diagonals $overline{ac}$ and $overline{bd}$ form right angles at point m in parallelogram abcd. prove abcd is a rhombus.
statements
1. abcd is a parallelogram
2. $angle amb$, $angle bmc$, $angle cmd$, and $angle dma$ are right angles
3. $angle ambcongangle bmccongangle cmdcongangle dma$
4. ac bisects bd;
bd bisects ac;
5. $overline{am}congoverline{mc}$, $overline{mb}congoverline{md}$
6.
reasons
1. given
2. given
3. right angles are congruent
4. diagonals of a parallelogram bisect each other
5. definition of a bisector
6. sas congruency theorem

Question

Accepted Answer

# Explanation:
## Step1: Recall properties of parallelogram
Given that $ABCD$ is a parallelogram, its diagonals $AC$ and $BD$ bisect each other at $M$ (diagonals of a parallelogram bisect each other), so $AM = MC$ and $MB=MD$. Also, $\angle AMB=\angle BMC = \angle CMD=\angle DMA = 90^{\circ}$ (given).
## Step2: Consider triangles $\triangle AMB$ and $\triangle BMC$
In $\triangle AMB$ and $\triangle BMC$, we have $AM = MC$ (from diagonal - bisecting property), $\angle AMB=\angle BMC = 90^{\circ}$, and $MB$ is common to both triangles.
## Step3: Apply SAS congruency theorem
By the Side - Angle - Side (SAS) congruency theorem ($AM = MC$, $\angle AMB=\angle BMC$, $MB = MB$), $\triangle AMB\cong\triangle BMC$.
## Step4: Use congruent - triangle property
Since $\triangle AMB\cong\triangle BMC$, then $AB = BC$ (corresponding parts of congruent triangles are congruent).
## Step5: Recall the definition of a rhombus
A parallelogram with adjacent sides equal is a rhombus. Since $ABCD$ is a parallelogram and $AB = BC$, $ABCD$ is a rhombus.

# Answer:
In statement 6, the two triangles that can be used to apply the SAS congruency theorem are $\triangle AMB$ and $\triangle BMC$ (or $\triangle BMC$ and $\triangle CMD$, or $\triangle CMD$ and $\triangle DMA$, or $\triangle DMA$ and $\triangle AMB$). Since $\triangle AMB\cong\triangle BMC$, $AB = BC$. A parallelogram with adjacent - sides equal is a rhombus, so $ABCD$ is a rhombus.

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QUESTION IMAGE

Question

Explanation:

Step1: Recall properties of parallelogram

Step2: Consider triangles \(\triangle AMB\) and \(\triangle BMC\)

Step3: Apply SAS congruency theorem

Step4: Use congruent - triangle property

Step5: Recall the definition of a rhombus

Answer: