Question

part 2 of 4
prove the following statement using the given pieces of information.
given: ∠gph ≅ ∠dph, $overline{ph} perp overline{gd}$
prove: $\triangle hgp cong \triangle hdp$
complete the two - column proof by filling in the blanks
statements\treasons

1. $overline{ph} perp overline{gd}$\t1. given
2. ∠phg and ∠phd are right angles\t2. definition of ⊥
3. ∠phg ≅ ∠phd\t3.

(options for reason 3: right angles congruent theorem, vertical angles theorem, corresponding angles converse theorem, third angles theorem)
(there is also a diagram of triangle gpd with a perpendicular segment from p to gd at h, forming two right triangles gph and dph)

Explanation:

Step1: Analyze the Diagram and Given

We have two right triangles \( \triangle HGP \) and \( \triangle HDP \) with \( \angle GPH \cong \angle DPH \) and \( \overline{PH} \perp \overline{GD} \), so \( \angle PHG \) and \( \angle PHD \) are right angles (by definition of perpendicular lines, they measure \( 90^\circ \)).

Step2: Identify the Theorem for Congruence of Angles

We need to prove \( \angle PHG \cong \angle PHD \). Since both are right angles, the Right Angles Congruent Theorem states that all right angles are congruent. So for the third step (statement 3: \( \angle PHG \cong \angle PHD \)), the reason is the Right Angles Congruent Theorem.

Answer:

Right Angles Congruent Theorem