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Unit 4: Transformational Geometry Lesson 16 Part I
CPCTC
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Date ________________________________ Block _______
Lesson 16: Corresponding Parts of Congruent Triangles Are Congruent Warm-‐up
1. Create a picture of right triangles where you would have to use HL to prove the triangles are congruent. State the givens.
2. Create a picture of right triangles where you would have to use SAS to prove the triangles are congruent. State the givens.
3. a. Why does the theorem stated below make sense? Explain in your own words / include diagrams if helpful.
If two angles are complementary to two congruent angles, then the angles are congruent.
b. How is the theorem in part a different then the theorem written below:
If two angles are complementary to the same angle, then the angles are congruent.
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Unit 4: Transformational Geometry Lesson 16 Part I
CPCTC
In many proofs, you will not simply be proving two triangles are congruent. Instead, you will be using congruent triangles to prove other statements about a diagram. We have already seen examples of these, but NOW it will be your task to:
a) Decide what triangles you need to prove congruent, and b) Actually prove them to be congruent, then c) Use the congruent triangles to reach the final conclusion.
Some helpful hints:
• Mark up your diagram with the given information • Think first, then write.
o Start with a mental outline and/or write an outline of your steps • Start with what you want to prove and work backwards
o What information do you need to make your final claim? o Which triangles may be helpful to prove congruent?
• Think of the additional information you can bring in to help reach your conclusion Two Definitions you will need in this section:
Median of a Triangle – extends from a vertex to the midpoint of the opposite side. It therefore, bisects the opposite side. Altitude of a Triangle -‐-‐ an altitude is a segment drawn from a vertex of a triangle to the line containing the opposite side, extended if necessary, so that it is perpendicular to the line containing the opposite side.
Class Examples Example 1
Given: OQ bisects ∠NQP
QO is an altitude of ΔNPQ
Prove: ΔNQP is isosceles
ON P
Q
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Unit 4: Transformational Geometry Lesson 16 Part I
CPCTC
Example 2 Given:
C is the midpoint of Prove:
∠B ≅ ∠YBY
AB ≅ ZYC
Y
BA
Z
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Unit 4: Transformational Geometry Lesson 16 Part I
CPCTC
Practice Problems Of course, they can get more complicated….(this is similar to #7 from lesson 15)
1. Given: 𝐽𝐾 = 𝐽𝐿, 𝐽𝑋 = 𝐽𝑌. Prove: 𝐾𝑋 = 𝐿𝑌 (What two triangles must congruent in order to use CPCTC as a reason as you work towards your final claim?)
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Unit 4: Transformational Geometry Lesson 16 Part I
CPCTC
2. Given: T is the midpoint of RW
RS ≅ TV RS ⊥ ST , TV ⊥ VW Prove: ∠RTS ≅ ∠TWV
R
Q W
S T
V
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Unit 4: Transformational Geometry Lesson 16 Part I
CPCTC
Sometimes, you may need to use TWO sets of congruent triangles.
3. Given: ∠A ≅ ∠C AR ≅ CS TR ⊥ AB, TS ⊥ BC Prove: TB bisects ∠RBS
(Hint, what angles do you need to prove congruent, in order to prove that the given angle is bisected?)
SR
TA C
B
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Unit 4: Transformational Geometry Lesson 16 Part I
CPCTC
4. Given: !1≅!2 and !7≅!8 Prove: !5≅!6
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Unit 4: Transformational Geometry Lesson 16 Part I
CPCTC
5. Given: 𝐴𝐵 = 𝐴𝐶, 𝑅𝐵 = 𝑅𝐶,
Prove: 𝑆𝐵 = 𝑆𝐶
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Unit 4: Transformational Geometry Lesson 16 Part I
CPCTC
6. Given: 𝐴𝐷 ⊥ 𝐷𝑅,𝐴𝐵 ⊥ 𝐵𝑅,
𝐴𝐷 = 𝐴𝐵. Prove: ∠𝐷𝐶𝑅 ≅ ∠𝐵𝐶𝑅.
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Unit 4: Transformational Geometry Lesson 16 Part I
CPCTC
7. Given: 𝑚∠𝑤 = 𝑚∠𝑥 and 𝑚∠𝑦 = 𝑚∠𝑧.
Prove: i. △ 𝐴𝐵𝐸 ≅△ 𝐴𝐶𝐸. ii. 𝐴𝐵 = 𝐴𝐶 and
iii. 𝐴𝐷 ⊥ 𝐵𝐶.
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Unit 4: Transformational Geometry Lesson 16 Part I
CPCTC
Homework Problem Set #1 I. A few review problems: 1. Find the value of 𝒚 in the diagram.
2. Find the values of 𝒘,𝒙,𝒚, 𝐚𝐧𝐝 𝒛.
II. Proofs
3. Given: Circle with center O. RO ⊥ MP
Prove: MR ≅ PR
OM P
R
(y)°(x+25)°(2x+7)°
z°
w°
y°x° 50°
25°
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Unit 4: Transformational Geometry Lesson 16 Part I
CPCTC
4. Given: 𝑨𝑪 ⊥ 𝑫𝑩, 𝑬𝑭 ⊥ 𝑫𝑩, 𝑨𝑪 ≅ 𝑬𝑭,∡𝑨 ≅ ∡𝑬
Prove: !B ≅!D
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Unit 4: Transformational Geometry Lesson 16 Part I
CPCTC
5. Given: 𝑩𝑫,𝑨𝑫 ≅ 𝑪𝑫,∡𝟑 ≅ ∡𝟒 Prove: 𝑩𝑫 bisects !ABC
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Unit 4: Transformational Geometry Lesson 16 Part I
CPCTC
Homework Problem Set #2
1. Find the values of 𝑥,𝑦,𝑎, 𝑏, 𝑐,𝑑, 𝑒, 𝑓,𝑔, and ℎ.
2. Given: 𝐵𝐹 ⊥ 𝐴𝐶,𝐶𝐸 ⊥ 𝐴𝐵.
𝐴𝐸 = 𝐴𝐹
Prove: CE ≅ BF
g°
h°e°
f°c° d°
b°
a°(6x)°
(5x+25)°
(7y+5)°(3y-15)°
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Unit 4: Transformational Geometry Lesson 16 Part I
CPCTC
3. Given: 𝐴𝐵 ⊥ 𝐵𝐶, 𝐵𝐶 ⊥ 𝐷𝐶. 𝐷𝐵 bisects ∠𝐴𝐵𝐶, 𝐴𝐶 bisects ∠𝐷𝐶𝐵.
Prove: AB ≅ DC
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Unit 4: Transformational Geometry Lesson 16 Part I
CPCTC
4. Given: ∠1 = ∠2,∠3 = ∠4. Prove: 𝐴𝐶 = 𝐵𝐷.
