Question
Quadrilateral CDEF is inscribed in circle A.
D
E
F
C
Complete the proof that LD and LF are supplementary.
1. Since LD is an inscribed angle that intercepts arc CFE, the measure of arc CRE is
2. Since LF is an inscribed angle that intercepts arc EDC, the measure of arc EDC is
3. The sum of arc CFE and arc EDC is
→, which is
the sum of LD and LF.
By definition, LD and LF are supplementary.
4. Therefore, the sum of LD and LF is
the measure of LD. the measure of LP.
180°
twice
360°
equal to
half
D
E
F
C
Complete the proof that LD and LF are supplementary.
1. Since LD is an inscribed angle that intercepts arc CFE, the measure of arc CRE is
2. Since LF is an inscribed angle that intercepts arc EDC, the measure of arc EDC is
3. The sum of arc CFE and arc EDC is
→, which is
the sum of LD and LF.
By definition, LD and LF are supplementary.
4. Therefore, the sum of LD and LF is
the measure of LD. the measure of LP.
180°
twice
360°
equal to
half
Answers
Answered by
GPT-4o mini
To complete the proof that the inscribed angles \( LD \) and \( LF \) are supplementary, we need to follow the reasoning based on the properties of inscribed angles and arcs intercepted by those angles. Let's fill in the proof step by step:
1. Since \( LD \) is an inscribed angle that intercepts arc \( CFE \), the measure of arc \( CFE \) is \( 2 \times \text{measure of } LD \).
2. Since \( LF \) is an inscribed angle that intercepts arc \( EDC \), the measure of arc \( EDC \) is \( 2 \times \text{measure of } LF \).
3. The sum of arc \( CFE \) and arc \( EDC \) is \( 360^\circ \) (as they comprise the entire circle), which means:
\[
\text{measure of arc } CFE + \text{measure of arc } EDC = 360^\circ
\]
Hence, we can substitute the measures of the arcs:
\[
2 \times \text{measure of } LD + 2 \times \text{measure of } LF = 360^\circ
\]
4. Dividing the entire equation by 2 gives:
\[
\text{measure of } LD + \text{measure of } LF = 180^\circ
\]
Since the sum of the measures of angles \( LD \) and \( LF \) is \( 180^\circ \), by definition, \( LD \) and \( LF \) are supplementary angles.
Thus we have completed the proof that the angles \( LD \) and \( LF \) are supplementary.
1. Since \( LD \) is an inscribed angle that intercepts arc \( CFE \), the measure of arc \( CFE \) is \( 2 \times \text{measure of } LD \).
2. Since \( LF \) is an inscribed angle that intercepts arc \( EDC \), the measure of arc \( EDC \) is \( 2 \times \text{measure of } LF \).
3. The sum of arc \( CFE \) and arc \( EDC \) is \( 360^\circ \) (as they comprise the entire circle), which means:
\[
\text{measure of arc } CFE + \text{measure of arc } EDC = 360^\circ
\]
Hence, we can substitute the measures of the arcs:
\[
2 \times \text{measure of } LD + 2 \times \text{measure of } LF = 360^\circ
\]
4. Dividing the entire equation by 2 gives:
\[
\text{measure of } LD + \text{measure of } LF = 180^\circ
\]
Since the sum of the measures of angles \( LD \) and \( LF \) is \( 180^\circ \), by definition, \( LD \) and \( LF \) are supplementary angles.
Thus we have completed the proof that the angles \( LD \) and \( LF \) are supplementary.