1. C.P.C.T.C Corresponding Parts of a Congruent Triangles are Congruent
<AMB & <BMC are right angles
<ABM< ≅ <MBC
Corresponding Parts of a Congruent Triangles are Congruent
C.P.C.T.C
Find a pair of ∆
with parts you are
trying to prove are ≅
Prove the two
triangles are
congruent
by C.P.C.T.C
corresp. parts
are congruent
proving a bisector
or angle bisector
proving lines are parallel or perpendicular
prove AB ∥ CD
prove the 2 angles
it bisects
are congruent. B A C B
Statements Reasons
1) BE ∥ DF, BE ≅ DF,
CE ≅ AF
1) given C E A D F M
Statements Reasons
1) <AMB and <BMC are right angles,
<ABM ≅ <MBC
1) given

2. 1) <AMB and <BMC are right angles, <ABM ≅ <MBC 1) given
prove the 2 angles
it bisects
are congruent. B A C M
Statements Reasons
1) <AMB and <BMC are right angles,
<ABM ≅ <MBC
1) given
2) <AMB ≅ <BMC
2) Right angle congruence theorem
3) BM ≅ BM
3) Reflexive Property
4) ∆AMB ≅ ∆CMB
4) A S A
5) AM ≅ MC
5) CPCTC
6) BM is a bisector of AC
7) Definition of a segment bisector

3. prove AB ∥ CD B V V Statements Reasons C F E D A 1) given
1) BE ∥ DF, BE ≅ DF,
CE ≅ AF
2) <BEA ≅ <DFC
2) alternate exterior angles
3) AE + EF = AF
CF + EF = CE
3) Betweenness of points
4) AE + EF = CF + EF
4) substitution
5) subtraction
5) AE = CF
6) ∆BEA≅∆DFC
6) SAS
7) <BAE ≅ <DCF
7) <C.P.C.T.C
8) AB ∥ CD
8) If 2 lines form ≅ alt. int. angles, they are ∥

4. Things that can be proven with C.P.C.T.C
What am I
trying to prove??
Here are
the steps to
do it!
Show triangles are ≅
Prove segments or angles within the ∆ are ≅ by C.P.C.T.C
segments or angles are congruent
Show triangles are ≅
Are the parts of bisecting segment or angle contained in the ∆?
Prove those parts are ≅ by C.P.C.T.C (this proves the bisector)
segments or angles bisector
Show triangles are ≅
Are the angles in the ∆ alt. interior, corresponding, or alt. exterior angles of lines?
Prove those parts are ≅ by C.P.C.T.C (this proves the bisector)
Lines are
parallel ∥
Show triangles are ≅
Are the angles contained in ∆ and formed by the lines linear pairs?
Prove those angles are ≅ by C.P.C.T.C (this proves you have right angles and lines must be parallel)
Lines are
perpendicular ⊥

5. perpendicular bisector
Altitude
The altitude is a ____________ segment
from one ___________
to the other side.
perpendicular
vertex
Every triangle has exactly ______ altitudes. 3
An altitude can be inside OR ___________
of a triangle.
altitude
outside
You may need to extend the bottom in order to draw the altitude.
A median is a segment
from a ______________
to the ___________ of the other side.
Median
vertex
midpoint
median
An altitude and median combined into one is a
perpendicular bisector
Every triangle has exactly ______ medians and all 3 ____________ at a single point
called a centroid. 3
intersect

7. You already know “x” so, substitute it in.
Rolling Ball Challenge #1
What do you need to know?
You need to find the value of “x” so you can determine what the measurement for angle (7x) is. This will tell you which way the horizontal line will tilt.
2) What do already know?
Because we have parallel lines...
angles (12x) and (12y +6)
are congruent, alternate, interior angles.
So...
12x=12y +6
Using algebra to solve for “x” we find x=(y+0.5)
3) Now what?
Our parallel lines also tell us
angle (8y) and angle(4x+10)
are supplementary interior angles on the same side as a transversal.
8y + 4x+10 = 1800
You already know “x” so, substitute it in.
8y + 4(y+0.5) +10 = 180 Remember: x=(y+0.5)
8y+4y+2+10= solve for x
12y +12=180 x=(y+0.5)
12y=168 x=( )
y= x = 15
4) Now, back to the angle that determines tilt... (7x)
<7x = 1050
Let’s think about this: but this is what
This is what a horizontal our 7x angle would look like.
line would look like Can you see how it would tilt
to the left??