Download Warm-Up: Visual Approach Lesson Opener

Geometry A Unit 3 Day 4 Notes
5 Shortcuts for Proving
Congruence
Warm-Up
In each of the following, determine the segments or angles that are congruent based on
the given information. If two segments or angles are congruent, circle their names. If a
second set of segments or angles is congruent, put a box around them.
1. In the diagram below, TS bisects YL.
2. In the diagram below, M is the
midpoint of TS.
T
Y
M
T
M
Y
S
L
Which segments are congruent?
S
Which segments are congruent?
TM
TM
MS
YM
ML YL ST
YM
ML YL ST
4. In the diagram below, AB  MN.
A
3.
Y
MS
L
T
M
S
L
M
Which angles are congruent?
 YMT
 LMT
 LMS
 YMS
N
1
3T
4 2
B
Which angles are congruent?
1
2
3
4
6. In the diagram, TR bisects  WRY
5. In the diagram, MY | | LT
M
3 1 2
4
Y
T
L
5 78
6
1 2
L
3
R
T
Which angles are congruent?
1
2
3
4
5
Y
W
6
7
8
Which angles are congruent?
1
2
3
I. Proving That Two Triangles are Congruent
A. If you KNOW that two triangles share the right number of parts, in the right order,
you can be sure they are congruent and use the resulting symmetry as you see fit.
1. ________________ - ___________________ - ________________ Congruence
a. Every problem in this chapter is like a proof, but not all will be. We will
however start with one.
Given:  RST is isosceles with base RT
and RV = VT
Prove:  RSV   TSV
Statement
S
Reason
1. __________________________ 1. _______________
2. __________________________ 2. _______________
3. __________________________ 3. ___________________
R
T
V
4. ___________________________ 4. __________________
5. ___________________________ 5. ___________________
Examples: Determine if enough information is given to be sure that the two triangles are
congruent. If so, complete the congruence statement. IF not, tell what other pair of parts
would need to match.
a.
b. Given:  RST is isosceles with base RT
and RV = TV
S
Given: AB = DE,
BC = EG and AC = DG
A
C
D
B
E
Yes:  ABC   ___ ___ ___
No: We need ______  ________
G
R
V
T
Yes:  SVR   ___ ___ ___
No: We need ______  ________
d.  EQL is Equilateral
E
c. M is the midpoint of NP,
NP bisects QR and NR = QP.
P
T
Q
L
M
N
Q
R
Yes:  MQP   ___ ___ ___
Yes:  ETQ   ___ ___ ___
No: We need ______  ________
No: We need ______  ________
2. __________________ - ________________ - _________________ congruence
(Notice that the angle is included between the two sides)
Examples: Determine if enough information is given to be sure that the two triangles are
congruent. If so, complete the congruence statement. IF not, tell what other pair of parts
would need to match.
b. Given:  A   P
AB  PT
a. Given AB  DE
B is the midpoint of DE
A
A
C
P
B
E
B
D
T
 ABD   ___ ___ ___
 CBA   ___ ___ ___
No: We need ______  ________
No: We need ______  ________
V
d. AT | | CU, AT  CU, CU bisects AM.
c. M is the midpoint of NP
NP bisects QR.
N
A
R
T
C
M
P
U
M
Q
 MRN   ___ ___ ___
 TCA   ___ ___ ___
No: We need ______  ________
No: We need ______  ________
In the drawing below, three parts are used to start two “different” triangles.
I.
A
46
28
B
C 46
28
D
A. List pairs of equal angles.
 ____   ____ and  ____   ____
B. List a pair of equal sides.
_______ = ________
C. Complete the drawings of the two triangles. Are the resulting triangles
identical?
_____________________
If the answer is yes, then three parts in the order we started with are enough to
guarantee two triangles are congruent.
_____ _____ _____
Ex. 1: Determine which figure shows triangles that can be proven to be congruent.
In the figure where we do not know enough information, give a pair of parts that
would convince us the triangles are congruent.
a. Given:  1   2
snd AC bisects  BCD
b. Given: AB  BD and EC  CD
E
B
B
D
A
1
2
C
C
A
D
YES
(circle the reason)
SSS SAS
ASA
or NO we need ________  __________
II.
YES
(circle the reason)
SSS SAS
or NO we need ________  __________
The remaining two shortcuts work because they lead directly to one of the
three.
A. Angle – Angle – Side
1. Find the missing angle in each triangle.
28
46
ASA
46
28
2. If two of the angles in a triangle match two angles in another
triangle, the third has to match.
So, AAS leads to ____ _____ ____. Therefore, we’ll add AAS to
the list.
B. Hypotenuse – Leg
1. If we call the sides of a triangle hypotenuse and leg, then the
triangle must be a _______________ triangle.
2. Below, two triangles are drawn that have matching hypotenuses
and a matching leg. The third side HAS to be 12 in each.
(Pythagorean Theorem).
5
13
13
5
3. The sides you found should have matched. Therefore, HL leads
to _____ _____ _____. So we should add HL to the list.
C. In case you were wondering why we don’t use AAA or SSA, those parts
do not necessarily establish one triangle. That is…
A picture can be drawn to make it look like triangles are the same because of
AAA…
But it is possible to draw two triangle of different size even if their angles
match.
Triangles can be drawn to look like they are the same because of SSA…
But it is possible that two different triangles can have two matching sides and
a matching angle, so long as the angle between the sides doesn’t match.
Therefore the complete list of congruent triangle shortcuts is…
1) __________ 2) _________ 3) __________ 4) __________ 5) _________
HW: Handout