1. Lesson 4.3 and 4.4 Proving Triangles are Congruent

2. Learning Target
I can list the conditions (SAS, SSS) to prove triangles are congruent.
I can identify and use reflexive, symmetric and transitive property in my proof.

3. How To Find if Triangles are Congruent
Two triangles are congruent if they have:
exactly the same three sides and
exactly the same three angles.
But we don't have to know all three sides and all three angles ...usually three out of the six is enough.
There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

4. 1. SSS (side, side, side) SSS stands for "side, side, side“
and means that we have two triangles
with all three sides equal.
For example:
is congruent to:  
If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

5. 2. SAS (side, angle, side) SAS stands for "side, angle, side"
and means that we have two triangles
where we know two sides and the included angle are equal.
For example:
is congruent to:  
If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent.

6. 3. ASA (angle, side, angle) ASA stands for "angle, side, angle“
and means that we have two triangles
where we know two angles and the
included side are equal.
For example:
is congruent to:  
If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

7. 4. AAS (angle, angle, side) For example: is congruent to:
AAS stands for "angle, angle, side“
and means that we have two triangles
where we know two angles and the
non-included side are equal.
For example:
is congruent to:  
If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

8. 5. HL (hypotenuse, leg) HL applies only to right angled-triangles!
HL stands for "Hypotenuse, Leg" (the longest side of the triangle is called the "hypotenuse", the other two sides are called "legs")
and  
HL applies only to right angled-triangles!

9. 5. HL   (hypotenuse, leg)
It means we have two right-angled triangles with
the same length of hypotenuse and
the same length for one of the other two legs.
It doesn't matter which leg since the triangles could be rotated.
For example:
is congruent to 
If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent.

10. Caution ! Don't Use "AAA" ! AAA means we are given all three
angles of a triangle, but no sides.
This is not enough information to decide if two triangles are congruent!
Because the triangles can have the same angles but be different sizes:
For example:
is congruent to 
Without knowing at least one side, we can't be sure if two triangles are congruent..

11. Proving Triangles are Congruent
You have learned to prove that two triangles are congruent by the definition of congruence – that is, by showing that all pairs of corresponding angles and corresponding sides are congruent.
THEOREM A B C
Theorem 4.4 Properties of Congruent Triangles
Reflexive Property of Congruent Triangles
Every triangle is congruent to itself. D E F
Symmetric Property of Congruent Triangles
If  , then 
ABC
DEF J K L
Transitive Property of Congruent Triangles
If  and  , then 
JKL
ABC
DEF
Goal 2

12. Using the SAS Congruence Postulate
Prove that
 AEB  DEC. 2 1
Statements
Reasons
AE  DE, BE  CE Given 1
1  2 Vertical Angles Theorem 2
 AEB   DEC SAS Congruence Postulate 3

13. Proving Triangles Congruent
MODELING A REAL-LIFE SITUATION
Proving Triangles Congruent
ARCHITECTURE You are designing the window shown in the drawing. You
want to make  DRA congruent to  DRG. You design the window so that
DR AG and RA  RG.
Can you conclude that  DRA   DRG ? D G A R
SOLUTION
GIVEN
DR AG
RA RG
PROVE
 DRA  DRG

14. Proving Triangles CongruentD
GIVEN
PROVE
 DRA  DRG
DR AG
RA RG A R G
Statements
Reasons 1
Given
DR AG
If 2 lines are , then they form
4 right angles.
DRA and DRG
are right angles. 2 3
Right Angle Congruence Theorem
DRA  DRG 4
Given
RA  RG 5
Reflexive Property of Congruence
DR  DR 6
SAS Congruence Postulate
 DRA   DRG

15. Given: SP  QR; QP  PR Prove  SPQ  SPR
Statements Reasons
1. Given
1. SP  QR; QP  PR
2. QPS and RPS are right ’s.
2. Def. of 
3. QPS  PRS
3. Rt.   Thm.
4. SP  SP
4. Reflexive POC
5.  SPQ  SPR
5. SAS  Post.

16. Pair-share Work on classwork on “Congruence Triangle”
Sage and Scribe on #21 to #24