Name__________________________________________________
1.) Given:
̅̅̅̅
̅̅̅̅ ≅ 𝑇𝑆
𝑇𝑅
̅̅̅̅
𝑇𝑄 bisects ∡𝑅𝑇𝑆
Prove: Q is a midpoint of
̅̅̅̅
𝑅𝑆
Statements
Reasons
̅̅̅̅
̅̅̅̅ ≅ 𝑇𝑆
1.) 𝑇𝑅
̅̅̅̅
𝑇𝑄 bisects ∡𝑅𝑇𝑆
1.) Given
2.) ∡𝑅𝑇𝑄 ≅ ∡𝑆𝑇𝑄
2.) Definition of bisector
3.) ∡𝑅 ≅ ∡𝑆
3.) If 2 sides of a ∆ are ≅, then the
opposite angles are ≅.
4.) ∆𝑅𝑇𝑄 ≅ ∆𝑆𝑇𝑄
̅̅̅̅ ≅ 𝑆𝑄
̅̅̅̅
5.) 𝑅𝑄
4.)𝐴𝑆𝐴 ≅ 𝐴𝑆𝐴
6.) Q is a midpoint of ̅̅̅̅
𝑅𝑆
6.) A midpoint divides a segment
5.) CPCTC
into 2 congruent segments.
Statements
̅̅̅̅ ∥ 𝑄𝑅
̅̅̅̅
2.) Given: 𝑃𝑆
̅̅̅̅
𝑅𝑆 bisects ̅̅̅̅
𝑃𝑄
Prove: ∆𝑃𝑆𝑇 ≅ ∆𝑄𝑅𝑇
1.) ̅̅̅̅
𝑃𝑆 ∥ ̅̅̅̅
𝑄𝑅
̅̅̅̅ bisects 𝑃𝑄
̅̅̅̅
𝑅𝑆
Reasons
1.) Given
2.) ∡𝑃𝑆𝑇 ≅ ∡𝑄𝑅𝑇; ∡𝑆𝑃𝑇 ≅ ∡𝑅𝑄𝑇 2.) Alternate interior angles
theorem
̅̅̅̅
̅̅̅̅ ≅ 𝑄𝑇
3.) 𝑃𝑇
3.) Definition of bisector
4.) ∆𝑃𝑆𝑇 ≅ ∆𝑄𝑅𝑇
4.) 𝐴𝐴𝑆 ≅ 𝐴𝐴𝑆
3.)
Given:
̅̅̅̅̅̅
𝐷𝐸𝐹 , ̅̅̅̅̅̅
𝐴𝐸𝐶
̅̅̅̅ and 𝐷𝐹
̅̅̅̅ bisect each other
𝐴𝐶
Prove: ∡𝐷𝐴𝐸 ≅ ∡𝐹𝐶𝐸
4.) Given: ∆𝑅𝐴𝑀 and ∆𝑃𝐵𝑀
M is the midpoint of
̅̅̅̅
𝑅𝑃 and ̅̅̅̅
𝐴𝐵
Prove: ̅̅̅̅
𝑅𝐴 ∥ ̅̅̅̅
𝑃𝐵
Statements
Reasons
1.)
̅̅̅̅̅̅
𝐷𝐸𝐹 , ̅̅̅̅̅̅
𝐴𝐸𝐶
̅̅̅̅ and 𝐷𝐹
̅̅̅̅ bisect each other
𝐴𝐶
1.) Given
̅̅̅̅ ; ̅̅̅̅
̅̅̅̅
2.) ̅̅̅̅
𝐴𝐸 ≅ 𝐶𝐸
𝐷𝐸 ≅ 𝐹𝐸
2.) Definition of bisector
3.) ∡𝐷𝐸𝐴 ≅ ∡𝐹𝐸𝐶
3.) Vertical angles theorem
4.) ∆𝐷𝐴𝐸 ≅ ∆𝐶𝐹𝐸
4.) 𝑆𝐴𝑆 ≅ 𝑆𝐴𝑆
5.) ∡𝐷𝐴𝐸 ≅ ∡𝐹𝐶𝐸
5.) CPCTC
Statements
Reasons
1.) ∆𝑅𝐴𝑀 and ∆𝑃𝐵𝑀
1.) Given
̅̅̅̅
̅̅̅̅
M is the midpoint of 𝑅𝑃 and 𝐴𝐵
2.) ̅̅̅̅̅
𝑅𝑀 ≅ ̅̅̅̅̅
𝑃𝑀; ̅̅̅̅̅
𝐴𝑀 ≅ ̅̅̅̅̅
𝐵𝑀
2.) Definition of midpoint
3.) ∡𝑅𝑀𝐴 ≅ ∡𝑃𝑀𝐵
3.) Vertical angles theorem
4.) ∆𝑅𝑀𝐴 ≅ ∆𝑃𝑀𝐵
4.) 𝑆𝐴𝑆 ≅ 𝑆𝐴𝑆
5.) ∡𝑀𝑅𝐴 ≅ ∡𝑀𝑃𝐵
5.) CPCTC
̅̅̅̅ ∥ 𝑃𝐵
̅̅̅̅
6.) 𝑅𝐴
6.) Converse of the alternate interior
angles theorem
5.) Given: Parallelogram PQRS
̅̅̅̅
𝑃𝐸 ⊥ ̅̅̅̅
𝑆𝑄; ̅̅̅̅
𝑅𝐹 ⊥ ̅̅̅̅
𝑆𝑄
Prove: ̅̅̅̅
𝑆𝐸 ≅ ̅̅̅̅
𝑄𝐹
6.) Given: Quadrilateral ABCD
Diagonal ̅̅̅̅̅̅̅̅
𝐴𝐹𝐸𝐶
̅̅̅̅
̅̅̅̅
𝐴𝐸 ≅ 𝐹𝐶
̅̅̅̅ , ̅̅̅̅
̅̅̅̅
̅̅̅̅ ⊥ 𝐴𝐶
𝐵𝐹
𝐷𝐸 ⊥ 𝐴𝐶
∡1 ≅ ∡2
Prove: ABCD is a parallelogram.
Statements
Reasons
1.) Parallelogram PQRS
1.) Given
̅̅̅̅ ; 𝑅𝐹
̅̅̅̅
̅̅̅̅ ⊥ 𝑆𝑄
̅̅̅̅ ⊥ 𝑆𝑄
2.) 𝑃𝐸
2.) Given
3.) ∡𝑃𝐸𝑆, ∡𝑅𝐹𝑄 are right ∡𝑠
3.) Definition of ⊥ lines
4.) ∡𝑃𝐸𝑆 ≅ ∡𝑅𝐹𝑄
̅̅̅̅
5.) ̅̅̅̅
𝑃𝑆 ≅ 𝑅𝑄
4.) All right angles are ≅
̅̅̅̅ ∥ 𝑅𝑄
̅̅̅̅
6.) 𝑃𝑆
6.) In a parallelogram,
opposite sides are
parallel.
7.) ∡𝑃𝑆𝑄 ≅ ∡𝑅𝑄𝑆
7.) Alternate interior angles
theorem
8.) ∆𝑃𝐸𝑆 ≅ ∆𝑅𝐹𝑄
9.) ̅̅̅̅
𝑆𝐸 ≅ ̅̅̅̅
𝑄𝐹
8.) 𝐴𝐴𝑆 ≅ 𝐴𝐴𝑆
5.) In a parallelogram,
opposite sides are ≅.
9.) CPCTC
Statements
Reasons
1.) Quadrilateral ABCD
1.) Given
̅̅̅̅̅̅̅̅
2.) Diagonal 𝐴𝐹𝐸𝐶
3.) ̅̅̅̅
𝐴𝐸 ≅ ̅̅̅̅
𝐹𝐶
2.) Given
4.) ̅̅̅̅
𝐹𝐸 ≅ ̅̅̅̅
𝐹𝐸
4.) Reflexive Property
5.) ̅̅̅̅
𝐴𝐸 − ̅̅̅̅
𝐹𝐸 ≅ ̅̅̅̅
𝐹𝐶 − ̅̅̅̅
𝐹𝐸
3.) Given
5.) Subtraction Property
or
̅̅̅̅
𝐴𝐹 ≅ ̅̅̅̅
𝐶𝐸
6.) ̅̅̅̅
𝐵𝐹 ⊥ ̅̅̅̅
𝐴𝐶 , ̅̅̅̅
𝐷𝐸 ⊥ ̅̅̅̅
𝐴𝐶
6.) Given
7.) ∡𝐵𝐹𝐴, ∡𝐷𝐸𝐶 are right angles 7.) Definition of ⊥ Lines
8.) ∡𝐵𝐹𝐴 ≅ ∡𝐷𝐸𝐶
8.) All right angles are ≅
9.) ∡1 ≅ ∡2
9.) Given
10.) ∆𝐵𝐹𝐴 ≅ ∆𝐷𝐸𝐶
10.) 𝐴𝐴𝑆 ≅ 𝐴𝐴𝑆
̅̅̅̅; ∡𝐵𝐴𝐶 ≅ ∡𝐷𝐶𝐴
11.) ̅̅̅̅
𝐴𝐵 ≅ 𝐶𝐷
̅̅̅̅
̅̅̅̅
12.) 𝐴𝐵 ∥ 𝐶𝐷
11.) CPCTC
12.) Converse of the
Alternate Interior ∡𝑠 Theorem
13.) ABCD is a parallelogram
13.) If one pair of sides are
both ≅ and parallel, then
a quadrilateral is a
parallelogram
7.) The diagram below shows
rectangle ABCD with points E and F on
side ̅̅̅̅
𝐴𝐵 . Segments 𝐶𝐸 and 𝐷𝐹
intersect at G, and ∡𝐴𝐷𝐺 ≅ ∡𝐵𝐶𝐺.
̅̅̅̅
Prove: ̅̅̅̅
𝐴𝐸 ≅ 𝐵𝐹
Statements
Reasons
1.) rectangle ABCD with points E 1.) Given
̅̅̅̅
and F on side 𝐴𝐵
2.) Segments 𝐶𝐸 and 𝐷𝐹
intersect at G
2.) Given
3.) ∡𝐴𝐷𝐺 ≅ ∡𝐵𝐶𝐺
3.) Given
4.) ∡𝐴, ∡𝐵 are right angles
4.) A rectangle has right angles
5.) ∡𝐴 ≅ ∡𝐵
̅̅̅̅
6.) ̅̅̅̅
𝐷𝐴 ≅ 𝐶𝐵
5.) All right angles are ≅
7.) ∆𝐷𝐴𝐹 ≅ ∆𝐶𝐵𝐸
̅̅̅̅ ≅ 𝐵𝐸
̅̅̅̅
8.) 𝐴𝐹
7.) 𝐴𝑆𝐴 ≅ 𝐴𝑆𝐴
9.) ̅̅̅̅
𝐸𝐹 ≅ ̅̅̅̅
𝐸𝐹
9.) Reflexive Property
10.) ̅̅̅̅
𝐴𝐹 − ̅̅̅̅
𝐸𝐹 ≅ ̅̅̅̅
𝐵𝐸 − ̅̅̅̅
𝐸𝐹
10.) Subtraction Property
6.) In a rectangle, opposite
sides are ≅
8.) CPCTC
or
̅̅̅̅
𝐴𝐸 ≅ ̅̅̅̅
𝐵𝐹
8.) In the accompanying diagram of
circle O, diameter ̅̅̅̅̅̅
𝐴𝑂𝐵 is drawn,
̅̅̅̅ is drawn to the circle at B,
tangent 𝐶𝐵
E is a point on the circle, and
̅̅̅̅
𝐵𝐸 ∥ ̅̅̅̅̅̅
𝐴𝐷𝐶 .
Statements
Reasons
̅̅̅̅̅̅ is drawn,
1.) diameter 𝐴𝑂𝐵
̅̅̅̅
tangent 𝐶𝐵 is drawn to the
circle at B, E is a point on the
circle, and ̅̅̅̅
𝐵𝐸 ∥ ̅̅̅̅̅̅
𝐴𝐷𝐶 .
1.) Given
Prove: ∆𝐴𝐵𝐸~∆𝐶𝐴𝐵
2.) ∡𝐷𝐴𝐵 ≅ ∡𝐸𝐵𝐴
2.) Alt Interior ∡𝑠 Theorem
3.) ∡𝐶𝐵𝐴 is a right angle
3.) Tangent is ⊥ to a radius
4.) ∡𝐴𝐸𝐵 is a right angle
4.) An angle inscribed in a
semicircle is a right angle.
5.) ∡𝐶𝐵𝐴 ≅ ∡𝐴𝐸𝐵
5.) All right angles are ≅
6.) ∆𝐴𝐸𝐵~∆𝐶𝐴𝐵
6.) 𝐴𝐴~𝐴𝐴