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clankjewishElectronics - Devices

Oct 10, 2013 (3 years and 8 months ago)

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T
ARGET
3
A
:

U
NDERSTANDING

D
EDUCTIVE
R
EASONING


I can …

Sample Question

Sample Solutions

1.

I can use deductive
reasoning to verify
and explain
theorems.
(
Not
tested
)

A. “The sum of the interior angles of a triangle is
.” Use dedu
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瑨敯r敭⸠


B. “Alternate interior angles are equal when formed
by parallel lines and a transversal.”


啳攠T敤ec瑩t攠牥r獯n楮g⁴o v敲楦e/e硰x慩a⁴U楳i
瑨敯r敭.











䄮†䝩癥n
Ⱐ灲,ve⁴U

.



Draw CD parallel to AB.




and

because alternate interior angles
formed by parallel lines are congruent, so they have the same
measure.



becaus
e they form a straight angle.



, by substitution.


B. Given:
. Prove
.



because corresponding angles formed by parallel lines
are congruent.

because vertical angles are congruent.


by the transitive property.

2.

I can use deductive
reasoning to verify
and explain
conjectures or
geometric
statements

In the diagram, AB = AC,
, and
.
Sam thinks that
. Explain why he is correct.


Since AB = AC
is isosceles, so the base angles are
congruent
.
Since
,

(The sum of the angles of a triangle is
.
.) Since
,
,
because corresponding angles are
congruent

only when formed by
parallel lin
es.

3.

I can use
transformations to
justify the truth of
a conjecture.


Since lengths are preserved over a reflection, I know that

and
. Since the resulting quadrilateral has exactly two pairs of
conse
cutive sides that are equal,

is a kite.







Since a
ngles are preserved over a rotation, I know that

and
. Since both of these are pairs
of alternative angles and the pair
s are equal, the lines forming them
must be parallel:

and
. Since the triangle was rotated
over the midpoint of
, I know that

and its rotation,
form a quadrilateral. A quadrilateral with two sets of parallel
sides is a parallelogram.

4.

I can
explain the
role of defin
i
tions,
undefined terms,
postulates
(axioms), and
theorems.

Explain the similarities and differences between

definitions, undefined terms, postulates (axioms), and
theorems and how they are used.







Undefined terms and postulates give us words and assumptions to
build on, because we have to start with something. Definitions,
postulates, and theorems are all used to justify statements.
Definitions simply define words. Postulates are statements t
hat we
assume to be true, but can’t prove. Theorems are statements that we
have proven to be true.




Which of the following statements are true in
geometry? Explain your thinking.

a.

Every term can be defined and every true
statement can be proved true.

b.

Every term
can be defined, but it is necessary
to assume certain statements are true.

c.

Some terms must be left undefined, but every
true statement can be proved true.

d.

Some terms must be left undefined, and it is
necessary to have some statements which are
assumed true
.


Statement “d” is correct. Some terms (point, line, and plane) are
undefined, as they cannot be defined without using other figures.
Postulates and axi
oms make assumptions about core geometry
concepts that that cannot be proved without using other theorems.


T
ARGET
3
B
:

U
NDERSTANDING

A
NGLE
R
ELATIONSHIPS WITH
L
INES


I can …

Sample Question

Sample Solution

1.

I can identify
angle pairs
(corresponding,
alternate
interior,
alternate
exterior, and
same
-
side
interior)


Given lines
a

and
b

are intersected by a transversal
t
,

identify 2 sets of each of the following angle pairs










Corresponding ___________________________

Alternate interior_________
__________________

Alternate exterior___________________________

Same
-
side interior__________________________



Corresponding:


Alternate interior:

Alternate exterior:

Same
-
side interior:


2.

I can draw
conclusions
about angle pairs
formed by
parallel lines and
a transversal.

Given line
j

is parallel to line
k,
what is the measure of
angle 1?












Since the
ang
le and

are corresponding angles, they are equal.


Since

are a linear pair, they are supplementary.




a

b

t

1


2


3


4


5


6


7


8

3.

I can use angle
relationships to
determine
when
lines are parallel.

In the diagram below
,
Angles 1 and 2 are
supplementary.


Can you conclude
? If so, prove it; if not, find
a counterexample.







Given the diagram to the left and the fact that angles 1 and 2 are
sup
plementary.



Def. of Supplementary Angles


Angles of a linear pair are
supplementary


Substitution


Subtraction property


Alt. Interior Angles



lines


4.

I can apply
theorems about
parallel and
perpendicular
lines.

Write at least four statements that can be used to justify that
j

is
parallel to
k.








Some possi
ble responses are shown below:



If
, then

by Corresponding Angles.



If
, then

by Alternate Interior Angles.



If
, then

by Alternate Exterior Angles.



If

are supplementary, then

by Same
-
Side
Interior Angles.

5.

I can solve
problems
involving angles
and parallel lines.

Use

what you know about the
angles formed by parallel lines and
a transversal to find
.





Since
are Same
-
Side Interior angles, they
are supplementary.








6.

I can use and
explain
constructions
related to
parallel and
perpendicular
lines.

Construct a line parallel to line
m

through point
A
.










T
ARGET
3
C
:

U
NDERSTANDING
A
NGLES AND
S
EGME
NTS IN A
T
RIANGLE


I can …

Sample Question

Sample Solutions

1.

I can use the
exterior angle
theorem to
solve problems.

Find
.



Find
x

and
.






Since exterior angles of a triangle are equal to the
sum

of the two remote interior angles,












Since exterior angles of a triangle are equal to the
sum of the two remote interior angles,





15 = x



2.

I can find the
sum
s

of the
interior angles

and exterior
angles

of a
convex
polygon
.

a.
Find the sum of the interior angles of a heptagon.


b.
Find the sum of the exter
ior angles of a heptagon.



c. The sum of the interior angles of a figure is 1800. What type of
polygon is this figure?




d.
Find the sum of the exterior angles of a kite.

a.
Heptagon


7
-
sided polygon

Sum of Interior Angles = 180 (7
-
2) =


b. Sum

of Exterior Angles =


c. Sum of Interior Angles = 180(n



2) =
1800


n
-
2=10


n = 12

The figure is a
12
-
sided polygon.


d.
Sum of Exterior Angles

of a kite

=

3.

I can find
the
measure of
interior and

exterior angles
of a
convex
polygon
.

Find the value of
n
.


Figure ABCDEF is a regular hexagon. Find
x

and
y
.




Since the sum of the exterior angles for all convex
polygons is
,













Since the figure is a regular hexagon, the interior angles
are all congruent and the exterior angles are all
congruent.


Sum of Interior Angles = 180(6
-
2) =


x

=
Each

interior angle =


Sum of Exte
rior Angles =


y

= Each exterior angle =


Solve for x and

the measure of all the unknown interior angles in the
pentagon below.



Since the figure is a pentagon, the sum of the interior
angles is equal to
.












4.

I can explain
the
characteristics
of the
midsegments
,
medians, and
altitudes

of a
triangle.

A. Explain what

you know about the midsegments of a triangle.


B. Explain what you know about the medians of a triangle.


C. Explain what you know about the altitudes of a triangle.


D. Find the
angles x, y, and z in the figure. Find the length of AD, AC,
and BE, FD.




A. A midsegment is formed by connecting the
midpoints of two sides of a triangle. It is parallel to the
third side, and also half the length of the third side.


B. A median is a segment from a vertex of the triangle
to the midpoint of the opposit
e side. The three
midpoints meet at one point in every triangle. This point
is called the centroid.


C. An altitude is a segment from a vertex of a triangle
that is perpendicular to the opposite side. The three
altitudes of a triangle meet at one point
. This point is
called the orthocenter.


D.
:
are alternate angles
formed by parallel lines.

:
is the third angle of
, and 180
-
62
-
25=93, so
.

because they
are alternate angl
es formed by parallel line.



because
and
are corresponding

angles formed by parallel lines.

AD=5 in., AC=10in., BE=3 in., FD=3 in.: Justification: the
midsegment is half the value of th
e side opposite it. T
he
p
oints E, F, and D are midpoints, as indicated by the tick
marks.

T
ARGET
3

C
ONNECTIONS

I can …

Samp
le Question

Sample Solutions

1.

I can solve
problems by
reasoning with
geometry and
algebra.


Find the value of x.

y
+101=180

because they are a linear pair.

Therefore y=79˚.


砫y=2x⬲0

b散慵獥so映fU攠數瑥物o爠慮r汥⁴Ueo牥mⰠ慮T
y=79˚, so x+79=2x+20
.


59 = x

2.

I can use
transformations
to solve
problems

A. MNOP is an isosceles trapezoid
, with
. Use symmetry
and other g
eometric ideas to find the measure of all the interior
angles.



B. Use transformations to explain why the formula for the area of a
parallelogram is
(base)x(
height
)
.












A. An isosceles trapezoid has a line of symmetry through
the midpoints of
its bases. Therefore

and
. Since
,
are
supplementary, so
.
Because of the
symmetry,

.





B.
If you translate
to the right,
the fig
ure formed is
a rectangle, and
has the same area as the original
parallelogram. The area of a rectangle is
(base)x(height).
.
Justification:
When translated,
coincides with
because they were

parallel and the same length. Also,


.

because they
are same
-
side interior angles formed by parallel lines, so
is a straight li
ne.

because

is
perpendicular to one of two parallel lines, so it must be
perpendicular to the other.

because
.









3.

I can

use
coordinate
geometry
to
solve problems.

Find the orthocenter of the triangle below using coordinate
geometry.









Find the equations for two altitudes:


1.

Altitude from

to B:



x=3 (Vertical Line)

2.

Altitude from

to C:

Slope of

is 3, so the slope of the altitude is
.

Point C (11, 1)




3.

Solve for the intersection
of the two lines:


So, the orthocenter is at (3,
).

4.

I can use
systems of
equations to
find points on a
coordinate
plane.

Figure FHIJ is a kite.
and
. Solve for
x

and
y
.



Since the figure is a kite, HI = JI and GH = GJ.























T
ARGET
3

C
OMMUNICATIONS



Students give evidence of clear communication by showing their thinking on each problem.