Question

Draw two-line segments, $\overline{\mathrm{KL}}$and $\overline{\mathrm{MN}}$, that bisect each other at O. Mark a point P on$\overline{\mathrm{KN}}$ and let Q be the point where$\overline{\mathrm{PO}}$intersects $\overline{\mathrm{ML}}$. Prove that O is the midpoint of $\overline{\mathrm{PQ}}$(First state what is given and what is to be proved.)

Short answer

$\mathrm{\Delta }\mathbf{KNO}\cong \mathrm{\Delta }\mathbf{MLO}$	SAS postulate
$\overline{\mathrm{MO}}\cong \overline{\mathrm{ON}}$	Corr. Parts of$\cong \mathrm{\Delta }$are $\cong$.
$\mathrm{\Delta }\mathbf{PON}\cong \mathrm{\Delta }\mathbf{QOM}$	ASA postulate
O is the midpoint of$\overline{\mathrm{PQ}}$	Corr. Parts of$\cong \mathrm{\Delta }$are $\cong$.

Step-by-step solution

Step 1. Draw a diagram.

Draw a diagram with given instructions.

Draw two-line segments,$\overline{KL}\text{and}\overline{MN}$that bisect each other at $O$. Mark a point$P$ on$\overline{KN}$ and let$Q$ be the point where$\overline{PQ}$ intersects $\overline{ML}$.

Step 2. Description of step.

Consider$\overline{KO}\cong \overline{ON},\overline{MO}\cong \overline{ON}$ and$\angle KON\cong \angle MOL$ then by SAS postulate, $\mathrm{\Delta }KNO\cong \mathrm{\Delta }MLO$.

Step 3. Description of step.

Consider$\mathrm{\Delta }PON$ and$\mathrm{\Delta }QOM$ in which $\overline{MO}\cong \overline{ON}$,$\angle MOQ\cong \angle PON$ as they are vertically opposite angle and$\angle PNO\cong \angle OMQ$ then by ASA postulate, $\mathrm{\Delta }PON\cong \mathrm{\Delta }QOM$.

Step 4. Description of step.

As $\mathrm{\Delta }PON\cong \mathrm{\Delta }QOM$by the corresponding parts of congruent triangles, $\overline{PO}\cong \overline{QO}$. Which implies that$O$ is the midpoint of $\overline{PQ}$.

Step 5. Key steps of a proof.

Make a list of key steps of a proof.

$\mathrm{\Delta }KNO\cong \mathrm{\Delta }MLO$	SAS postulate
$\overline{MO}\cong \overline{ON}$	Corr. Parts of$\cong \mathrm{\Delta }$are $\cong$.
$\mathrm{\Delta }PON\cong \mathrm{\Delta }QOM$	ASA postulate
$O$is the midpoint of$\overline{PQ}$	Corr. Parts of$\cong \mathrm{\Delta }$are $\cong$.