Question

rhombus proofs assignment
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assignment:
given: parallelogram abcd is a rhombus. side bc = 10 and segment ao = 6. answer the questions below. (2 points each)

1. true or false · diagonal bd bisects angle d.
2. what is the measure of angle $d_1$?
3. true or false · diagonal bd bisects diagonal ac.
4. what is the measure of segment oc?
5. true or false · diagonal bd and ac are congruent.
6. true or false · diagonals bd and ac are perpendicular.
7. true or false · triangle boc is a right triangle.
8. what is the measure of diagonal bd? show your work. click here to watch a help video.

Explanation:

Step1: Recall rhombus angle properties

In a rhombus, opposite angles are equal, consecutive angles are supplementary, and diagonals bisect the angles. Also, $\angle ADB = 37^\circ$, and diagonal $BD$ bisects $\angle ADC$.

Step2: Calculate $\angle d_1$

Since $BD$ bisects $\angle ADC$, $\angle ADC = 2\times\angle ADB = 2\times37^\circ = 74^\circ$. $\angle d_1$ is $\angle ADC$, so $\angle d_1 = 74^\circ$.

Step3: Evaluate True/False 1

Diagonal $BD$ bisects $\angle ADC$ (angle D), so this is True.

Step4: Evaluate True/False 2

Diagonals of a rhombus bisect the angles, not each other. So this is False.

Step5: Find length of $OC$

In a rhombus, diagonals bisect each other, so $OC = AO = 6$ cm.

Step6: Evaluate True/False 3

Diagonals of a rhombus are not congruent (unless it's a square), so this is False.

Step7: Evaluate True/False 4

Diagonals of a rhombus are perpendicular, so this is True.

Step8: Evaluate True/False 5

Diagonals of a rhombus are perpendicular, so $\angle BOC = 90^\circ$, making $\triangle BOC$ a right triangle. This is True.

Step9: Calculate length of $BD$

First, find $BO$ using Pythagoras in $\triangle BOC$: $BC=10$ cm, $OC=6$ cm, so $BO = \sqrt{BC^2 - OC^2} = \sqrt{10^2 - 6^2} = \sqrt{100-36} = \sqrt{64} = 8$ cm. Diagonals bisect each other, so $BD = 2\times BO = 2\times8 = 16$ cm.

Answer:

1. True
2. $\boldsymbol{74^\circ}$
3. False
4. $\boldsymbol{6}$ cm
5. False
6. True
7. True
8. $\boldsymbol{16}$ cm