M1
Lesson 9A
Hart Interactive – Geometry
T
GEOMETRY
Basic Properties Answer Chart
Property
Meaning
Geometry Example
Reflexive Property
A quantity is equal to itself.
‫ܤܣ = ܤܣ‬
Transitive Property
If two quantities are equal to the same
quantity, then they are equal to each
other.
If ‫ ܥܤ = ܤܣ‬and ‫ܨܧ = ܥܤ‬,
then ‫ܨܧ = ܤܣ‬.
Symmetric Property
If a quantity is equal to a second
quantity, then the second quantity is
equal to the first.
If ܱ‫ܤܣ = ܣ‬, then ‫ܣܱ = ܤܣ‬.
Addition Property of Equality
If equal quantities are added to equal
quantities, then the sums are equal.
If ‫ ܨܦ = ܤܣ‬and ‫ܦܥ = ܥܤ‬,
then
‫ ܤܣ‬+ ‫ ܨܦ = ܥܤ‬+ ‫ܦܥ‬.
Subtraction Property of Equality
If equal quantities are subtracted from
equal quantities, the differences are
equal.
If ‫ ܤܣ‬+ ‫ ܦܥ = ܥܤ‬+ ‫ ܧܦ‬and
‫ܧܦ = ܥܤ‬, then ‫ܦܥ = ܤܣ‬.
Multiplication Property of
Equality
If equal quantities are multiplied by
equal quantities, then the products are
equal.
If ݉‫ܼܻܺס݉ = ܥܤܣס‬, then
2(݉‫ = )ܥܤܣס‬2(݉‫)ܼܻܺס‬.
Division Property of Equality
If equal quantities are divided by equal
quantities, then the quotients are
equal.
If ‫ܻܺ = ܤܣ‬, then
Substitution Property of Equality
A quantity may be substituted for its
equal.
If ‫ ܧܦ‬+ ‫ ܧܥ = ܦܥ‬and ‫= ܦܥ‬
‫ܤܣ‬,
then ‫ ܧܦ‬+ ‫ܧܥ = ܤܣ‬.
Segment Addition Postulate
The whole segment is equal to the sum
of its parts.
If point ‫ ܥ‬is on തതതത
‫ܤܣ‬, then
‫ ܥܣ‬+ ‫ܤܣ = ܤܥ‬.
Angle Addition Postulate
The whole angle is equal to the sum of
its parts.
If ray BD lies in the interior of
‘ABC, then ‘ABD + ‘DBC =
‘ABC.
Lesson 9A:
Introduction to Writing Proofs
Thi workk iis d
This
derived
i d ffrom EEureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M1-TE-1.3.0-07.2015
஺஻
ଶ
=
௑௒
.
ଶ
139
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 9A
Hart Interactive – Geometry
M1
T
GEOMETRY
Handout for Students – Each student needs one sheet, scissors and glue or tape.
Meanings & Examples
Meaning
Geometry Example
If equal quantities are multiplied by
equal quantities, then the products are
equal.
‫ܤܣ = ܤܣ‬
A quantity may be substituted for its
equal.
If ‫ ܨܦ = ܤܣ‬and ‫ܦܥ = ܥܤ‬,
then
‫ ܤܣ‬+ ‫ ܨܦ = ܥܤ‬+ ‫ܦܥ‬.
If a quantity is equal to a second
quantity, then the second quantity is
equal to the first.
If ‫ܻܺ = ܤܣ‬, then
If equal quantities are added to equal
quantities, then the sums are equal.
If ‫ ܥܤ = ܤܣ‬and ‫ܨܧ = ܥܤ‬,
then
‫ܨܧ = ܤܣ‬.
If two quantities are equal to the same
quantity, then they are equal to each
other.
If ‫ ܤܣ‬+ ‫ ܦܥ = ܥܤ‬+ ‫ ܧܦ‬and
‫ܧܦ = ܥܤ‬, then ‫ܦܥ = ܤܣ‬.
The whole angle is equal to the sum of
its parts.
തതതത, then
If point ‫ ܥ‬is on ‫ܤܣ‬
‫ ܥܣ‬+ ‫ܤܣ = ܤܥ‬.
If equal quantities are divided by equal
quantities, then the quotients are
equal.
If ray BD lies in the interior of
‘ABC, then ‘ABD + ‘DBC =
‘ABC.
If equal quantities are subtracted from
equal quantities, the differences are
equal.
If ‫ ܧܦ‬+ ‫ ܧܥ = ܦܥ‬and ‫= ܦܥ‬
‫ܤܣ‬,
then ‫ ܧܦ‬+ ‫ܧܥ = ܤܣ‬.
The whole segment is equal to the sum
of its parts.
If ݉‫ܼܻܺס݉ = ܥܤܣס‬, then
2(݉‫ = )ܥܤܣס‬2(݉‫)ܼܻܺס‬.
A quantity is equal to itself.
Lesson 9A:
Introduction to Writing Proofs
Thi workk iis d
This
derived
i d ffrom EEureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M1-TE-1.3.0-07.2015
஺஻
ଶ
=
௑௒
.
ଶ
If ܱ‫ܤܣ = ܣ‬, then ‫ܣܱ = ܤܣ‬.
140
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 9A
Hart Interactive – Geometry
M1
T
GEOMETRY
*HRPHWU\(QYHORSH3URRIV
1. Cut apart each proof.
2. *OXHWKH*LYHQDQG3URYH6WDWHPHQWVRQWKHIURQWRIWKHHQYHORSH.
3. 3lace the mixed up statements and reasons in the HQYHORSH
5*LYHQ 3( x 5)
2 x 20
3URYH x
5
3( x 5)
2 x 20
*LYHQ
3 x 15
2 x 20
'LVWULEXWLYH3URSHUW\
x 15
x
6XEWUDFWLRQ3URSHUW\RI(TXDOLW\
20
6XEWUDFWLRQ3URSHUW\RI(TXDOLW\
5
6*LYHQ 9 w 5
9w 5
4(3w 1)
4(3w 1)
3URYH w
3
*LYHQ
'LVWULEXWLYH3URSHUW\
9 w 5 12 w 4
5
3w 4
6XEWUDFWLRQ3URSHUW\RI(TXDOLW\
9
3w
$GGLWLRQ3URSHUW\ RI(TXDOLW\
3
w
'LYLVLRQ3URSHUW\RI(TXDOLW\
6\PPHWULF3URSHUW\RI(TXDOLW\
w 3
Lesson 9A:
Introduction to Writing Proofs
Thi workk iis d
This
derived
i d ffrom EEureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M1-TE-1.3.0-07.2015
141
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 9A
Hart Interactive – Geometry
M1
T
GEOMETRY
7*LYHQ 4( p 5)
3URYH p
5( p 7)
15
4( p 5)
5( p 7)
*LYHQ
4 p 20
5 p 35
'LVWULEXWLYH3URSHUW\
20
p 35
15
p
p
15
6XEWUDFWLRQ3URSHUW\RI(TXDOLW\
$GGLWLRQ3URSHUW\RI(TXDOLW\
6\PPHWULF3URSHUW\RI(TXDOLW\
8*LYHQ
1
(3 x 6)
2
1
(3 x 6)
2
4 2x 6
4 2x 6
3URYH x
2
*LYHQ
1
(3 x 6) 10 2 x
2
Simplify
3x 6
20 4 x
0XOWLSOLFDWLRQ3URSHUW\RI(TXDOLW\
7x 6
20
$GGLWLRQ3URSHUW\RI(TXDOLW\
7 x 14
6XEWUDFWLRQ3URSHUW\RI(TXDOLW\
x
'LYLVLRQ3URSHUW\RI(TXDOLW\
2
Lesson 9A:
Introduction to Writing Proofs
Thi workk iis d
This
derived
i d ffrom EEureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M1-TE-1.3.0-07.2015
142
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.