Question

1. $overline{eg} perp$ ; $overline{fg} perp$ 3. thrm. 7.2a
2. $angle$ and $angle pfg$ 4. def. of $perp$ lines

form $90^circ angle$s.

1. $angle peg$ and $angle$ 5. def. of

are rt. $angle$s.

1. $\triangle peg$ and $\triangle pfg$ are rt. $\triangle$s. 6.
2. $overline{ep} cong overline{fp}$ 7. all radii on the same circle are
3. $overline{pg} cong overline{pg}$ 8.
4. $\triangle peg cong \triangle pfg$ 9. hl
5. $overline{eg} cong overline{fg}$ 10. cpctc

a. $peg$
b. symmetric property of $cong$
c. segment
d. $overline{ep}$

Explanation:

To solve this, we analyze each step:

Step 3:

Theorem 7.2A likely relates to tangents or perpendiculars to radii. So \(\overline{EG} \perp \overline{EP}\) and \(\overline{FG} \perp \overline{FP}\) (since \(EP\) and \(FP\) are radii, and a tangent is perpendicular to the radius at the point of contact).

Step 4:

By the definition of perpendicular lines, \(\angle PEG\) and \(\angle PFG\) form \(90^\circ\) angles. So the angle is \(\angle PEG\).

Step 5:

A right angle (\(rt. \angle\)) is \(90^\circ\), so \(\angle PEG\) and \(\angle PFG\) are right angles. So the angle is \(\angle PFG\), and the definition is "right angle".

Step 6:

A triangle with a right angle is a right triangle. So \(\triangle PEG\) and \(\triangle PFG\) are right triangles by the definition of a right triangle.

Step 7:

All radii of the same circle are congruent. So \(\overline{EP} \cong \overline{FP}\) because they are radii.

Step 8:

\(\overline{PG} \cong \overline{PG}\) by the Reflexive Property of Congruence (a segment is congruent to itself), but the option here is the Symmetric Property? Wait, no—wait, the options include "Symmetric Property of \(\cong\)" but actually, the Reflexive Property is \(a \cong a\). But looking at the options, option b is "Symmetric Property of \(\cong\)"—wait, maybe a typo, but the Reflexive Property is what applies here. However, the given options: let's check the options again. The options are:

a. \(PEG\)

b. Symmetric Property of \(\cong\)

c. segment

d. \(\overline{EP}\)

Wait, for step 8, \(\overline{PG} \cong \overline{PG}\) is the Reflexive Property, but the option b is Symmetric? Wait, maybe the question is about step 7 or 8. Wait, step 7: "All radii on the same circle are [congruent]". So step 7's blank is "congruent", but the options: let's re - evaluate.

Wait, the user might be asking for a specific part. Let's assume we are to answer one of the blanks. Let's take step 4: the angle is \(\angle PEG\), which is option a.

Final Answer (for step 4 as an example):

a. \(PEG\)

Answer:

a. \(PEG\)