Transcript Hypotenuse-Leg Congruence Theorem: HL

Hypotenuse-Leg
Congruence Theorem: HL
Section: 5.4
Objective:
• Use the HL Congruence Theorem and
summarize congruence postulates and
theorems.
Key Vocabulary - Review
• Hypotenuse
• Leg of a right triangle
Theorems
• 5.2 Hypotenuse-Leg Congruence
Theorem (HL)
Right Triangles
• Right Triangles: side across from right
angle is the hypotenuse, the remaining two
are legs.
• The longest side of a right triangle is the
hypotehuse.
Congruent Right Triangles
A
• There is one additional congruence rule
that applies only to right triangles
5
3
– The two right triangles have two congruent
4
sides, the hypotenuse and one leg
T
• Is this enough information to show that the two
4
triangles are congruent?
• Remember, SSA is not a congruence rule
E
C
– Since ∆TEA and ∆CUP are both right triangles,
we can use the Pythagorean Theorem to find
the length of the other leg
3
• ∆TEA  ∆CUP by SSS (or SAS)
U
5
P
Theorem 5.2
Hypotenuse-Leg Congruence Theorem (HL)
Words; If the hypotenuse and a leg of a right triangle are
congruent to the hypotenuse and a leg of a second right
triangle, then the two triangles are congruent..
Symbols;
A
D
If ∆ABC and ∆DEF are right
triangles, and
H AC  DF , and
L BC  EF ,
then ∆ABC  ∆DEF
B
C
E
F
A
Examples
M
leg
P
C
N
B
leg
OR
A
leg
M
P
N
leg
C
B
HL Theorem
• To use the HL Theorem, you must show
that three conditions are met.
1) There are two right triangles.
2) The triangles have congruent hypotenuses.
3) There is one pair of congruent legs.
Example 1
Is it possible to show that ∆JGH  ∆HKJ
using the HL Congruence Theorem?
Explain your reasoning.
SOLUTION
In the diagram, you are given that ∆JGH and ∆HKJ are
right triangles.
By the Reflexive Property, you know JH  JH (hypotenuse)
and you are given that JG  HK (leg). You can use the HL
Congruence Theorem to show that ∆JGH  ∆HKJ.
Your Turn:
Tell whether the pair of triangles is congruent or not and
why.
ANSWER
Yes; HL ≅ Thm.
Your Turn:
Is there enough given information to prove the
triangles congruent? If there is, state the postulate or
theorem.
FGH, HJK
ANSWER
HL ≅ Thm.
Review:
Triangle Congruence Shortcuts
Triangle Congruence Shortcut
PRACTICE
Does the diagram give enough information to show that
the triangles are congruent? If so, state the postulate or
theorem you would use.
a.
b.
SOLUTION
a.
From the diagram, you know that BAC  DAC,
B  D, and BC  DC. You can use the AAS
Congruence Theorem to show that ∆BAC  ∆DAC.
b.
From the diagram, you know that FG  HG, EG  EG,
and EFG  EHG. Because the congruent angles
are not included between the congruent sides, you
cannot show that ∆FGE  ∆HGE.
Does the diagram give enough information to show that
the triangles are congruent? If so, state the postulate or
theorem you would use.
1.
ANSWER
yes; HL Congruence
Theorem
2.
ANSWER
no
3.
ANSWER
yes; SSS Congruence
Postulate, SAS Congruence
Postulate, or HL Congruence
Theorem
Is there enough given information to prove the
triangles congruent? If there is, state the postulate or
theorem.
1.
ABE, CBD
ANSWER
SAS ≅ Post.
State a third congruence that would allow you to
prove ΔRST ≅ ΔXYZ by the SAS Congruence
postulate.
2. ST ≅ YZ, RS≅ XY
ANSWER
∠S ≅ ∠Y
Joke Time
• What do you call a man who lives in an
envelope?
• Bill.
• What dog can't bark?
• A hot dog.
• What do you call a monkey on a mine
field?
• A Ba-Boom.
Assignment
• Pg 260 – 263:#1 – 39 odd