Question

complete the proof that $m\angle 1 = 90^{\circ}$.

step	statement	reason
2	$df \cong ef$	both circles have radii of the same length.
3	$cf = cf$	theyre lengths of the same segment.
4	$\square$	side - side - side congruence (1, 2, 3)
5	$\angle efc \cong \angle dpc$	corresponding parts of congruent triangles are congruent, (4).
6	$fg = fg$	theyre lengths of the same segment.
7	$\triangle pdg \cong \triangle peg$	$\square$ congruence (2, 5, 6)
8	$\angle 1 \cong \angle 2$	corresponding parts of congruent triangles are congruent, (7).
9	$m\angle 1 = m\angle 2$	angles are congruent if and only if they have the same measure (8).
10	$m\angle 1 + m\angle 2 = 180^{\circ}$	the measures of angles that form a linear pair sum to $180^{\circ}$.
11	$m\angle 1 + m\angle 1 = 180^{\circ}$	substitution $\square$
12	$2(m\angle 1) = 180^{\circ}$	collect like terms (11).
13	$m\angle 1 = 90^{\circ}$	divide by 2 (12).

Explanation:

Step 1: Determine Step 4 Statement

We know from steps 1, 2, 3 that \( CD = CE \), \( DF = EF \), and \( CF = CF \). By SSS (Side - Side - Side) congruence criterion, \(\triangle DFC\cong\triangle EFC\). So the statement for step 4 should be \(\triangle DFC\cong\triangle EFC\).

Step 2: Determine Step 7 Reason

We have \( DF = EF \) (step 2), \(\angle EFC\cong\angle DFC\) (step 5), and \( FG = FG \) (step 6). The congruence criterion that uses two sides and the included angle is SAS (Side - Angle - Side). So the reason for step 7 is "Side - angle - side" (SAS).

Step 3: Determine Step 11 Substitution

We know that \( m\angle1=m\angle2 \) (step 9), so we substitute \( m\angle2 \) with \( m\angle1 \) in the equation \( m\angle1 + m\angle2=180^{\circ} \) (step 10) to get \( m\angle1 + m\angle1 = 180^{\circ} \). The substitution is based on \( m\angle2=m\angle1 \) from step 9.

Answer:

• Step 4 Statement: \(\boldsymbol{\triangle DFC\cong\triangle EFC}\)
• Step 7 Reason: \(\boldsymbol{\text{Side - angle - side (SAS)}}\)
• Step 11 Substitution: Substitute \( m\angle2 \) with \( m\angle1 \) (from \( m\angle1 = m\angle2 \) in step 9) into \( m\angle1 + m\angle2=180^{\circ} \)