Question

prove the following statement using the given pieces of information
given: ∠gph ≅ ∠dph, ph⊥gd
prove: δhgp ≅ δhdp
complete the two - column proof by filling in the blanks.
statements\treasons

1. ph⊥gd\t1. given
2. ∠phg and ∠phd are right angles.\t2. definition of ⊥
3. ∠phg ≅ ∠phd\t3. right angles congruent theorem
4. ph ≅ ph\t4.

(options for reason 4: symmetric property of ≅, reflexive property of ≅, transitive property of ≅)

Explanation:

Step1: Analyze the Proof Structure

We are proving triangle congruence (ΔHGP ≅ ΔHDP) using a two - column proof. We know that PH is a common side to both triangles, ∠GPH ≅ ∠DPH (given), and ∠PHG and ∠PHD are right angles (from the diagram and the "right angles" statement). For the last step, we have the statement \( \overline{PH}\cong\overline{PH} \).

Step2: Identify the Property

The reflexive property of congruence states that any segment (or angle, shape) is congruent to itself. Since \( \overline{PH} \) is a side of both triangles and is congruent to itself, we use the reflexive property of congruence.

Answer:

Reflexive property of \( \cong \)