Download HL Triangle Congruence

LESSON
6.3
Name
HL Triangle Congruence
6.3
G.6.B
Explore
Prove two triangles are congruent by applying the . . . Hypotenuse-Leg
congruence conditions. Also G.2.B, G.5.C
Is There a Side-Side-Angle
Congruence Theorem?
Follow these steps to draw △ABC such that m∠A = 30°, AB = 6 cm, and BC = 4 cm.
The goal is to determine whether two side lengths and the measure of a non-included
angle (SSA) determine a unique triangle.
G.1.F
Analyze mathematical relationships to connect and communicate
mathematical ideas.

Use a protractor to draw a large 30° angle on a separate sheet of paper.
Label it ∠A.

Use a ruler to locate point B on one ray of ∠A so that AB = 6 cm.
Language Objective
1.A, 2.C.3, 2.C.4, 2.D.1
Explain the HL Congruence Theorem in your own words.
A
_
Now draw BC so that BC = 4 cm. To do this, open a compass to a
distance of 4 cm. Place the point of the compass on point B and draw
_
an arc. Plot point C where the arc intersects the side of ∠A. Draw BC

ENGAGE
View the Engage section online. Discuss the photo,
asking students to describe the shape of the kite in
terms of angles, triangles, and any other geometrical
terms that seem relevant. Then preview the Lesson
Performance Task.
What do you notice? Is it possible to draw only one △ABC with the
given side length? Explain.
→
‾ in two places. So, it is
If extended, the arc would intersect AC
© Houghton Mifflin Harcourt Publishing Company

possible to draw two different triangles with side length BC.
6 cm
A
30°
B
6 cm
A
30°
4 cm
C
Reflect
1.
Do you think that SSA is sufficient to prove congruence? Why or why not?
No. SSA is not sufficient to determine congruence because a given set of values does not
necessarily describe a unique triangle. It is possible to draw two different triangles that
have two congruent sides and a congruent non-included angle.
2.
Discussion Your friend said that there is a special case where SSA can be used to prove
congruence—namely, if the non-included angle is a right angle. Is your friend right? Explain.
Yes; if the congruent non-included angle were a right angle, then SSA would work. Given a right
angle, one set of congruent sides would be legs and the other set the hypotenuses. Given a leg
and the hypotenuse of a right triangle, the Pythagorean theorem guarantees a unique triangle.
Module 6
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
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
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© Houghto
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Reflect
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Module 6
6L3 331
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ESE3538
GE_MTX
Lesson 6.3
30°
B
to complete △ABC.
6.3
331
Resource
Locker
You have already seen several theorems for proving that triangles are congruent. In this
Explore, you will investigate whether there is a SSA Triangle Congruence Theorem.
Mathematical Processes
PREVIEW: LESSON
PERFORMANCE TASK
HL Triangle Congruence
G.6.B Prove two triangles are congruent by applying the ... Hypotenuse-Leg congruence
conditions. Also G.2.B, G.5.C
The student is expected to:
If a leg and the hypotenuse of one right triangle are
congruent to the corresponding leg and
hypotenuse of another right triangle, the triangles
are congruent.
Date
Essential Question: What does the HL Triangle Congruence Theorem tell you about two
triangles?
Texas Math Standards
Essential Question: What does the HL
Triangle Congruence Theorem tell you
about two triangles?
Class
04/02/15
11:54 AM
04/02/15 11:54 AM
Explain 1
Justifying the Hypotenuse-Leg Congruence Theorem
In a right triangle, the side opposite the right angle is the hypotenuse.
The two sides that form the sides of the right angle are the legs.
EXPLORE
hypotenuse
You have learned four ways to prove that triangles are congruent.
Is there a Side-Side-Angle
Congruence Theorem?
legs
• Angle-Side-Angle (ASA) Congruence Theorem
• Side-Angle-Side (SAS) Congruence Theorem
• Side-Side-Side (SSS) Congruence Theorem
• Angle-Angle-Side (AAS) Congruence Theorem
INTEGRATE TECHNOLOGY
The Hypotenuse-Leg (HL) Triangle Congruence Theorem is a special case that allows you to show that two right
triangles are congruent.
Students can use geometry software to explore what
you can do with two sides and a non-included angle.
Hypotenuse-Leg (HL) Triangle Congruence Theorem
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.
Example 1
QUESTIONING STRATEGIES
B
Prove the HL Triangle Congruence Theorem.
E
c
a
Given: △ABC and △DEF are right triangles;
∠C and ∠F are right angles.
_ _
_ _
AB ≅ DE and BC ≅ EF
C
b
d
A
Could you use SAS on the triangles we are
discussing here? Explain. No; SAS requires
the angle to be included between the two pairs of
congruent sides.
f
F
D
e
Prove: △ABC ≅ △DEF
2
2
By the Pythagorean Theorem, a 2 + b 2 = c 2 and d + e = f 2. It is given that
2
2
_ _
AB ≅ DE, so AB = DE and c = ƒ. Therefore, c 2 = f 2 and a 2 + b 2 = d + e . It is given that
_ _
BC ≅ EF, so BC = EF and a = d. Substituting a for d in the above equation, a 2 + b 2 = a
2
2
+ e
EXPLAIN 1
2
2
the SSS Triangle Congruence Theorem .
Your Turn
3.
V
Determine whether there is enough information to prove that
triangles △VWX and △YXW are congruent. Explain.
Yes. △VWX and △YXW are
_right
_ triangles
_
that share hypotenuse WX . WX ≅ WX by the
Reflexive
of Congruence. It is given
_ Property
_
that WV ≅ XY , therefore △VWX ≅ △YXW by
the HL Triangle Congruence Theorem.
Module 6
332
W
Z
Y
X
© Houghton Mifflin Harcourt Publishing Company
Therefore, △ABC ≅ △DEF by
Justifying the Hypotenuse-Leg
Congruence Theorem
.
Subtracting a from each side shows that b = e , and taking the square root of each side, b = e .
_
_
This shows that AC ≅ DF .
2
AVOID COMMON ERRORS
Students may use hypotenuse to describe the longest
side of any triangle. Remind them that the term is
used only with right triangles.
QUESTIONING STRATEGIES
With what kind of triangles can you use the
Pythagorean Theorem? right triangles only
Lesson 3
PROFESSIONAL DEVELOPMENT
GE_MTXESE353886_U2M06L3.indd 332
Integrate Mathematical Processes
1/22/15 4:07 AM
This lesson provides an opportunity to address Mathematical Process
TEKS G.1.F, which calls for students to “analyze mathematical relationships to
connect and communicate mathematical ideas.” Students look at pairs of triangles
that have two congruent sides and congruent non-included angles. They analyze
these relationships to determine that this information is sufficient only to prove
right triangles congruent.
HL Triangle Congruence 332
Explain 2
EXPLAIN 2
Example 2
Applying the Hypotenuse-Leg
Congruence Theorem

Use the HL Congruence Theorem to prove that the triangles are
congruent.
_ _
Given: ∠P and ∠R are right angles. PS ≅ RQ
P
Q
S
R
Prove: △PQS ≅ △RSQ
QUESTIONING STRATEGIES
Statements
Are all right angles congruent? Explain. Yes; a
right angle is an angle that measures 90°, so
all right angles have the same measure. This means
that all right angles are congruent.

AVOID COMMON ERRORS
Students may assume that the hypotenuses of two
given right triangles are congruent. Remind them not
to assume anything is true, and to use only the
information that is given.
Reasons
1. ∠P and ∠R are right angles.
_ _
2. PS ≅ RQ
_ _
3. SQ ≅ SQ
1. Given
4. △PQS ≅ △RSQ
4. HL Triangle Congruence Theorem
2. Given
3. Reflexive Property of Congruence
Given: ∠J and ∠L are right angles. K is the midpoint
_
_
of JL and MN.
Prove: △JKN ≅ △LKM
M
J
K
L
N
Statements
© Houghton Mifflin Harcourt Publishing Company
CONNECT VOCABULARY
Have students label the parts of a right triangle,
identifying the hypotenuse, the legs, and the right
angle, and then measure and write the measures of
the other two angles.
Applying the HL Triangle Congruence Theorem
Reasons
1. ∠J and ∠L are right angles.
_
_
2. K is the midpoint of JL and MN .
_ _
_ _
3. JK ≅ LK and MK ≅ NK
1. Given
4. △JKN ≅ △LKM
4. HL Triangle Congruence Theorem
2. Given
3. Definition of midpoint
Reflect
4.
Is it possible to write the proof in Part B without using the HL Triangle Congruence Theorem? Explain.
_ _
Yes, you can use the SAS Triangle Congruence Theorem (∠J ≅ ∠L, JK ≅ LK, and
vertical angles JKN and LKM are congruent) or the SAS Triangle Congruence
_ _
_ _
Theorem (JK ≅ LK and MK ≅ NK, and vertical angles JKN and LKM are congruent).
Your Turn
Use the HL Congruence Theorem to prove that the triangles are congruent.
_ _
5. Given: ∠CAB and ∠DBA are right angles. AD ≅ BC
A
Prove: △ABC ≅ △BAD
_ _
It is given that and ∠CAB and ∠DBA are right angles and AD ≅ BC.
_ _
AB ≅ AB by the Reflexive Property of Congruence. Then
C
D
Module 6
Lesson 3
B
△ABC ≅ △BAD by the HL Triangle Congruence Theorem.
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COLLABORATIVE LEARNING
GE_MTXESE353886_U2M06L3.indd 333
Small Group Activity
Have students work in small groups to illustrate the differences between the HL
and SAS Triangle Congruence Theorems. They may convey the information in
any way, including making a poster, writing an essay, or creating a model. Have
each group present their project to the class.
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Lesson 6.3
1/22/15 4:07 AM