Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

ST
...

PHOENIX Math for the 21st Century Learners: Second Edition, PHOENIX Publishing House Inc
...

Prove two triangles are congruent applying the triangle congruence postulates
...

These can now be used to prove the congruence of triangles using deductive reasoning
...

𝐵
EXAMPLE 1: Given: ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐴𝐷, ̅̅̅̅
𝐵𝐶 ≅ ̅̅̅̅
𝐷𝐶
Prove: ∆𝐴𝐵𝐶 ≅ ∆𝐴𝐷𝐶
𝐶
Proof:
STATEMENT
̅̅̅̅ ≅ 𝑨𝑫
̅̅̅̅
1
...
𝑩𝑪
̅̅̅̅ ≅ 𝑨𝑪
̅̅̅̅
3
...
∆𝐴𝐵𝐶 ≅ ∆𝐴𝐷𝐶

𝐴

REASONS
1
...
Given
3
...
SSS Postulate

𝐷

The 𝑆𝐴𝑆 Congruence Postulate states that if two sides and the included angle of one triangle are congruent to two
sides and the included angle of another triangle, then the triangles are congruent
...

EXAMPLE 2: Given: ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐶𝐷, ̅̅̅̅
𝐴𝐵 ⊥ 𝐵𝐷
𝐶𝐷 ⊥ ̅̅̅̅
𝐵𝐷, 𝐸 is the midpoint of 𝐵𝐷
Prove: ∆ABE ≅ ∆CDE
𝐴

𝐵

Proof:

𝐶

𝐸

𝐷

STATEMENT
̅̅̅̅ ≅ 𝑪𝑫
̅̅̅̅
1
...
∠𝐵 is a right angle
̅̅̅̅ ⊥ 𝐵𝐷
̅̅̅̅
3
...
∆𝐴𝐵𝐶 ≅ ∆𝐴𝐷𝐶
5
...

̅̅̅̅ and 𝐵𝐸
̅̅̅̅
...

𝐶𝐷

̅̅̅̅̅
6
...

7
...
∆ABE ≅ ∆CDE

REASONS
1
...
Definition of Perpendicular
3
...
Definition of Perpendicular
5
...

6
...
Definition of midpoint
8
...


𝐴
EXAMPLE 3: Given: ∠1 ≅ ∠2, 𝐷 is the midpoint of ̅̅̅̅
𝐵𝐶
...

1

2

Prove: ∆𝐴𝐷𝐵 ≅ ∆𝐴𝐷𝐶
𝐵
Proof:

STATEMENT

𝐷

𝐶

REASONS

1
...
∠𝐴𝐵𝐷 and ∠1 form a linear pair
∠𝐴𝐶𝐷 and ∠2 form a linear pair
3
...
∠𝑨𝑩𝑫 ≅ ∠𝑨𝑪𝑫
5
...


1
...
̅̅̅̅̅
𝑩𝑫 ≅ ̅̅̅̅
𝑪𝑫
7
...

∠𝐴𝐷𝐶 is a right angle
...
∠𝑨𝑫𝑩 ≅ ∠𝑨𝑫𝑪

6
...
∆𝐴𝐷𝐵 ≅ ∆𝐴𝐷𝐶

9
...
Definition of a linear pair
3
...
Supplements to congruent angles are congruent
...
Given

7
...
Any two right angles are congruent
...


KINDLE YOUR IDEAS

ACTIVITY
Direction: Complete each proof
...

CHOICES:
• Any two right angles are congruent
...
Given:

Prove:
Proof:

̅̅̅̅ ≅ ̅̅̅̅
𝐴𝐶
𝐷𝐸
̅̅̅̅ ⊥ 𝐶𝐵
̅̅̅̅ , 𝐷𝐸
̅̅̅̅ ⊥ 𝐸𝐵
̅̅̅̅
𝐴𝐶
̅̅̅̅ ≅ 𝐸𝐵
̅̅̅̅
𝐶𝐵
∆𝐴𝐶𝐵 ≅ ∆𝐷𝐸𝐵

STATEMENT
̅̅̅̅ ≅ 𝐷𝐸
̅̅̅̅
1
...
𝐴𝐶
3
...


𝐵
𝐴

𝐷

REASONS
1
...

3
...
∠𝐶 ≅ ∠𝐸
5
...


6
...


CHOICES:
• Alternate-Interior Angles are equal
• Given
• Definition of Vertical Angles
̅̅̅̅
• ̅̅̅̅
𝐴𝐵||𝐷𝐸
• Angle-Side-Angle Postulate
• ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐶

𝐸

5
...
Given:
Prove:
Proof:

𝐵

STATEMENT

𝐷

REASONS

1
...
Given

̅̅̅̅ ≅ ̅̅̅̅
2
...


3
...


4
...


5
...


𝐶

𝐴

𝐸

ADDITIONAL NOTES TO REVIEW:
SPECIAL RIGHT TRIANGLES
In a right triangle, if one angle measures 30˚, the length of the hypotenuse is twice the length of the side
opposite the angle with measure 30˚
...


If one angle of a right triangle measures 30˚, the length of the side opposite the angle with measure 60˚ is
equal to√𝟑 times the length of the side opposite the 30˚ angle
...


4𝑥 + 6

8𝑥 − 18

Solve for the value of x
...

Then transpose to combine similar terms, 6 + 18 = 8𝑥 − 4𝑥
...

24 4𝑥
=
4
4
6=𝑥
,



---