Download 4.4 - Prove Triangles Congruent by SAS and HL

4.4 - Prove Triangles Congruent by SAS and HL
Included Angle:
Angle in-between two congruent sides
1. Use the diagram to name the included angle
between the given pair of sides.
GH and HI
H
1. Use the diagram to name the included angle
between the given pair of sides.
HI and IG
HIG
1. Use the diagram to name the included angle
between the given pair of sides.
JG and IG
JGI
Side-Angle-Side (SAS) Congruence Postulate
E
A
4cm
AB  CD
A  C
AE  CF
F
B
C
4cm
D
included
If two sides and the _____________
angle of
congruent to two sides and
one triangle are __________
the included angle of a second triangle, then the
congruent
two triangles are ____________
Right Triangles:
hypotenuse
leg
leg
Hypotenuse-Leg (HL) Congruence Theorem:
hypotenuse
If the _______________
and a ________
of a
leg
right
congruent
___________
triangle are ____________
to
the _____________
of a second
hypotenuse and ________
leg
_________
right
triangle, then the two
triangles are _________________.
congruent
2. Decide whether the triangles are
congruent. Explain your reasoning.
Yes, SSS
2. Decide whether the triangles are
congruent. Explain your reasoning.
Yes, SSS
2. Decide whether the triangles are
congruent. Explain your reasoning.
Yes, SAS
2. Decide whether the triangles are
congruent. Explain your reasoning.
No, AD ≠ CD
2. Decide whether the triangles are
congruent. Explain your reasoning.
Yes, SAS
2. Decide whether the triangles are
congruent. Explain your reasoning.
Yes, HL
2. Decide whether the triangles are
congruent. Explain your reasoning.
No, Not a right
triangle
2. Decide whether the triangles are
congruent. Explain your reasoning.
Yes, SSS
3. State the third congruence that must be given to
prove ABC  DEF.
BA  ______.
ED
GIVEN: B  E, BC  EF , ______
Use the SAS Congruence Postulate.
3. State the third congruence that must be given to
prove ABC  DEF.
DF
AC  ______.
GIVEN: AB  DE , BC  EF , ______
Use the SSS Congruence Postulate.
3. State the third congruence that must be given to
prove ABC  DEF.
GIVEN: AC  DF , A is a right angle and A  D.
Use the HL Congruence Theorem.
BC  EF
4. Given: G is the midpoint of RT and HI
Prove: ∆RGI  ∆TGH
1. G is the midpoint of RT
2. RG  GT
3. HG  GI
4. RGI  TGH
5. ∆RGI  ∆TGH
and HI
1. given
2. Def. of midpt
3. Def. of midpt
4. Vertical angles
5. SAS
A
5. Given:
Prove:
AB CD
AB  CD
∆ABD  ∆CDB
Statements
1.
AB CD
B
D
C
Reasons
1. Given
A
B
D
C
5. Given:
A
AB CD
AB  CD
Prove:
D
∆ABD  ∆CDB
Statements
1.
B
AB CD
Reasons
1. Given
2. CDB  ABD
2.
3. AB  CD
3. Given
4. DB  DB
4. Reflexive
5. ∆ABD  ∆CDB
5. SAS
C
Alternate Interior Angles
A
6. Given: AC bisects DAB
DA  BA
Prove: ∆ACD  ∆ACB
Statements
D
C
B
1. AC bisects DAB
Reasons
1. Given
2. DAC  BAC
2. Def. of Angle Bisector
3. DA  BA
3. Given
4. AC  AC
4. Reflexive
5. ∆ACD  ∆ACB
5. SAS
A
7. Given: AD  AB
AC  BD
Prove: ∆ACD  ∆ACB
Statements
1. AD  AB
2. AC  BD
D
C
B
Reasons
1. Given
2. Given
3. ACD and ACB are
right angles
4. ∆ACD and ∆ACB are
right triangles
5. ACD  ACB
3. Def. of perp. lines
6. AC  AC
7. ∆ACD  ∆ACB
6. Reflexive
7. HL
4. Def. of right ∆’s
5. All right angles are 
HW Problem
4.4 243 1-7odd, 9-11, 13, 14, 20, 21, 25-27, 34, 35, 37, 38
# 26
Ans: AB  DE