Download Foundations of Math III Unit 4: Logical Reasoning (Part 2)

Foundations of Math III
Unit 4: Logical
Reasoning (Part 2)
in Geometry
1.12 Practice Making Mini-Proofs
Fill in the missing pieces for each mini-proof.
̅̅̅̅
M is the midpoint of 𝐴𝐵
A
̅̅̅̅ bisects ̅̅̅̅
𝐸𝐷
𝐴𝐵
M
C
Definition of
Segment
Bisector
̅̅̅̅̅ ≅̅̅̅̅̅
If 𝐴𝑀
𝑀𝐶
Definition of
Congruent
Segments
then ________________
̅̅̅̅ ≅ ̅̅̅̅
𝐴𝐶
𝐶𝐵
1
________ ⊥ ________
Definition of
Perpendicular
Lines
∠𝐹𝐻𝐼 and ∠𝐼𝐻𝐺 are supplementary angles
∠𝐹𝐻𝐼 and ∠𝐼𝐻𝐺 are
supplementary angles
∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4
2
⃡𝑀𝑁 is the perpendicular
bisector of ̅̅̅
𝐽𝐾
𝑇𝑉 bisects ∠𝑆𝑇𝑈
Definition of
Midpoint
𝑚∠_________ = 𝑚∠_________
JL = LK
3
̅̅̅̅ is the perpendicular
𝐶𝐺
bisector of ̅̅̅̅
𝐷𝐸
Definition of
Segment Bisector
̅̅̅̅
𝐺 is the midpoint of 𝐷𝐸
𝐵 is the midpoint of ̅̅̅̅̅
𝑊𝐴
_______________ ≅ _____________
_______________ = _____________
4
1.13 Homework - Triangle Congruence Theorems
Determine which theorem can be used to prove that the triangles are congruent. If it is not
possible to prove that they are congruent, write not possible.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
5
Flow Proof Examples
1)
C is the
midpoint of
̅̅̅̅
𝐴𝐷
A  D
1  2
̅̅̅̅
𝐴𝐶  ̅̅̅̅
𝐷𝐶
ABC  ___________
2)
̅̅̅̅̅
𝑀𝑁 ∥ ̅̅̅̅
𝑃𝑇
3  4
̅̅̅̅
𝑁𝑂  ̅̅̅̅
𝑇𝑂
1  2
MNO  ___________
3)
̅̅̅̅ bisects
𝐵𝐷
ABC
̅̅̅̅
𝐵𝐷  ̅̅̅̅
𝐵𝐷
̅̅̅̅
𝐴𝐵  ̅̅̅̅
𝐶𝐵
ABD  CBD
ABD  ___________
6
4)
E is the
midpoint of
̅̅̅̅
𝐴𝐷
A  D
B  C
̅̅̅̅
̅̅̅̅
𝐴𝐸  𝐷𝐸
ABE  ___________
5)
̅̅̅̅ bisects
𝐴𝑆
̅̅̅̅̅
𝑀𝑃
Reflexive
Property of
Congruence
Given
MAS  ___________
6)
Given
Definition of
Angle Bisector
ABD  ___________
7
1.14 Homework - Triangle Congruence Theorems
Determine which theorem can be used to prove that the triangles are congruent. If it is not
possible to prove that they are congruent, write not possible.
1.
2.
3.
State what additional pair of angles or sides need to be congruent order to prove that the triangles
congruent using the given theorem.
4. ASA
5. SAS
6. SSS
7. Fill in the blanks in the following flow proof.
Given
̅̅̅̅
̅̅̅
𝑅𝑆 ≅ ̅𝑇𝑆
RVS  __________
8
Reflexive
Property
1.15 Examples – Flow Proofs using CPCTC
Example 1
Hutchins Lake is a long, narrow lake. Its length is represented by ̅̅̅̅
𝐴𝐵 in the diagram shown below. Dmitri
̅̅̅̅ .
designed the following method to determine its length. First, he paced off and measured ̅̅̅̅
𝐴𝐶 and 𝐵𝐶
Then, using a transit, he made 𝑚∠𝑃𝐶𝐴 = 𝑚∠𝐴𝐶𝐵. He then marked point D on 𝐶𝑃 so that 𝐷𝐶 = 𝐵𝐶, and
̅̅̅̅. Dmitri claims that 𝐴𝐵 = 𝐴𝐷. Is he correct? Justify your answer.
he measured 𝐴𝐷
Example 2
To measure the width of a sinkhole on his property, Harry marked off congruent triangles as shown in
̅̅̅̅ . How does he know that the
the diagram. He then measures the length of ̅̅̅̅̅̅
𝐴′𝐵′ to find the length of 𝐴𝐵
̅̅̅̅̅̅
̅̅̅̅
lengths of 𝐴′𝐵′ and 𝐴𝐵 are equal? Justify your answer.
Example 3
̅̅ ≅ ̅̅̅̅
Given: Δ𝐻𝐽𝐾 is an isosceles triangle with ̅̅
𝐻𝐽
𝐻𝐾 . An auxiliary line is drawn from vertex H to the
̅̅̅ to form two triangles.
midpoint N of 𝐽𝐾
Prove: ∠𝐽 ≅ ∠𝐾
9
1.15 Homework – Flow Proofs using CPCTC
1) According to legend, on of Napoleon’s officers used congruent triangles to estimate the width of a
river. On the riverbank, the officer stood up straight and lowered the visor of his cap until the farthest
thing he could see was the edge of the opposite bank (Point F). He then turned and noted the spot on his
side of the river that was in line with his eye and the tip of his visor (Point G). The officer then paced off
the distance along the riverbank from where he was standing to the spot he sighted (EG). He declared
that distance to be the same as the width of the river. Prove that he was correct.
Given: ∠𝐷𝐸𝐺 and ∠𝐹𝐸𝐺 are right angles (We assume that the
officer was standing perpendicular to the ground)
∠𝐸𝐷𝐺 ≅ ∠𝐸𝐷𝐹 (because he sighted both points F and G
from the same angle)
Prove: ̅̅̅̅
𝐸𝐹 ≅ ̅̅̅̅
𝐸𝐺
Fill in the empty spots in the flow proof below.
∠𝐷𝐸𝐺 and ∠𝐹𝐸𝐺
are right angles
Given
̅̅̅̅
𝐷𝐸  ̅̅̅̅
𝐷𝐸
________  ________
FDE  ___________
̅̅̅̅
𝐸𝐹 ≅ ̅̅̅̅
𝐸𝐺
10
Fill in the missing reasons or statements in each proof.
1.
2.
11
1.16 Warm Up
12
1.16 Review for Quiz
For each example, determine which triangle congruence postulate is needed to prove the two triangles
are congruent. Then write the congruence statement.
Complete the following flow proof.
13
1.17 Warm Up – Missing Parts
14