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Geometry – Unit 4 Targets & Info Name: This Unit’s theme – Proving Triangles Congruent Approximately October 28 – November 19 Use this sheet as a guide throughout the chapter to see if you are getting the right information in reaching each target listed. By the end of Unit 4, you should know how to…
Target found in…
Did I reach the Target?
DIAGRAMS & EXAMPLES!
Identify and use correct vocabulary: SSS, SAS, ASA, AAS, HL, CPCTC (what does it stand for), postulate, corresponding angles, equiangular, equilateral, auxiliary line, alternate interior angles, congruent triangles
Chapter 4
Determine if triangles are congruent using SSS, SAS, ASA, AAS, and HL
Chapter 4 Sections 2-3, 6 pages 226-243 258-264
Calculate the measures of sides and angles of a triangle using the isosceles triangle theorem and its converse, 180o in a triangle, and equilateral triangles
Chapter 4 Section 5, pages 250-256
Prove triangles are congruent by providing statements and reasons to complete a partially completed two column proof
Chapter 4
Complete a blank two column proof about congruent triangles using given information and a diagram.
Chapter 4 Section 7 Pages 265-271
Lesson 1: Congruent Figures and Triangles Classifying Triangles:
I. Side Lengths Equilateral Isosceles Scalene
II. Angle Measurements Acute Equiangular Right Obtuse Be as specific as possible: Classify: Right and Isosceles Triangles: Given: ABC, AB || CD Prove: m∠1+ m∠2 + m∠3 = 180
1 2
3 4 5
A B
C D
Statements Reasons
Theorem: The sum of the interior angles of a triangle is 180o.
Corollary: The acute angles of a right triangle are complementary. Congruence in Triangles: Writing Corresponding Parts: Proving Triangles are Congruent: p. 222 10 – 20, 30 – 34, 36, 38 – 40, 45, 54, 57, 58,
1 2
3
Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.
Lesson 1 Practice: Congruent Figures and Triangles
Lesson 2: Proving Triangles are Congruent Using SSS, SAS, ASA, AAS Congruent Triangles All six parts of a triangle are exactly the same as the corresponding six parts of another triangle Ex: XWY ≅ YZX
Ways to Prove Two Triangles Congruent
1) SSS (Side-‐Side-‐Side) If three sides of one triangle are congruent to the corresponding three sides of another triangle, then the two triangles are congruent. 2) SAS (Side-‐Angle-‐Side) If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the two triangles are congruent. 3) ASA (Angle-‐Side-‐Angle) If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. 4) AAS (Angle-‐Angle-‐Side) If two angles and a non-‐included side of a triangle are congruent to two angles and the non-‐included side of another triangle, then the two triangles are congruent.
Z X
Y W
Name the included angle between the pair of sides given. 1) AB and CB 2) DC and BD 3) AE and CE Determine if there is enough information to prove the triangles congruent. If yes, then state the postulate or theorem that would prove them congruent.
4) 5) 6)
Statements Reasons
A B
C D
E
A B
C D
M N
P
Q
R
X
Y
Z
W
V
Given: AB ≅ CD,AB CD Prove: ABC ≅CDA 1
2
A
B
D
C
Is it possible to prove the triangles congruent? If yes, state what postulate or theorem you would use to prove it. 7) 8) 9) 10) Given: AD || EC, BD ≅ BC Prove: ABD ≅ EBC Statements Reasons
1 2
3 4
A
B
C
D E
Lesson 2 Practice: Proving Triangles are Congruent Using SSS, SAS, ASA, AAS
Lesson 3: Using Congruent Triangles
𝛥𝐴𝐵𝐶 ≅ 𝛥𝑋𝑌𝑍
∠𝐴 ≅ ________ ∠𝐵 ≅ ________ ∠𝐶 ≅ ________
𝐴𝐵 ≅ _________ 𝐵𝐶 ≅ _________ 𝐴𝐶 ≅ ________
CPCTC:
Corresponding Parts of Congruent Triangles are Congruent
Given: 𝐹𝐿||𝐻𝑊,𝐹𝑌 ≅𝑊𝑌
Prove: 𝐹𝐿 ≅𝑊𝐻
Statements Reasons
A
B C X
Y Z F
L
Y
H
W
Auxiliary Line: a line added to a picture to help with a proof
Given: 𝛥𝐹𝐺𝐻 is an isosceles triangle with vertex ∠𝐺
Prove: ∠𝐹 ≅∠𝐻
Statements Reasons
ISOSCELES TRIANGLE THEOREM: If two sides of a triangle are congruent, then the angles opposite from those two sides are also congruent. x = ____________ y = _____________ x = ____________ y = _____________ z = ____________
G
H F
xo
yo
65o xo
xo
zo
THEOREM If a triangle is equilateral, it is also equiangular. Given: ∠𝐹 ≅ ∠𝐻 Prove: 𝐹𝐺 ≅ 𝐻𝐺 CONVERSE OF THE ISOSCELES TRIANGLE THEOREM If two angles of a triangle are congruent, then the sides opposite those two angles are also congruent.
x = __________ x = __________
If a triangle is equiangular it is also equilateral.
G
H F
63o 63o
5x – 8 2x + 7 4x -‐ 6
14
18
12
y
x
x = __________ y = __________ z = __________
Given: ∠1 ≅ ∠2,∠3 ≅ ∠4 Prove: 𝐹𝐺 ≅ 𝐻𝐽 Statements Reasons
1 2
4 31
F1
J1
H G
Lesson 3 Practice: Using Congruent Triangles
1. x = __________ 2. x = __________ y = __________ 3. x = __________ 4. x = __________ y = __________ y = __________ z = __________ 5. x = __________ 6. x = __________
40°
x° y° 4x + 12
5x -‐ 2 6x -‐ 4 x°
y° z°
x°
y° 65°
(3x)°
x°
100° x° 60° 60°
75°
7. Given: AB ≅ AC
AD bisects ∠BAC
Prove: BD ≅ CD
Statements Reasons 1. AB ≅ AC 1. 2. ∠B ≅ ∠C 2. 3. AD bisects ∠BAC 3. 4. ∠1 ≅ ∠2 4. 5. ΔABD ≅ ΔACD 5. 6. BD ≅ CD 6. 8. Given: ∠1 ≅ ∠4 Prove: AB ≅ AC Statements Reasons
D B C
A
1 2
C B 4 3 2 1
A
9. Given: RV || ST QS ≅ QT Prove: ∠1 ≅ ∠3 Statements Reasons 10. Given: ∠ ≅∠ ∠ ≅∠1 2 3 4, Prove: BCE ≅ DCE Statements Reasons
4
3
2
1 V R
S T
Q
1 2
3 4 A
B
C
D
E
Lesson 4: Hypotenuse-‐Leg Theorem
Can you prove that the triangles below are congruent?
Hypotenuse Leg Theorem (HL) 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.
Are the following triangles congruent? If they are, state the reason you would use to prove them congruent.
1. __________ 2. __________ 3. ____________
4. __________ 5. __________ 6. ___________
A
B C
X
Y Z
RQ
P
S
RQ
P
S
PLEASE NOTE: To use HL in a proof, you must have 1) a right triangle 2) the hypotenuses congruent 3) a pair of corresponding legs congruenT
Given: PR ⊥ QR
PS ⊥ QS
QR ≅ QS
Prove: ΔPRQ ≅ ΔPSQ
Given: PR ⊥ QR
PS ⊥ QS
PQ bisects ∠SPR
Prove: ΔPRQ ≅ ΔPSQ
y°
x°
140°
y°150° x°
60°150°(3x+15)°
(2x-15)°
(2x-10)°
Lesson 4 Practice: Hypotenuse-‐Leg Theorem Determine if each pair of triangles is congruent. If they are, tell which postulate or theorem you could use to prove them congruent. Mark the picture with congruence markings if necessary. 1. ____________ 2. ____________ 3. ____________ 4. ____________ 5. ____________ 6. ____________ 7. ____________ 8. ____________ 9. ____________
10. ____________ 11. ____________ 12. ____________ 13. x = __________ 14. x = __________ 15. x = __________ y = __________ y = __________
Y
X
Z
W
21
K
F A
L
J
16. Given: ∠W is a right angle
∠Y is a right angle
WX ≅ YX
Prove: WZ ≅ YZ
Statements Reasons 1. ∠W is a right angle 1. ∠Y is a right angle 2. ΔXWZ and ΔXYZ are right triangles 2. 3. WX ≅ YX 3. 4. XZ ≅ XZ 4. 5. ΔXWZ ≅ ΔXYZ 5. 6. WZ ≅ YZ 6. 17. Given: LF ≅ KF
LA ≅ KA
Prove: LJ ≅ KJ
Statements Reasons 1. LF ≅ KF 1.
LA ≅ KA 2. FA ≅ FA 2. 3. ΔFAL ≅ ΔFAK 3. 4. ∠1 ≅ ∠2 4. 5. FJ ≅ FJ 5. 6. ΔFJL ≅ ΔFJK 6. 7. LJ ≅ KJ 7.
H
E
G
F
(2x - 12) ft (x + 5) ft z°y°
x°y°
x°40°
18. x = __________ 19. x = __________ 20. x = __________ y = __________ y = __________ z = __________ 21. Given: EF ⊥ EG
HG ⊥ EG
EH ≅ GF
Prove: ∠H ≅ ∠F
Statements Reasons
Unit 4 Test Review Tell if each pair of triangles can be proved congruent. If the answer is yes, state a reason you would use to prove them congruent. Mark each pictures with the appropriate congruence signs. 1. ____________ 2. ____________ 3. ____________ 4. ____________ 5. ____________ 6. ____________ 7. ____________ 8. ____________ 9. ____________ 10. ____________ 11. ____________ 12. ____________
z°
y°
x°x°6
66
(3x-10)°(x+30)°
60°3x+24
5x
y°
x°
120°
55°
x° 50°
x
37 cm
33 cm
54° 63°
x°(2x)°
(4x+10)°
(5x+5)°
y°x°25°65°
70°
y°x°
52°
87°
31°
13. x = __________ 14. x + y + z = ________ 15. x =__________ 16. x = __________ 17. x = ____________ 18. x =__________ 19. x = __________ 20. x = ____________ 21. x =____________ y = __________ y = ____________ y = ___________ 22. x = __________ 23. x =__________
432
1ZX
Y
T F 24. A corollary is a statement that is very easy to prove from a theorem. T F 25. A theorem is a statement that we accept without proof. T F 26. If two sides of one triangle are congruent to two sides of another triangle, then the third
sides are congruent also. T F 27. An equilateral triangle is also equiangular. T F 28. We can draw auxiliary lines in a diagram to help us with a proof. 29. Name 5 ways to prove triangles congruent. 1. 2. 3. 4. 5. 30. CPCTC stands for ______________________________________________________ 31. The measure of the vertex angle of an isosceles triangle is 118°. What is the measure of a base
angle? 32. In ΔQRS, ∠Q ≅ ∠S, QR = 3t + 4, RS = 5t – 8, and QS = 4t – 12. Find t and the perimeter of the
triangle. 33. Given: XY ≅ ZY Prove; ∠1 ≅ ∠4 Statements Reasons
4321
Y
QS
X
R
C
MB D
A
34. Given: ∠1 ≅ ∠3 RX ≅ RY Prove: ΔRXS ≅ ΔRYS Statements Reasons 35. Given: AM ≅ CM AB ⊥ BD CD ⊥ BD M is the midpoint of BD Prove: ∠A ≅ ∠C Statements Reasons