Download 4.5 HL and overlapping triangles ink.notebook

4.5 HL and overlapping triangles ink.notebook
November 14, 2016
Page 168
Page 167
Page 166
4.5 HL and
Overlapping
Triangles
Lesson Objectives
Standards
Lesson Notes
4.5 HL and Overlapping Triangles
Press the tabs to view details.
Lesson Objectives
Standards
Lesson Notes
After this lesson, you should be able to successfully use HL to prove triangles are congruent. You will also learn how to prove overlapping triangles are congruent. Press the tabs to view details.
1
4.5 HL and overlapping triangles ink.notebook
Lesson Objectives
Standards
Lesson Notes
G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
November 14, 2016
HYPOTENUSE­LEG (HL)
CONGRUENCE THEOREM
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second triangle, then the two triangles are ______________.
B
E
G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
G.CO.10 Prove theorems about triangles.
A
C
D
F
For each diagram, determine which pairs of triangles can be proved congruent by the HL Theorem.
a)
b)
c)
2
4.5 HL and overlapping triangles ink.notebook
November 14, 2016
E
1. Given: AC ¤ EC, AB ¼ BD, ED ¼ BD, and AC is a bisector of BD A
Prove: ÆABC ¤ ÆEDC B
Statements
D
C
Reasons
1. AC ¤ EC,AB ¼ BD, ED ¼ BD, 1.
and AC is a bisector of BD
2. ÚB and ÚD are 2. Defn of _____________
3. ÆABC and ÆEDC are 3. Defn of _____________ 4. BC ¤ DC 4. Defn of _____________
5. ÆABC ¤ ÆEDC 5.
Sometimes when triangles overlap each other, they share corresponding parts.
When trying to picture the corresponding parts, try redrawing the triangles separately.
B
C
30¡
A
D
B
A
C
30¡
D
A
D
2. In the figure ΔABD ¤ ΔDCA and mÚBDA = 30°.
Find mÚCAD = _____ and mÚCDA = ______
3
4.5 HL and overlapping triangles ink.notebook
November 14, 2016
3. In the figure ΔABC ¤ ΔDCB, mÚABC = 110, and mÚD = 20. Find mÚACB = _______ and mÚDCB = ________
D
A
B
C
4. Given: ÚA and ÚB are rt Ús, ÚACD ¤ ÚBDC, AD ¤ BC, AC ¤ BD A
B
Prove: ÆADC ¤ ÆBCD D
Statements
30
C
Reasons
1. 1.
2. 2.
3. 3. 4. 4.
5. 5.
4
4.5 HL and overlapping triangles ink.notebook
November 14, 2016
PRACTICE
On the
Worksheet
Using Congruent Triangles CPCTC
1.
R
P
A
2.
1
B
C
D
S
Y
A
2
E
5
4.5 HL and overlapping triangles ink.notebook
Tell whether the HL Theorem can be applied to prove the triangles congruent. If possible, write the triangle congruence.
3. B
4.
A
R
G
Tell whether the HL Theorem can be applied to prove the triangles congruent. If possible, write the triangle congruence.
U
5.
T
M
F
D
Z
C
O
L
K
8.
H
6.
T
E
H
F
N
W
Tell whether the HL Theorem can be applied to prove the triangles congruent. If possible, write the triangle congruence.
7.
November 14, 2016
Y
9. Given: ÚYEB & ÚYET are rt angles and ÚYBE ¤ ÚYTE
Prove: èBYE ¤ èTYE
I
B
N
K
H
E
T
J
A
6
4.5 HL and overlapping triangles ink.notebook
State if the two triangles are congruent. If they are, state how you know.
10.
11.
November 14, 2016
State if the two triangles are congruent. If they are, state how you know.
12.
13.
State if the two triangles are congruent. If they are, state how you know.
14.
15.
Missing
Parts
7
4.5 HL and overlapping triangles ink.notebook
November 14, 2016
16. Given: Ú1 ¤ Ú2 Prove: èABD ¤ èCBD
By SAS
Name the Part or Parts needed to prove the è’s ¤ by the given è¤ theorem. You may mark the reflexive sides or angles without stating them.
A
D
1 2
B
C
Need: _______________________
17. Given: ÚA ¤ ÚC Prove: èEAB ¤ èDC
By SAS
E
A
18. Given: Ú1 ¤ Ú2 Prove: èDAB ¤ èDCB
By AAS
D
B
C
D
A
12
B
C
Need: _______________________
& _______________________
Need: _______________________
8
4.5 HL and overlapping triangles ink.notebook
November 14, 2016
20.
19.
A
1
B
C
2
Prove: èABD ¤ èBAC
By SSS
Hint: Draw separate è’s B
A
If needed.
E
D
D
C
E
Need: _______________________
Need: _______________________
& _______________________
21. Given: NO = NM
Prove: èMNQ ¤ èONP
By SSS
M
Q
N
O
22. Given: AB = BC
Prove: èABD ¤ èCBD
By HL
A
B
1 2
C
D
P
Need: _______________________
Need: _______________________
& _______________________
& _______________________
9
4.5 HL and overlapping triangles ink.notebook
November 14, 2016
24. Given: Ú1 ¤ Ú2 Prove: èABD ¤ èCBD
By ASA
23. Given: ÚB & ÚD are right Ú’s
Prove: èADC ¤ èCBD
By HL
A
D
34
B
D
C
A
Need: _______________________
(2 possible answers)
D
25.
5
C
Need: _______________________
26. Given: èPQR ¤ èSQV
Name all pairs of corresponding sides and angles.
34
6
12
B
P
Q
V
R
5
A
1 2
B
6
C
_____ ¤ _____ Ú_____ ¤ Ú _____ S
_____ ¤ _____ Ú_____ ¤ Ú _____ Need: _______________________
_____ ¤ _____ Ú_____ ¤ Ú _____ & _______________________
10
4.5 HL and overlapping triangles ink.notebook
November 14, 2016
Answers:
1. Given, ÚPSY ¤ ÚASR, Vert Ú Thm, èPSY ¤ èASR, AAS, CPCTC 3. Yes, èABD ¤ èDCA 5. Yes, èMNU ¤ èMNL 7. Yes, èOKN ¤ èAHN 9. Given, all rt Ú’s are ¤, reflexive, AAS 11. No 13. No 15. Yes, SAS 19. ÚD ¤ ÚE, Ú1 ¤ Ú2 25. Ú1 ¤ Ú2, Ú5 ¤ Ú6 11