Question

point ( m ) is the midpoint of ( overline{pq} ), and ( overleftrightarrow{lm} ) is the perpendicular bisector of ( overline{pq} ). write a two - column proof to show that ( lp = lq )

Explanation:

Step1: Reason for $\overline{LM}\cong\overline{LM}$

Any segment is congruent to itself. This is the Reflexive Property of Congruence.

Step2: Statement for Right Angle Congruence Theorem

Since $\overleftrightarrow{LM}\perp\overline{PQ}$, $\angle PML$ and $\angle QML$ are right angles. By the Right - Angle Congruence Theorem, $\angle PML\cong\angle QML$.

Step3: Reason for $\triangle PML\cong\triangle QML$

We have $\overline{PM}\cong\overline{QM}$ (given), $\angle PML\cong\angle QML$ (from Step 2), and $\overline{LM}\cong\overline{LM}$ (from Step 1). By the SAS (Side - Angle - Side) Congruence Theorem, $\triangle PML\cong\triangle QML$.

Step4: Statement for Corresponding parts of congruent triangles

Since $\triangle PML\cong\triangle QML$, their corresponding sides $\overline{LP}$ and $\overline{LQ}$ are congruent. So, $\overline{LP}\cong\overline{LQ}$.

Step5: Reason for $LP = LQ$

If two segments $\overline{LP}$ and $\overline{LQ}$ are congruent ($\overline{LP}\cong\overline{LQ}$), then by the Definition of segment congruence, their lengths are equal, i.e., $LP = LQ$.

Answer:

STATEMENTS	REASONS
2. $\overline{LM}\cong\overline{LM}$	2. Reflexive Property of Congruence
3. $\angle PML\cong\angle QML$	3. Right Angle Congruence Theorem
4. $\triangle PML\cong\triangle QML$	4. SAS Congruence Theorem
5. $\overline{LP}\cong\overline{LQ}$	5. Corresponding parts of congruent triangles are congruent
6. $LP = LQ$	6. Definition of segment congruence