Question

in exercises 13 and 14, write a proof to verify that the construction is valid. (see example 4.)

1. line perpendicular to a line through a point not on the line

plan for proof show that $\triangle apq \cong \triangle bpq$ by the sss congruence theorem (theorem 5.8). then show that $\triangle apm \cong \triangle bpm$ using the sas congruence theorem (theorem 5.5). use corresponding parts of congruent triangles to show that $\angle amp$ and $\angle bmp$ are right angles.

Explanation:

Step1: Prove \( \triangle APQ \cong \triangle BPQ \) (SSS)

From the construction, \( AP = BP \) (arcs drawn with equal radius from \( P \)), \( AQ = BQ \) (arcs drawn with equal radius from \( A, B \)), and \( PQ = PQ \) (common side). By SSS Congruence Theorem, \( \triangle APQ \cong \triangle BPQ \).

Step2: Prove \( \triangle APM \cong \triangle BPM \) (SAS)

From Step 1, \( \angle APQ = \angle BPQ \) (CPCTC). We know \( AP = BP \) (from construction), \( PM = PM \) (common side), and \( \angle APM = \angle BPM \). By SAS Congruence Theorem, \( \triangle APM \cong \triangle BPM \).

Step3: Show \( \angle AMP, \angle BMP \) are right angles

From Step 2, \( \angle AMP = \angle BMP \) (CPCTC). Since \( \angle AMP + \angle BMP = 180^\circ \) (linear pair), \( \angle AMP = \angle BMP = 90^\circ \). Thus, \( PQ \perp AB \), verifying the construction.

Answer:

The construction is valid as shown by proving triangle congruences (SSS, SAS) and using CPCTC to show right angles, confirming the perpendicular line through \( P \) to \( AB \).