or segments) then the differences are congruent.

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By Mark Ryan

There are four subtraction theorems you can use in geometry proofs: two are for segments and two are for angles. Each of these corresponds to one of the addition theorems.

Here are the subtraction theorems for three segments and three angles (abbreviated as segment subtraction, angle subtraction, or just subtraction):

Segment subtraction (three total segments): If a segment is subtracted from two congruent segments, then the differences are congruent.

Angle subtraction (three total angles): If an angle is subtracted from two congruent angles, then the differences are congruent.

Check out the above figure, which provides the visual aids for these two theorems.

Last but not least, here are the subtraction theorems for four segments and for four angles (abbreviated just like the subtraction theorems for three things):

Segment subtraction (four total segments): If two congruent segments are subtracted from two other congruent segments, then the differences are congruent.

Angle subtraction (four total angles): If two congruent angles are subtracted from two other congruent angles, then the differences are congruent.

The above figure illustrates these two theorems.

Before reading the formal solution of the next proof, try to think through your own game plan or commonsense argument about why the prove statement has to be true.

Statement 1:
Reason for statement 1: Given.

Reason for statement 2: If an angle (angle QUR) is subtracted from two congruent angles (angles PUR and SUQ), then the differences are congruent.
Statement 3:
Reason for statement 3: Given.
Statement 4:
Reason for statement 4: If a ray bisects an angle, then it divides it into two congruent angles (definition of bisect).

Reason for statement 5: If two congruent angles (the angles from statement 2) are added to two other congruent angles (the ones from statement 4), then the sums are congruent.

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Using Subtraction Theorems in Proofs - dummies

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There are four subtraction theorems you can use in geometry proofs: two are for segments and two are for angles. Each of these corresponds to one of the addition theorems. Here are the subtraction theorems for three segments and three angles (abbreviated as segment subtraction, angle subtraction, or just subtraction): Segment subtraction (three total segments): ...

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LT 12: I can recognize vertical angles. 

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LT 12: I can recognize vertical angles. 

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Two angles are vertical angles if the rays forming the sides of one and the rays forming the sides of the other are opposite rays.
Vertical angles are congruent.

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Two rays that share a common endpoint and extend in different directions.

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LT 3: I can recognize complementary and supplementary angles. 

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LT 3: I can recognize complementary and supplementary angles. 

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Complementary Angles: If two (adjacent) angles sum to 90 degrees (right angle) then they are complementary.
90-x
Supplementary Angles: If two (adjacent) angles sum to 180 degrees (straight angle) then they are supplementary.
180-x

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LT 2: I understand the concept of perpendicularity. 

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LT 2: I understand the concept of perpendicularity. 

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Definition: If two lines (rays, segments) intersect at right angles, then the lines are perpendicular.
Converse: If lines are perpendicular, then they intersect at right angles.
DO NOT assume perpendicularity!!!
In the picture, line A is perpendicular to line B.