Question

given: j k l m is a rhombus;
overline{k p} cong overline{j q} ; overline{l p} cong overline{k q}
prove: j k l m is a square.

1. ( m angle j + m angle j = 180 ) or ( 2 m angle j = 180 )
2. click here to insert
3. click here to insert
4. ( m angle m = 90 ), ( m angle k l m = 90 )

Explanation:

Step 1: Find the measure of ∠J

We have \(2m\angle J = 180\). Dividing both sides by 2 (using the Division Property), we get \(m\angle J=\frac{180}{2}=90\).

Step 2: Substitute the value of \(m\angle J\)

Since \(JKLM\) is a rhombus, opposite angles are equal and consecutive angles are supplementary. In a rhombus, if one angle (\(\angle J\)) is \(90^{\circ}\), then all angles are \(90^{\circ}\) (because in a rhombus \(m\angle J=m\angle L\) and \(m\angle K = m\angle M\), and consecutive angles are supplementary. If \(m\angle J = 90\), then \(m\angle K=90\) (since \(m\angle J+m\angle K = 180\)), \(m\angle L = 90\) (since \(m\angle J=m\angle L\)) and \(m\angle M=90\) (since \(m\angle K=m\angle M\))). A rhombus with all angles equal to \(90^{\circ}\) is a square.

Answer:

1. Consecutive angles of a rhombus are supplementary. Since \(JKLM\) is a rhombus, \(\angle J\) and \(\angle K\) are consecutive angles. But from the given congruences and triangle - congruence (not shown in full here, but in the context of the rhombus properties), we can also use the fact that adjacent angles in a parallelogram (a rhombus is a parallelogram) sum to \(180^{\circ}\).
2. \(m\angle J = 90\) (after applying the Division Property \(2m\angle J=180\Rightarrow m\angle J = 90\))
3. Using the properties of a rhombus (opposite angles are equal: \(m\angle J=m\angle L\) and \(m\angle K=m\angle M\), and consecutive angles are supplementary \(m\angle J + m\angle K=180\)). If \(m\angle J = 90\), then \(m\angle K = 90\), \(m\angle L=90\) and \(m\angle M = 90\)
4. A rhombus with all angles equal to \(90^{\circ}\) is a square. A square is a special case of a rhombus where all angles are right angles.