Quiz 6D Proof answers with extra kite proof answer. C
D
4. Given: ABCD is a quadrilateral &
AC and BD bisect each other.
Prove: ABCD is a parallelogram.
E
A
Statement 1. ABCD is a quadrilateral &
B
Reason 1. Given AC and BD bisect each other 2. 𝐷𝐸 ≅ 𝐸𝐵 and 𝐴𝐸 ≅ 𝐶𝐸 3. ∠𝐴𝐸𝐷 ≅ ∠𝐵𝐸𝐷 4. ∆𝐴𝐸𝐷 ≅ ∆𝐵𝐸𝐶 5. ∠EDA ≅ ∠EBC and ∠ABD ≅ ∠CDB 6. 𝐴𝐷 ∥ 𝐵𝐶 and 𝐶𝐷 ∥ 𝐴𝐵 7. 𝐴𝐵𝐶𝐷 is a parallelogram 2. Definition of Bisect 3. Vertical Angles 4. SAS 5. CPCTC 6. AIA → Parallel Lines 7. Opposite sides are parallel or by definition of a parallelogram 5. Prove that a rhombus has opposite angles congruent. You may only assume what is stated in the definition of a
rhombus, “A rhombus is a quadrilateral with 4 congruent sides.”
B
Given: ABCD is a Rhombus
Prove: ∠𝐴 ≅ ∠𝐶 𝑎𝑛𝑑 ∠𝐵 ≅ ∠𝐷 A
Statement 1. ABCD is a Rhombus 2. 𝐴𝐷 ≅ 𝐵𝐶 ≅ 𝐴𝐵 ≅ 𝐷𝐶 3. 𝐴𝐶 ≅ 𝐴𝐶 4. ∆𝐴𝐵𝐶 ≅ ∆𝐶𝐷𝐴 5. ∠B ≅ ∠𝐷 6. 𝐵𝐷 ≅ 𝐵𝐷 7. ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐷𝐵 8. ∠𝐴 ≅ ∠𝐶 Reason 1. Given 2. Definition of Rhombus 3. Reflexive 4. SSS 5. CPCTC 6. REFLEXIVE 7. SSS 8. CPCTC C
D
H
Given:
Prove:
MJ is the angle bisector of both ∠HMK and
∠HJK
M
G
J
HGKJ is a kite
K
Statement 1. MJ is the angle bisector of both ∠HMK and
∠HJK
2. ∠𝐻𝑀𝐽 ≅ ∠𝐾𝑀𝐽 and ∠𝐻𝐽𝑀 ≅ ∠𝐾𝐽𝑀 3. 𝑀𝐽 ≅ 𝑀𝐽 4.∆𝐻𝐽𝑀 ≅ ∆𝐾𝐽𝑀 5. 𝐻𝐽 ≅ 𝐾𝐽 6. 𝐺𝐽 ≅ 𝐺𝐽 7. .∆𝐺𝐻𝐽 ≅ ∆𝐺𝐾𝐽 8. 𝐺𝐻 ≅ 𝐺𝐾 9. HGKJ is a kite
Reason 1. Given 2. Def of angle bisector 3. Reflexive 4. ASA 5. CPCTC 6. Reflexive 7. SAS 8. CPCTC 9. Two Pairs of consecutive sides are congruent (Def of Kite)