Question

question 5
the diagonals of rhombus abcd form several triangles.
given: abcd is a rhombus
prove: each diagonal bisects a pair of opposite angles.

1. abcd is a rhombus 1. given
2. abcd is a parallelogram 2. select an answer
3. \\( \angle a b c \cong \angle a d c \\) and \\( \angle b a d \cong \angle b c d \\) 3. select an answer
4. \\( \overline{a b} \cong \overline{b c} \cong \overline{c d} \cong \overline{d a} \\) 4. sss
5. \\( \triangle a b c \cong \triangle a d c \\) 5.
6. \\( \angle 5 \cong \angle 6 \\) and \\( \angle 7 \cong \angle 8 \\) 6. cpctc
7. \\( \triangle b a d \cong \triangle b c d \\) 7. definition of angle bisector
8. \\( \angle 1 \cong \angle 2 \\) and \\( \angle 3 \cong \angle 4 \\) 8. aas
9. each diagonal bisects a pair of opposite angles 9. sas

asa
opposite angles are congruent in a parallelogram

Explanation:

Step1: Reason for statement 2

A rhombus is a special type of parallelogram. By definition, a rhombus has all the properties of a parallelogram. So, the reason for "ABCD is a parallelogram" (statement 2) is "Definition of a rhombus".

Step2: Reason for statement 3

In a parallelogram, opposite angles are congruent. Since ABCD is a parallelogram (from statement 2), $\angle ABC\cong\angle ADC$ and $\angle BAD\cong\angle BCD$. So, the reason for statement 3 is "Opposite angles are congruent in a parallelogram".

Step3: Reason for statement 4

A rhombus has all four sides equal. So, $AB\cong BC\cong CD\cong DA$. The reason for statement 4 is "Definition of a rhombus".

Step4: Reason for statement 5

We have $AB = AD$, $BC = CD$, and $AC=AC$ (common side). By the SSS (Side - Side - Side) congruence criterion, $\triangle ABC\cong\triangle ADC$. So, the reason for statement 5 is "SSS".

Step5: Reason for statement 6

Since $\triangle ABC\cong\triangle ADC$, by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), $\angle5\cong\angle6$ and $\angle7\cong\angle8$. So, the reason for statement 6 is "CPCTC".

Step6: Reason for statement 7

We have $AB = BC$, $AD = CD$, and $BD = BD$ (common side). By the SSS congruence criterion, $\triangle BAD\cong\triangle BCD$. So, the reason for statement 7 is "SSS".

Step7: Reason for statement 8

Since $\triangle BAD\cong\triangle BCD$, by CPCTC, $\angle1\cong\angle2$ and $\angle3\cong\angle4$. So, the reason for statement 8 is "CPCTC".

Step8: Reason for statement 9

Since $\angle5\cong\angle6$, $\angle7\cong\angle8$, $\angle1\cong\angle2$, and $\angle3\cong\angle4$, each diagonal bisects a pair of opposite angles. The reason for statement 9 is "Definition of angle bisector".

Answer:

1. Definition of a rhombus; 3. Opposite angles are congruent in a parallelogram; 4. Definition of a rhombus; 5. SSS; 6. CPCTC; 7. SSS; 8. CPCTC; 9. Definition of angle bisector.