Download Bulletin Parallel and perpendicular - Qu 1

Agenda 12/12/2016
Bulletin
Parallel and perpendicular - Qu 1 - 5 on second
page of review packet, page 14, 3A workbook
Proofs with Similar Triangles (p. 41-42 in 3A
workbook)
Review Semester 1 (1)
1. What is the slope of a line parallel to the line
y = -5/2 x + 2
a) Which part of the equation tells you the
slope? y = mx + b, coefficient in front of x.
b) What do you know about the slope of lines
that are parallel to each other? They are the
same slope. (They will never cross.)
c) Slope of line parallel to the line above would
be -5/2
2. Write an equation for a line that
would be perpendicular to the line
defined by y = 3x + 2
a) What does perpendicular mean? 2 lines ar
perpendicular if they cross at an angle of 90
degrees.
b) What is the product of the slopes of 2
perpendicular lines? -1 (the slopes of 2
perpendicular lines are inverse reciprocals
of each other.
2. Write an equation for a line that
would be perpendicular to the line
defined by y = 3x + 2
c) Inverse reciprocal of 3 is -⅓
(3)(- ⅓) = -1
Possible equation
y = -⅓ x + 4
3. ABCD is a square
A(3, 6)
B(3, 3)
C(6, 3) and D (6, 6)
Side lengths
(3, 3) to (6, 3) x increases by 3, y stays same.
(3, 3) to (6, 6) x stays same, y increases by 3.
A to D is a horizontal line, 3 units long
AB is a vertical line segment 3 units long
4. Which equation is parallel to
y = -2x +4 and passes through (-2,6)
a) y = ½ x + 8 opposite reciprocal slopes ,
(-2)(½) = -1, perpendicular lines
b) y = ½ x + 5 same as a
c) y = -2x +10 parallel, same slope
When x is -2 is y = 6?
y = (-2)(-2) + 10 = 4 + 10 = 14 NOT 6
4. Which equation is parallel to
y = -2x +4 and passes through (-2,6)
d) y = -2x +2 parallel, same slope
When x is -2 is y = 6?
y = (-2)(-2) + 2 = 4 + 2 = 6
This is the equation of a line parallel to
y = -2x +2 that passes through (-2, 6)
5. Write in slope-intercept form the
equation of the line passing through
the point (4, 2) and perpendicular to
the line y = -3x + 3
5. Write in slope-intercept form the
equation of the line passing through
the point (4, 2) and perpendicular to
the line y = -3x + 3
Slope of new line = ⅓
⅓ (-3) = -1
5. Write in slope-intercept form the
equation of the line passing through
the point (4, 2) and perpendicular to
the line y = -3x + 3
y = ⅓ x + b,
to find y intercept use point
given
2 = ⅓ (4) + b
b = 2 - (4/3) = ⅔
y=⅓x+⅔
Unit 3A Workbook - page 14
13. Are the two lines parallel,
perpendicular or neither?
Parallel - same slope
Perpendicular slope x slope = -1
(slopes are negative reciprocals of each other)
a) y = 11 x + 8
5
-11x + 5y = 45
y = 11x + 45
Parallel
5
5
13. Are the two lines parallel,
perpendicular or neither.
Parallel - same slope
Perpendicular slope x slope = -1
(slopes are negative reciprocals of each other)
b) y = -1 x + 7
y = 2x + 5
2
-1 (2) = -1
perpendicular
2
13. Are the two lines parallel,
perpendicular or neither.
Parallel - same slope
Perpendicular slope x slope = -1
(slopes are negative reciprocals of each other)
c) y = 2 x
3
y = 3x + 5
2
Neither
2 (3) = 1
3 2
14. Is the quadrilateral ABCD a
rectangle?
mAB = 4/1
perpendicular
mAD = -¼
∠DAB = 90°
mBC = -¼ AB same slope as CD
mCD = 4/1 BC same slope as AD A
Quadrilateral with 90 degree angles
and 2 pairs of parallel sides, is a
rectangle.
B
C
D
14. Cont.
AB = √12 + 42 = √1 + 16 = √17
B
BC = √12 + 42 = √1 + 16 = √17
CD = √12 + 42 = √1 + 16 = √17
AD = √12 + 42 = √1 + 16 = √17
C
A
D
14. Cont.
All 4 sides are the same length, so
this is a square (which is a special
type of rectangle.)
Perimeter = √17 + √17 +√17+√17 A
= 4√17 (Exact answer)
Exact answer - means leave the sq.
rt. - once use calculator get approx.
B
C
D
14. Cont.
Area = length x width
B
Area = √17 ( √17)
= 17 sq. units
C
A
D
15.
25
5x
If the 2 lines are
parallel and cut by a
transversal,
consecutive interior
angles are
supplementary.
25 + 5x = 180
5x = 155
x = 31
16.
100
2x + 10
If the 2 lines are
parallel and cut by a
transversal, alternate
exterior angles are
congruent.
100 = 2x + 10
90 = 2x
x = 45
17.
120°
3x + 3
9y + 3
If this is a
parallelogram, alternate
angles are congruent.
120 = 9y+ 3
117 = 9y
y = 13
Consecutive angles are supplementary
120 + 3x + 3 = 180
3x =57
x = 19
Constructions (with straight edge and
compass)
For help with constructions click on this link for video clips for each one:
http://www.mathopenref.com/tocs/constructionstoc.html
Copying a line segment.
Constructions (with straight edge and
compass)
Copying an angle
Constructions (with straight edge and
compass)
Perpendicular bisector of a line segment (also
finds the midpoint of a line segment).
Constructions (with straight edge and
compass)
Bisect an angle
Construction of a line parallel to
another, by copying an angle
Geometry Review Semester 1 (2)
1. a) The construction shows that ray GE
bisects the angle FGH.
2. c) David is constructing a line that is parallel
to RS that goes through Q.
Geometry Review Semester 1 (2)
3. Lines m and n are parallel and intersected by
a transversal , p. What is the value of x?
For parallel lines cut by a transversal, alternate
exterior angles are congruent (can work out from
vertical angles are congruent, and corresponding
angles are congruent)
4x + 5 = 6x - 45
50 = 2x
x = 25
Geometry Review Semester 1 (2)
4. Lines l and m are parallel and intersected by a
transversal, t. What is the value of x?
For parallel lines cut by a transversal, same side
exterior angles are supplementary (linear pair,
supplementary, then corresponding angles are
congruent)
4x + 28 + 6x + 2 = 180
10x + 30 = 180
10x = 150
x = 15
Geometry Review Semester 1 (2)
5. Which of the following conditions will
guarantee that line m is parallel to line l in the
diagram below?
Need angle 1 and 2 to be supplementary for the
lines m and n to be parallel.
c) measure of angle 1 = 30 degrees, and
measure of angle 2 = 150 degrees.
Unit 3A workbook, page 41
1. Prove the following relationships.
a) Given TV || XP
Statement
Prove:△TVY
Reason
△PXY
1. Prove the following relationships.
P. 41
a) Given TV || XP
Prove:△TVY
Statement
Reason
1. TV || XP
1. Given
2.
2.
3.
3.
4. △TVY
△PXY
4. AA, SAS or SSS
△PXY
1. Prove the following relationships.
P. 41
a) Given TV || XP
Prove:△TVY
△PXY
Statement
Reason
1. TV || XP
1. Given
2. TYV = PYX
2. When lines cross, vertical angles are
congruent.
3. TVY =
PXY
3. For parallel lines cut by a transversal,
alternate interior angles are congruent.
4. △TVY
△PXY
4. Angle Angle Similarity postulate
1. Prove the following relationships.
P. 41
b) Given G = HIJ
Statement
1.
Angle G is congruent to angle
HIJ
Prove:△FGH
Reason
1.
Given
△JIH
1. Prove the following relationships.
P. 41
b) Given G =
HIJ
Statement
Prove:△FGH
△JIH
Reason
1. Angle G is congruent 1. Given
to angle HIJ
2. Angle H is
congruent to angle H
2. Reflexive property
3. Triangle FGH is
similar ot triangle JIH
3. Angle Angle Similarity postulate
c) Given ABCD is a parallelogram
Prove:△AHE
△FHG
Statement
Reason
1. ABCD is a parallelogram
1. Given
2. AB||DC
2. Definition of a parallelogram
3. Angle EAH is congruent to
angle GFA
3. Parallel lines cut by transversal,
alternate interior angles are congruent
4. Angle AHE is congruent to
angle FHG
4. When lines cross, vertical angles are
congruent.
5. Triangle AHE is similar to
5. Angle Angle Similarity postulate
d) Given angle 1 is congruent to angle 2
and AC= AE
Prove:△CBD
△EFD
Statement
Reason
1. Angle 1 is congruent to
angle 2 and AC = AE
1. Given
2. Angle C is congruent to
angle E
2. Isosceles triangle theorem
3. △CBD
3. Angle Angle Similarity postulate
△EFD
d) Given UW||VX
Prove:△TUW
△TVX
Statement
Reason
1. UW || VX
1. Given
2. Angle T is congruent to
angle T
2. Reflexive property
3. Angle TUW is
congruent to angle TVX
3. For parallel lines, cut by a
transversal, corresponding angles are
congruent.
4. △TUW
4. Angle Angle Similarity postulate
△TVX
e) Given LN = 4cm, KL = 5cm, LY = 12cm,
LH = 15cm
Prove:△KLN
△HLY
Statement
Reason
1. LN = 4, KL = 5, LY = 12, LH
= 15
1. Given
2. 5 = 4 = 1
15 12 3
2. Ratio of sides (corresponding sides
are proportional)
3. Angle KLN is
congruent to angle HLY
3. For lines that cross, vertical angles
are congruent
2a) Given PQ/ PT = PR/PS
Prove: Angle Q = angle T
Statement
Reason
1. PQ/PT = PR/PS
1. Given
2. Angle QPR is congruent to
angle TPS
2. Vertical angles are congruent
3. △QPR
3. Side Angle Side similarity postulate
△TPS
4. Angle Q is congruent
to angle T
4. Corresponding angles of similar
triangles are congruent.
2b) Given angle U is congruent to angle
ZTW
Prove: UV/TZ = WU/WT
Statement
Reason
1. Angle U is congruent to
angle ZTW
1. Given
2. Angle
W is congruent
to angle W
3. △TWZ
△UWV
4. UV/TZ = WU/WT
2. Reflexive property
3. Angle Angle similarity postulate
4. Corresponding sides of similar
triangles are proportional.
2c) Given AB || DC
Prove: GA GC = GB GH
Statement
Reason
1. AB||DC
1. Given
2. Angle BAG is congruent to angle
CHG and angle B is congruent to
angle GCH
3. △BAG
△CHG
2. For parallel lines cut by a
transversal, alternate interior anlges
are congruent
3. Angle Angle similarity postulate
4. GA = GB
GH GC
4. Corresponding sides of similar
triangles are proportional.
5. GA GC = GB GH
5. Cross multiplication
2d) Given FG = 7cm, GH = 8cm, FH = 10cm, CB = 14cm, BA =
16cm, AC = 20cm
Prove: Angle F = angle C
Statement
Reason
1. FG = 7, GH = 8, FH = 10, CB
= 14, BA = 16, AC = 20
1. Given
2. 7/14 = 8/16 = 10/20
2. Ratio of sides is proportional
3. △FGH
3. Side Side Side similarity postulate
△CAB
4. Angle F = angle C
4. Corresponding angles of similar
triangles are congruent.
2e) Given TH = TL
TJ TK
Prove: HL || KJ
Statement
Reason
1. TH/TJ = TL/TK
1. Given
2. Angle HTL = angle JTK
2. Vertical angles are congruent
3. △HTL
3. Side Angle Side similarity postulate
△JTK
4. Angle L = angle K
4. Corresponding angles of similar
triangles are congruent.
5. HL || KJ
5. Alternate interior angles are
congruent for parallel lines cut by a
transversal
2f) Given Angle TUW = angle TVX
Prove: TU/TV = TW/TX
Statement
Reason
1. Angle TUW is congruent to
angle TVX
1. Given
2. Angle T is congruent to angle T
2. Reflexive property
3. △TUW
3. Angle Angle similarity postulate
△TVX
4. TU/TV = TW/TX
4. Corresponding sides of similar
triangles are proportional.