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Proofs
Chloe Clegg, Georgina Platt,
William Snapp, Komal Parikh,
David Korsunskiy, Joseph
Higginbotham
Dilations
Dilation : a transformation that makes a figure larger or smaller than the
original figure based on a ratio given by a scale factor.
Scale Factor Rules :
x>1= figure is made larger
0<x<1 = figure is made smaller
Center of Dilation = Origin
You have to multiply each coordinate of the pre- image by the scale factor to
get the dilated figure.
Center of Dilations
If the center of Dilation is not on the origin
then you use the formula :
k(x-xc) + xc
k(y-yc) + yc
k= scale factor of dilation
(x,y) = point of dilation
(xc ,yc) = center point of dilation
Center of Dilations cont’d.
When a figure is transformed under a
dilation , the CORRESPONDING ANGLES
of the pre- image and the image have equal
measurement.
A’
Example :
A
triangle ABC and triangle A’B’C’
<A=<A’ <B=<B’ <C=<C’
B
C
B’
C’
Dilations cont’d
When a figure is dilated, a segment of the pre-image that does not pass
through the center of the dilation is parallel to its image. In the figure below, AC
A’C’ since neither segment passes through the center of dilation. The same is
true about AB and A’B’ as well as BC and B’C’
A’
B’
A
B
C
C’
Dilations cont’d
When the segment of a figure does pass through the center of dilation, the
segment of the pre-image and image are on the same line. In the figure below,
the center of dilation is on AC, so AC and A’C’ are on the same line.
A’
B’
A
B
C
C’
Dilations Examples
1. Dilate the following triangles by 3
A
B
A
C
B
C
Examples cont’d
What are the ordered pairs of the new larger triangles?
What are the angles of the new larger triangles?
Rigid Motions
Rigid Motion : is a transformation of points in space
consisting of a sequence of one or more translations,
reflections, and/or rotations ( in any order )
Congruent figures always have the same side lengths and
angle measures as each other.
Rule: Two triangles are congruent if and only if their
corresponding sides and corresponding angles are
congruent = CPCTC
CPCTC= Corresponding Parts of Congruent Triangles are
Congruent
Proof Theorems
If two triangles are similar, two
corresponding angles are congruent. This is
the Angle Angle(AA) Similarity.
Rules :
1) If two triangles are similar ,all corresponding pairs of angles are
congruent.
2) If two triangles are similar all corresponding pairs of sides are
proportional.
3) Congruent = similarity
Sides and Angles
There are four ways you can show two triangles are
congruent to each other:
ASA : Angle Side Angle
Two angles and the included side of a triangle are
congruent to the two angles and the included side of the
other triangle
SSS: Side-Side-Side
Three sides of a triangle have to be congruent to the other
three sides of another triangle
Sides and Angles Cont’d
SAS : Side-Angle-Side
Two sides and the included angle of a triangle have to be
congruent to two sides and the included angle of another
triangle.
AAS: Angle- Angle- Side
Two angles and a non included side of a triangle are
congruent to the angles and the corresponding nonincluded side of the other triangle
Sides and Angles
SSA : Side -Side Angle
● There is no way to show triangle congruence through
SSA!
● The two sides and a non included angle one triangle
are congruent to two sides of a non - included angle of
a second triangle does not make them congruent.
ALSO..
● Two triangles having three angles congruent to each
other does not make them congruent it only makes
them similar!
Geometric Theorems
Auxiliary line : a line drawn in a diagram that makes other figures such
as congruent triangles or angles formed by a transversal.
Key Ideas on lines and angles:
Vertical Angle Theorem : Vertical angles are congruent
Alternate Interior Angles Theorem: If two parallel lines are cut by a
transversal, then alternate interior angles formed by the transversal are
congruent
Corresponding Angles Postulate :If two parallel lines are cut by a
transversal then corresponding angles formed by the transversal are
congruent.
Geometric Theorems
Key Points about Triangles:
Triangle Angle Sum Theorem : The sum of the measure
of the angles of a triangle is 180 degrees.
Isosceles Triangle Theorem: If two sides of a triangle are
congruent , then the angles opposite those sides are also
congruent .
Triangle Midsegment Theorem: If a segment joins the
midpoints of two sides of a triangle , then the segment is
parallel to the third side and half its length
Geometric Theorems
●
●
●
●
Key Points on Parallelograms :
Opposite sides are congruent and opposite angles are
congruent
The diagonals of a parallelogram bisect each other
If the diagonals of a quadrilateral bisect each other ,
then the quadrilateral is a parallelogram
A rectangle is a parallelogram with congruent diagonals.
Examples
Lines AB and CD are parallel
and line EF is a
transversal,m∠BGH=2x and
m∠DHF=5x−51. Find the
measurement, in degrees, of
∠CHF.
E
G
B
A
2x
C
D
H
F
More examples
Find the degree of
each angle in the
Isosceles triangle
below
Find the value of
X in the Right
Triangle below.
60°
68°
3X 5X-40
2x
Two Column Proof
Two Column proof : is a series of statements and reasons
often displayed in a chart that works from given information
to the statement that needs to be proven. The reasons can
be given information.
Paragraph proof : also uses a series of statements and
reasons that works from a given information to the
statement that need to be proven , but the information is
presented as running text in paragraph form.
Two Column Proof Example
In the triangle below, line AC is parallel
to line DE:
Prove that line DE
divides line AB and CB
proportionally.
B
D
A
E
C
Steps
Statement
Justification
Step 1
DE
given
Step 2
< BDE ≅<BAC
two lines cut by transversal
corresponding angles are
congruent
Step 3
<DBE ≅<ABC
reflexive property of
congruence b/c same angle
Step 4
DBG ≅
AC
ABC
AA
Step 5
BA/ BD = BC/BE
corresponding sides of
similar triangles are
proportional
Step 6
BD + DA= BA
BE+ BC= BC
segment addition
Step 7
BD + DA / BD = BE +
EC/BE
substitution
Step 8
1+DA/BD = 1+ EC/BE
simplify
Steps Cont’d
Statement
Justification
Step 9
DA/BD = EC/ BE
subtraction
Step 10
DE divides AB and CB
proportionally
definition of proportionality
Lets Try Another One..
Geometric Constructions
Geometric constructions uses only a
straightedge and a compass to make
shapes with specific angles or side lengths
https://www.khanacademy.org/math/geometry/geometric-constructions/geo-bisectors/v/constructing-a-perpendicular-bisectorusing-a-compass-and-straightedge