Happy Tuesday 
Do Before the Bell Rings:
1. Pick up the paper from the front table
2. Take out notes from yesterday
3. Have your homework out on your desk with a red
pen.
4. Take out your whiteboard and whiteboard pens.
Math History Presentations will be TODAY!
Math History Presentations
Warm Up: Whiteboards
1. What are sides AC and BC called? Side AB?
legs; hypotenuse
2. Which side is in between A and C?
3. Given DEF and GHI, if D  G and E  H,
why is F  I?
Third s Thm.
Example from yesterday: Verifying Triangle
Congruence
Show that the triangles are congruent for the given value of the
variable.
∆MNO  ∆PQR, when x = 5.
PQ  MN, QR  NO, PR  MO
∆MNO  ∆PQR by SSS.
Whiteboards
Show that the triangles are congruent for the given value of the
variable.
∆STU  ∆VWX, when y = 4.
ST  VW, TU  WX, and T  W.
∆STU  ∆VWX by SAS.
4-4 Triangle Congruence: SSS and SAS
Example 3 ( from yesterday):
The Hatfield and McCoy families are feuding over some land.
Neither family will be satisfied unless the two triangular fields
are exactly the same size. You know that BC is parallel to AD
and the midpoint of each of the intersecting segments. Write
a two-column proof that will settle the dispute.
. Given: BC || AD, BC  AD
Prove: ∆ABD  ∆CDB
Proof:
4.5: Triangle Congruence: ASA, AAS,
and HL
Learning Objective
SWBAT prove triangles congruent by using
ASA, AAS, and HL.
Math Joke of the Day
• What do you call a broken angle?
• A rectangle!
4.5 Triangle Congruence: SSS and SAS
There are five ways to prove triangles are congruent:
1. SSS
yesterday
2. SAS
3. ASA
Today!
4. AAS
5. HL
Included Side
• Yesterday we learned what an included angle
is. What do you think an included side would
be?
Included side
• common side of two consecutive angles in a
polygon.
4-4 Triangle Congruence: SSS and SAS
Angle–Side–Angle Congruence (ASA)
• If two angles and the included side of one triangle are
congruent to two angles and the included side of another
triangle, then the triangles are congruent
• . What is a possible congruent statement for the figures?
ASA
• Examples
• Non-Examples
Example 1
Example 1:
(a) Use ASA to explain why ∆UXV  ∆WXV.
Whiteboard
Determine if you can use ASA to prove
NKL  LMN. Explain.
By the Alternate Interior Angles Theorem. KLN  MNL. NL  LN by
the Reflexive Property. No other congruence relationships can be
determined, so ASA cannot be applied.
Angle-Angle-Side Congruence
Angle-Angle-Side(AAS)
• If two angles and a non-included side of one triangle are
congruent to the corresponding angles and a side of a
second triangle, then the two triangles are congruent.
What is the possible congruence statement for the figures?
Example/ Non-Examples: AAS
• Example
• Non-Example
Proof of AAS
Example 2:
Example 3
Use AAS to prove the triangles congruent.
Given: JL bisects KLM, K  M
Prove: JKL  JML
Hypotenuse-Leg (HL) Congruence
Hypotenuse-Leg Congruence (HL)
• If the hypotenuse and a leg of a right triangle are congruent
to the hypotenuse and a leg of another right triangle, then
the triangles are congruent.
Examples/Non-Examples: HL
• Example
• Non-Example
Example 4
Prove ABC  DCB.
Yes; it is given that AC  DB. BC  CB by the Reflexive
Property of Congruence. Since ABC and DCB are right
angles, ABC and DCB are right triangles. ABC  DCB by
HL.
Whiteboards
Identify the postulate or theorem that proves the triangles
congruent.
HL
ASA
SAS or SSS
Homework
• Worksheet online!
– The answers are on the bottom but try it yourself
first
– I will not be here tomorrow