Geometry
Common Statements and Reasons
in Proofs
Some “Given” statements are very common in congruent triangle proofs. You should know what
each statement allows you to include in your proof.
Complete this list by stating what you know results when each item listed is stated in a
proof (in other words, “what” does the item listed do?)
Bisectors:
If you were given a “BISECTOR” what does it do?
What is the result of knowing that something was bisected:
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Use this example to demonstrate what you know happens;
1. IF KH bisects ∠GKJ , THEN what happens?
2. IF KH bisects ∠GHJ ∠GKJ , THEN what happens?
G
H
K
J
Mid Point:
If you were given a “MID POINT” what does it do?
What is the result of knowing that a segment has a mid point:
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Use this example to demonstrate what you know happens;
1. IF C is the mid point, THEN what happens to BE?
B
2. IF C is the mid point, THEN what happens to AD?
A
3. IF BE bisects AD , THEN what happens to AD?
D
C
E
Parallel Lines:
If you were given a “Parallel lines” what do they do ?
What is the result of knowing that two lines are parallel and cut by a transversal? :
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_______________________________________________________________________
If you knew that
QT RS , what would be the result?
If you knew that
TS QR , what would be the result?
Q
R
S
T
Perpendicular Lines:
If you were given a “Perpendicular lines” what do they do ?
What results from two lines that are perpendicular ?:
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If you knew that QT ⊥ QR , what would be the result?
Q
R
If you knew that TS ⊥ RS , what would be the result?
T
S
Bisectors:
If an angle or a segment is bisected, it is divided into two congruent segments. You must
include a statement that the two angles or segments are congruent in order to use it to prove the
triangles congruent. (The reason is definition of a bisector).
Midpoint:
The midpoint of a segment divides it into two congruent segments. You must list the
congruent segments as their own step (reason: definition of a midpoint.)
Parallel Lines:
If two lines are parallel, the corresponding angles are congruent and the alternate interior
angles are congruent. You must include a statement that the angles are congruent (reason:
alternate interior angles of parallel lines are congruent, or corresponding angles of parallel lines
are congruent) if you want to use the angles in your proof.
Perpendicular Lines:
Perpendicular lines tell you that you have right angles. You must include a statement
that the angles are right angles (reason: definition of perpendicular lines) separate from the
given statement. If you want to use HL to prove the triangles congruent, there needs to be
another statement that says the triangles are right triangles. (Reason: definition of a right
triangle.)
Do these…
Tell what statements, and their reasons, can follow from the following “Given” statements.
(Do not write a whole proof – just what you can conclude from each statement given.)
There might be more than one conclusion you can draw from a particular statement.
1. Given: SQ ⊥ PR
S
2. Given: Q is the midpoint of PR
P
3. Given: SQ bisects ∠PSR
Q
R