# Question Video: Using Theories of Parallel Chords and Tangent Relationships to Find the Measure of an Arc Mathematics

In the following figure, π is a circle. Segment π΄π΅ and segment πΆπ· are two chords of the circle, and line πΈπΉ is a tangent to the circle at πΈ. If segment π΄π΅ β₯ segment πΆπ· β₯ πΈπΉ, the measure of arc π΄πΆ = 30Β°, and the measure of arc π·πΈ = 74Β°, find the measure of arc π΄π΅.

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### Video Transcript

In the following figure, π is a circle. Segment π΄π΅ and segment πΆπ· are two chords of the circle, and line πΈπΉ is a tangent to the circle at πΈ. If segment π΄π΅ is parallel to segment πΆπ· which is parallel to line πΈπΉ, the measure of arc π΄πΆ equals 30 degrees, and the measure of arc π·πΈ equals 74 degrees, find the measure of arc π΄π΅.

Letβs add some information weβre given to the figure. Arc π΄πΆ measures 30 degrees. Arc π·πΈ measures 74 degrees. The key point here is that weβre working with parallel chords and a parallel tangent line. The measures of the arcs between parallel chords in a circle are equal. Therefore, arc π΅π· will be equal in measure to arc π΄πΆ. And similarly, the measures of the arcs between a parallel chord and a tangent of that circle are equal. In our circle, that means that arc π·πΈ will be equal in measure to arc πΆπΈ. We can therefore add the measure of arc π΅π· equals 30 degrees and the measure of arc πΆπΈ equals 74 degrees onto our figure.

Our missing arc is π΄π΅. We know that the sum of the measures of all the arcs in a circle must equal 360 degrees. Therefore, the measure of arc π΄π΅ will be equal to 360 degrees minus the sum of all the other four arcs we know: 74 degrees plus 74 degrees plus 30 degrees plus 30 degrees. 360 degrees minus 208 degrees equals 152 degrees. The measure of arc π΄π΅ in this figure is 152 degrees.