Download OTHER ANGLES FROM PARALLEL LINES When given two parallel

FOUNDATIONS OF MATH 11
Ch. 2 – Day 2: ANGLES FORMED BY PARALLEL LINES (Part 1)
OTHER ANGLES FROM PARALLEL LINES
When given two parallel lines and a transversal,
we defined angles.
• interior angles are inside the parallel lines; c, d,
e, and f are interior angles.
line 1
line 2
• exterior angles are outside the parallel lines; a,
b, g, and h are interior angles.
a b
d c
e f
h g
line 3
• corresponding angles are pairs of angles, an interior angle and the nonadjacent exterior on the same side of the transversal; their positions at the
intersection points correspond to each other. a and e, b and f, c and g, d and h
are pairs of corresponding angles. Two properties of corresponding angles are
“if two lines are parallel, then the corresponding angles are congruent”.
“if corresponding angles are congruent, then the lines are parallel”.
We can add to our list of angles when given two parallel lines and a transversal.
• alternate exterior angles are a pair of exterior angles on alternate sides of the
transversal; alternate refers to the transversal and exterior refers to the
parallel lines. a and g, b and h are pairs of alternate exterior angles. We can
prove another property and its converse to be true.
If two lines are parallel, then the alternate exterior angles are congruent.
statement
line 1  line 2
b = f
f = h
b = h
reason
given
corresponding angles
vertically opposite angles
transitive property
It has been proven that alternate exterior angles, b and h, must be congruent.
If alternate exterior angles are congruent, then the two lines are parallel.
Reverse the above proof to show this converse is also true.
Ch. 2: Day 2 notes – Angles Formed by Parallel Lines (Part 1)
• alternate interior angles are a pair of interior angles
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line 1
a b
d c
on alternate sides of the transversal; alternate refers
to the transversal and interior refers to the parallel
line 2
lines. c and e, d and f are pairs of alternate interior
angles. Another angle property from parallel lines
e f
h g
line 3
and its converse will be true.
If two lines are parallel, then the alternate interior angles are congruent.
If alternate interior angles are congruent, then the two lines are parallel.
• interior angles on the same side of the transversal; c and f, d and e. Another
angle property from parallel lines and its converse will be true.
If two lines are parallel, then the interior angles on the same side of the
transversal are supplementary.
statement
line 1  line 2
b = f
b + c = 180°
b = 180° − c
f = 180° − c
f + c = 180°
reason
given
corresponding angles
angles on a line are supplementary
subtraction property for equations
transitive property
addition property for equations
It has been have proven that the interior angles on the same side of the
transversal, b and h, must be supplementary.
If interior angles on the same side of the transversal are supplementary, then
the two lines are parallel.
These angle properties can be used to calculate the measures of other angles.
FOUNDATIONS OF MATH 11
Ch. 2 – Day 3: GUIDED PROOFS
A guided proof is a partially completed proof; the goal is to finish the proof.
exercise: Given:
L
JG = KG
∠KJG = 30°
Prove:
F
A
∠LFB = 60°
B
D
K
G
AB  CD
C
statement
JG = KG
J
reason
given
∠KJG = 30°
∠KJG = ∠JKG
∠JKG = 30°
transitive property
∠JGK =
°
∠FGD =
°
∠LFB = 60°
∠ FGD = ∠LFB
AB  CD
Q.E.D.
M
Ch. 2: Day 3 notes – Guided Proofs
exercise: Given:
Page 2 of 2
AB  CD
L
∠KJG = ∠JKG
G
C
∠KJG =
reason
given
°
∠KJG = ∠JKG
∠JKG =
°
∠JGK =
°
AB  CD
D
K
Prove: ∠AFK = 80°
statement
∠CJK = 140°
F
A
∠CJK = 140°
B
given
∠AFK + ∠JGK = 180°
∠AFK + 100° = 180°
∠AFK = 80°
Q.E.D.
J
M
FOUNDATIONS OF MATH 11
Ch. 2 – Day 4: ANGLES FORMED BY PARALLEL LINES (Part 2)
STRATEGIES FOR WRITING GEOMETRY PROOFS
• Understand the given information and the conclusion that is to be proven.
• Draw a diagram from the given information and mark on it all the given
information and indicate what is to be proven.
• Consider the various deductions that can be made from the given information.
• Consider the various ways the conclusion can be proven.
example: Given:
QP ⊥ QR
Q
P
R
S
QR ⊥ RS
QR  PS
Prove: QPSR is a parallelogram
Think, plan! What is a parallelogram? What are its properties?
It is two pairs of parallel sides, we already have one pair.
Can we show the other pair of sides are parallel? How?
Corresponding angles, alternate exterior angles, alternate interior
angles, or interior angles on the same side of the transversal.
The right angles are interior angles to sides QP and RS on the same
side of the transversal QR.
Reverse these "statements" for the proof.
statement
reason
QP ⊥ QR
∠PQR = 90°
given
definition of perpendicular
QR ⊥ RS
∠SRQ = 90°
given
definition of perpendicular
∠PQR and ∠SRQ are
supplementary
definition of supplementary
QP  RS
Interior angles on the same side of
the transversal
QR  PS
given
QPSR is a parallelogram
definition of parallelograms
Q.E.D.
Ch. 2: Day 4 notes – Angles Formed by Parallel Lines (Part 2)
exercise: Given:
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QP  RS
P
Q
RT bisects ∠QRS
QU bisects ∠PQR
Prove:
QU  RT
statement
U
T
S
reason
R