angle pair relationships warmup notes on parallel lines parallel lines construction activity
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Angle Pair Relationships Warmup Notes on Parallel Lines Parallel Lines Construction Activity Parallel Lines and Transversals ( kuta ) Exit Quiz. Parallel lines – Two lines are parallel lines if they are coplanar and do not intersect. - PowerPoint PPT PresentationTRANSCRIPT
Geometry First
Quarter G.CO.9
G.CO.11G.CO.10
Block 28: Daily Activities
Angle Pair Relationships WarmupNotes on Parallel Lines
Parallel Lines Construction ActivityParallel Lines and Transversals (kuta)
Exit Quiz
Complementary Angles Add to 90
DegreesRemember: Complementary Angles
Do not Have to Be Adjacent
Supplementary Angles Add to 180 Degrees
Remember: Supplementary Angles Do not Have to Be Adjacent
Vertical Angles are Congruent
Linear Pair Angles are
Supplementary
Parallel Lines
Obj: Be able to prove and use results about parallel lines and transversals.
Definitions
1. Parallel lines – Two lines are parallel lines if they are coplanar and do not intersect.
2. Skew lines—Lines that do not intersect and are not coplanar.
3. Parallel planes—two planes that do not intersect.
Identifying Relationships in Space
1) Think of each segment in the diagram. Which appear to fit the description?a. Parallel to AB and contains Db. Perpendicular to AB and
contains D.c. Skew to AB and contains D.
d. Name the plane(s) that contains D and appear to be parallel to plane ABE
A
B
D
C
F
E H
G
CD
AD
, ,DG DH DE
DCH
Postulate 13: Parallel Postulate
• If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.
P
l
Exterior Angles
1
32
4
65
7 8
8 7, 2, ,1
Interior Angles 6 5, 4, ,3
Consecutive Interior Angles or Same Side Interior 6 and 4
5 and 3
Alternate Exterior Angles 7 and 2
8 and 1
Alternate Interior Angles
Corresponding Angles
5 and 46 and 3
8 and 4 7, and 36 and 2 5, and 1
Identifying Angles Formed by Transversals
Parallel Lines Construction Activity
1. Using a straightedge, draw two nonparallel intersecting lines m and n.
2. At point A, construct a line parallel to line m, by copying the angle formed by the intersection of lines m and n.
m
n
m
n
A
A
3. Measure all eight angles formed by the parallel lines and transversal.
Corresponding Angles
1 2 If ||, then corr 's .
l m
12
If corr 's , then ||.
If 2 parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
Converse also holds true
Alternate Interior Angles
lm
1 2
If ||, then alt int 's .
If alt int 's , then ||.
1 2
If 2 || lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
Converse also holds true
Consecutive Interior Angles(Same Side Interior Angles)
lm
1 2
If ||, then con int 's supp. orIf ||, then ss int 's supp.
1 2 180m m If con int 's supp, then ||. orIf ss int 's supp, then ||.
Converse also holds true
If 2 || lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.
Alternate Exterior Angles
lm
1
2
If ||, then alt ext 's .
If alt ext 's , then ||. 1 2
If 2 || lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
Converse also holds true
If two lines are parallel to the same line, then they are parallel to each other.
p q rIf p║q and q║r, then p║r.
1. Find the measure of angle a and b if t // m. Justify your reasoning using transformations.
t
m
125°a
b
a = 125° by translating the given angle along the transversalb = 125° by rotating the given angle 180°
1. Find the measure of angle a and b if t // m. Justify your reasoning using transformations.
t
m
65°a
bThe given angle and b form a linear pair, therefore b
= 115°. Rotate b 180° and translate along the transversal
onto a. Therefore a = 115°
Exit Quiz: Parallel Lines
Identify the type of angle pair that is given.
Find the value of .3. 4.
x
Lines and Transversals WarmupReview of Parallel Lines
Parallel Lines Quiz
l
m
1 2
3
57
4
68
Correctly match the following and include the transformation that maps one angle onto its pair._________ 1. Alternate Interior Angles A. ∠4 and ∠6_________ 2. Alternate Exterior Angles B. ∠1 and ∠ 5________ 3. Corresponding Angles C. ∠2 and ∠7_________ 4. Consecutive Interior Angles D. ∠1 and ∠4_________ 5. Vertical Angles E. ∠3 and ∠6
Translate ∠3 along the transversal onto ∠7. Then rotate 180° onto ∠6Rotate ∠2 180° and then translate along the transversal onto ∠7Translate ∠1 along the transversal onto ∠5Not a transformation
Lines and Transversals Warmup
E
C
B
A
DRotate ∠1 180°
Alternate: TransversalExterior: Parallel Lines
Alternate Exterior Angles are Congruent
Same Position with Respects to Transversal and Parallel Lines
Corresponding Angles are Congruent
Same Side: TransversalInterior: Parallel Lines
Same Side Interior Angles are
Supplementary
Parallel Lines and Transversals WarmupSpecial Segments in Triangles Notes
Segments in Triangles WorksheetExit Quiz
Objective: 1) Be able to identify the median of a triangle.2) Be able to apply the Mid-segment Theorem.
3) Be able to use triangle measurements to find the longest and shortest side.
Special Segments in Triangles
Figure Picture DefinitionThe midsegment of a triangle is parallel to the side it does not touch and is half as long.
B
D E
A C
2DE AC
Midsegment Construction1. Using a straight edge, draw a triangle. Label the verticesA, B, and C.
2. Using a compass, construct the midpoint of and .Label the midpoints D and E, respectively.
3. What do you notice abo
AB CB
ut the relationship between and ?DE AC
Example
1) Given: JK and KL are midsegments. Find JK and AB.
10
6
J
C
K
B
A L
5JK 12AB
Example2) Find x.
73 x
73 x
67 x
2 3 7 7 6
6 14 78
6
x x
xx
x
3 7x
Figure Picture Definition IntersectionA segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.
The concurrence of the medians is called the centroid.
1. Using a straight edge, draw a triangle. Label the verticesA, B, and C.2. Using a compass, construct the midpoint of all 3 sides.3. Using a straightedge, draw a segment from each midpoint to its opposite vertice.3. What do you notice about the three segments?
28
Concurrency of Medians of a Triangle
The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side.
If P is the centroid of ∆ABC, then
AP = 2/3 AD, BP = 2/3 BF, and CP = 2/3 CE
PE
D
F
B
A
C
The longest side of a triangle is always opposite the largest angle and the smallest side is always opposite the smallest angle.
Exit Quiz: Special Segments in Triangles
_____5. QT_____6. QR
_____7.
Rotations:Computers (Pullout)
Special Segments in Triangles and Examining Midsegments
Worksheets
Algebra of Triangles Worksheet WarmupQuadrilateral Activity
Quadrilateral Activity
• Students will first cut out their set of triangles.• Mark the triangle on both sides if there are congruent
sides or angles. • Using 2 or more triangles, they must transform the
original triangles to form quadrilaterals• Then glue these quadrilaterals onto the butcher paper. • Next to the quadrilateral write down any
characteristics that are displayed on the diagrams.• Present your findings.
Quadrilateral ActivityNotes on Quadrilaterals
Who Want to Be a Quadrilateral Millionaire
QuadrilateralsObjectives: Be able to discover properties of
quadrilaterals.
ParallelogramsA parallelogram is a quadrilateral with both pairs of opposite sides parallel.
When you mark diagrams of quadrilaterals, use matching arrowheads to indicate which sides are parallel. For example, in the diagram above, PQ RS and QR SP. The ║ ║symbol PQRS is read “parallelogram PQRS.”
REMEMBER, If two lines are parallel, then:1) Alternate interior angles are
congruent2) Alternate exterior angles are congruent3) Corresponding angles are
congruent4) Same-side interior angles are supplementary.
P
RQ
S
Thm 6.2
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
Thm 6.3
If a quadrilateral is a parallelogram, then the opposite angles are congruent.
Thm 6.4
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
Thm 6.5
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
Theorems about Parallelograms
PQ RS
SP QR
P
R
S
Q
P
P
P
Q
Q
Q
S
S
S
R
R
R
P RQ S
180
180
180
180
m P m Q
m Q m R
m R m S
m S m P
QM SM
PM RM
M
Example
1) Find the value of each variable in the parallelogram below.
11y
2) is a parallelogram. Find the value of x.WXYZ
W Z
X Y 4 9x
3 18x
Example:
3) is a parallelogram. Find the value of x.PQRS
Q
R
S
Example:
1203xQ
P
Thm 6.6 If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram.
Thm 6.7 If both pairs of opposite angles are congruent, then the quadrilateral is a parallelogram.
Thm 6.8 If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.
Thm 6.9 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Theorems about ParallelogramsP
R
S
Q
P
P
P
Q
Q
Q
S
S
S
R
R
R
Thm 6.10
If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram
Theorems about ParallelogramsP
R
S
Q
SummaryProving Quadrilateral are Parallelograms
Show that both pairs of opposite sides are parallel
Show that both pairs of opposite sides are congruent
Show that both pairs of opposite angles are congruent
Show that one angle is supplementary to both consecutive angles
Show that the diagonals bisect each other
Show that one pair of opposite sides are congruent and parallel
Quadrilaterals
A parallelogram with four congruent sides.
A parallelogram with four right angles.
A parallelogram with four congruent sides, and four right angles.
Corollaries– Rhombus Corollary: A quadrilateral is a
rhombus if and only if it has four congruent sides.
– Rectangle Corollary: A quadrilateral is a rectangle if and only if it has four right angles.
– Square Corollary: A quadrilateral is a square if and only if it is a rhombus and a rectangle.
You can use these to prove that a quadrilateral is a rhombus, rectangle or square without proving first that the quadrilateral is a parallelogram.
1) Decide whether the statement is always, sometimes, or never.
A. A rectangle is a square.
B. A square is a rhombus.
Example:
TheoremsTheorem
6.11
A parallelogram is a rhombus if and only if its diagonals are perpendicular.
A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.
Theorem
6.12
Theorem
6.13
A parallelogram is a rectangle if and only if its diagonals are congruent.
2) Which of the following quadrilaterals have the given property?
All sides are congruent. All angles are congruent. The diagonals are
congruent. Opposite angles are
congruent.
A.Parallelogram
B.Rectangle
C.Rhombus
D.Square
Examples:
3) In the diagram at the right, PQRS is a rhombus. What is the value of y?
5y - 6
2y + 3
P
S
Q
R
Example:
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
Bases: The parallel sides of a trapezoid.Legs: The nonparallel sides of the trapezoid.
Base
Base
Leg LegBase Angles
Trapezoids
Isosceles Trapezoid: A trapezoid whose legs are congruent.Midsegment: A segment that connects the midpoints of the legs and that is parallel to each base. Its length is one half the sum of the lengths of the bases.
Midsegment
A trapezoid that has congruent legs.
Isosceles Trapezoids
Theorem 6.14
Theorem 6.15
Theorem 6.16
If a trapezoid is isosceles, then each pair of base angles is congruent.
A
D C
B
A B C D
If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. D C
BA
A trapezoid is isosceles if and only if its diagonals are congruent.
A B
CDABCD is isosceles if and only if AC .BD
4) is an isosceles trapezoid with
10 and 95 . Find , , , and .
CDEF
CE m E DF m Cm D m F
C
FE
D
95
Example
Theorem 6.17: Midsegment of a trapezoidThe midsegment of a trapezoid is parallel to each base and its length is one half the sums of the lengths of the bases.MN║AD, MN║BCMN = ½ (AD + BC)
NM
A D
CB
The midsegment of a trapezoid is the segment that connects the midpoints of its legs.
Midsegment of a trapezoid
5) Find the length of the midsegment RT.
Example
Definition• A kite is a
quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.
Kite TheoremsTheorem 6.18• If a quadrilateral is a kite,
then its diagonals are perpendicular.
• AC BD
B
C
A
D
Theorem 6.19• If a quadrilateral is a kite,
then exactly one pair of opposite angles is congruent.
• A ≅ C, B ≅ D
B
C
A
D
Example 6) Find the lengths of all four sides of the kite.
12
1220
12
U
X
Z
W Y
Example7) Find mG and mJ
in the diagram at the right.
J
G
H K132° 60°
Rotations:Quadrilateral Website Warmup
Organizing Quadrilaterals WorksheetComputers (Pullout)
Quadrilaterals WorksheetQuadrilaterals Quiz
Quadrilaterals Diagram WarmupProofs G.CO.11 (#1-3)
Exit Quiz
Quadrilaterals WarmupProofs G.CO.11 (#4-6)
Exit Quiz
1. 2.
4.3.
Quadrilaterals Group Quiz Quadrilateral Test Review
Triangles, Parallel Lines, Segments in Triangles, and
Quadrilaterals Review
Triangles, Parallel Lines, Segments in Triangles, and
Quadrilaterals Test