Elizabeth Pawelka
HL Theorem and CPCTC
2/28/12 p.1
Geometry
Lesson Plans
Section 4-6: Congruence in Right Triangles (HL Theorem)
Section 4-4: Using Congruent Triangles (CPCTC)
Section 4-7: Using CPCTC
2/28/12 (60 minutes today)
Statement of Objectives
The student will be able to…
 prove right triangles congruent using the HL Theorem,
 use triangle congruence and CPCTC to prove that parts of two triangles are congruent, and
 identify congruent overlapping triangles
 (prove two triangles congruent by first proving two other triangles congruent – if time)
Warm-up and Review of Homework (30 minutes)
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
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
Warm-up: Practice 4-3, p. 44 in Practice Book: 1 - 11 (10 minutes)
They do warm-up while I check homework from previous night to see if there are any common
errors I should address during review
Go over Warm-up (5 minutes)
Review Homework from last night: (15 minutes) pp. 197 – 199: 1 – 29, 31-34 (attached) ask for
any specific questions. Go over the Proofs for 31-34:
31. Given: ∠N ≅ ∠P; MO ≅ QO
Conclusion: ΔMON ≅ ΔQOP
MO ≅ QO
Given (S)
∠N ≅ ∠P
Given (A)
∠MON ≅ ∠QOP
Vert. ∠ (A)
ΔMON ≅ ΔQOP
AAS
∠F ≅ ∠H
Given (A)
FG || JH
Given
Elizabeth Pawelka
HL Theorem and CPCTC
32. Given: ∠F ≅ ∠H; FG || JH
Conclusion: ΔFGJ ≅ ΔHJG
33. Given: AE || BD ; AE ≅ BD ; ∠E ≅ ∠D
Conclusion: ΔAEB ≅ ΔBDC
34. Given: DH bisects ∠BDF; ∠1 ≅ ∠2
Conclusion: ΔBDH ≅ ΔFDH
2/28/12 p.2
∠FGJ ≅ ∠HJG
AIA (A)
GJ ≅ JG
Reflexive (S)
ΔFGJ ≅ ΔHJG
AAS
∠E ≅ ∠D
Given (A)
AE ≅ BD
Given (S)
AE || BD
Given
∠EAB ≅ ∠DBC
Corresponding ∠’s (A)
ΔAEB ≅ ΔBDC
ASA
∠1 ≅ ∠2
Given (A)
DH ≅ DH
Reflexive (S)
DH bisects ∠BDF Given
∠BDH ≅ ∠FDH
Def of angle bisector (A)
ΔBDH ≅ ΔFDH
ASA
Elizabeth Pawelka
HL Theorem and CPCTC
2/28/12 p.3
Teacher Input (25 minutes)
Section 4-6: HL Theorem
Activity (5 minutes) – if time allows
 Introduce HL Theorem with the Geogebra file Sec4-6-HL.ggb. It has two right triangles whose
lengths can be changed. Show that, if one pair of legs and the hypotenuses have the same length, the
other legs will be the same, too, giving SSS congruence.
 There may be some cases in which it doesn’t quite match up at first, but explain that is due to
rounding considerations.
 Also point out that this is a special case of SSA for right angles only.
Lecture/Examples: (10 minutes)
 Review vocabulary hypotenuse and legs of right triangles. Show example:
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Hypotenuse: longest side and across from right angle
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Legs: the other two sides
Introduce the Hypotenuse-Leg Theorem on same slide: “If the hypotenuse and a leg of one right
triangle are congruent to the hypotenuse and a leg of another right triangle, then the two triangles are
congruent.”
HL Theorem cont’d
Point out that while this looks like SSA it is
ONLY for right triangles. Explain that the sides
of a right triangle have a definite relationship due
to the Pythagorean Theorem and that is why we
know that if two sides are congruent, then the
third side also is.
Example 1: HL Theorem
Elizabeth Pawelka
HL Theorem and CPCTC
2/28/12 p.4
Given: ∠P and ∠R are right angles
PS ≅ RQ
Prove: ΔPQS ≅ ΔRSQ
Point out you have to always show they are right triangles AND H and L are congruent.
∠P and ∠R are right angles:
Given
ΔPQS & ΔRSQ are right triangles: Def of right triangles
QS = SQ (Hypotenuse)
Reflexive Prop
PS ≅ RQ (Leg)
Given
ΔPQS ≅ ΔRSQ
HL Thm
 Example 2: HL Theorem
Given: MJ ┴ NK
MN ≅ MK
Prove: ΔMJN ≅ ΔMJK
MJ ┴ NK
Given
∠MJN and ∠MJK are
right angles
Def of perpendicular
ΔMJN and ΔMJK
are right Δ’s
Def of right Δ’s
MN ≅ MK (H)
Given
MJ = MJ (L)
Reflexive Prop
Flow Proof for slide:
e
___________
ΔPQS ≅ ΔRSQ
HL Theorem
Elizabeth Pawelka
HL Theorem and CPCTC
2/28/12 p.5
e
______________
e
______________
e
______________
e
___________
e
______________
Section 4-4/4-7: CPCTC
Lecture/Examples (15 minutes)
 Section 4-4 and 4-7 Review:
4) CA=JS; AT=SD; CT=JD
5) ∠C≅∠J; ∠A≅∠S; ∠T≅∠D
6) WX=JK; XY=KL; YZ=LM; WZ=JM
7) ∠W≅∠J; ∠X≅∠K; ∠Y≅∠L; ∠Z≅∠M

Introduce CPCTC definition: Once you have congruent triangles, you can make conclusions about
other parts of the triangles because, by definition, Corresponding Parts of Congruent Triangles are
Congruent (CPCTC)
Elizabeth Pawelka
HL Theorem and CPCTC
2/28/12 p.6

Example 3: Real-life example: Using CPCTC to measure distances (Note this is in the textbook on
p. 204 if there is not time to go into this.)
Tell story of Napoleon’s officer measured the width of a river using CPCTC. “He used his
vision and set his visor so the farthest thing he could see was the edge of opposite bank. Then he turned
down the river to note the farthest spot he could see on the bank and declared that distance the width of
the river. Was he right? Let’s prove it” Show the picture and Statements and fill in the Reasons
together.

CPCTC Notes – emphasize these ideas as a difference in CPCTC proofs
 Watch for what you are being asked to prove!
 CPCTC proofs are asking for parts to be congruent as the final statement, not the triangles.
 BUT, you have to prove the triangles are congruent first.

Example 4: Go over it with them – point out to look for what you are trying to prove and that
CPCTC proofs aren’t looking for triangle congruence as the final statement. You have to prove
triangles are congruent to prove a segment or angle congruency.

Example 5: They fill in blanks
Elizabeth Pawelka
a)

HL Theorem and CPCTC
; b) Def of angle bisector; c) QB=QB; d) SAS; e) CPCTC
Example 6: Overlapping Triangles (be sure to exaggerate what the triangles are)
Given:
Prove:
Statements
1.
2.
3.
4.

2/28/12 p.7
Reasons
1. Given
2. Given
(S)
(A)
3. Reflexive Property of ≅
(S)
4. SAS
Example 7: Overlapping Triangles: They help me fill it in.
Prove: ΔACD ≅ ΔECB
Statements
Statements
1. AB = ED,
2. BC = DC
3. ∠C ≅ ∠C
4. AB+BC = ED+DC
5. AC = EC
Reasons
Given
Given
Reasons
(S)
Reflexive
(A)
Segment Addition Postulate
Substitution (S)
Elizabeth Pawelka
6. ΔACD ≅ ΔECB
HL Theorem and CPCTC
2/28/12 p.8
SAS
The next two examples can be used if there is time to also teach how to use two congruent triangles to
prove two other triangles are congruent:

Example 8: Overlapping Triangles (be sure to exaggerate what the triangles are)
Sometimes you can prove one pair of triangles congruent and then use corresponding parts of those
triangles to prove another pair congruent.
Given:
(proven in Example 6)
Statements
1.
Reasons
1. Given
2. Given
3. CPCTC
4. CPCTC
5. SAS
2.
3.
4.
5.

Example 9: Overlapping Triangles: They help me fill it in.
ΔACD ≅ ΔECB (proved in Example 7)
Statements
Statements
Reasons
____________ Reasons
1. ΔACD (blue) ≅ ΔECB (green)
Given (previously proven)
2. ∠A ≅ ∠E
CPCTC
(A)
(S)
(A)
(S)
Elizabeth Pawelka
HL Theorem and CPCTC
3. ∠AFB ≅ ∠EFD
4. AB = ED
5.
Vertical angles
Given
AAS
2/28/12 p.9
(A)
(S)
Closure (5 minutes)

Today we learned to
 prove right triangles congruent using the HL Theorem,
 use triangle congruence and CPCTC to prove that parts of two triangles are congruent,
 identify congruent overlapping triangles, and
 (prove two triangles congruent by first proving two other triangles congruent – if time)

Tomorrow we’ll learn how to use and apply properties of isosceles and equilateral triangles
Homework
pp. 219-220: 1-8, 11, 12, 16, 17, 19, 20, 28
pp. 205-207: 7-17, 19, 23
pp. 227: 10, 11