Question Video: Using Theories of Parallel Chords to Find the Measure of an Arc | Nagwa Question Video: Using Theories of Parallel Chords to Find the Measure of an Arc | Nagwa

# Question Video: Using Theories of Parallel Chords to Find the Measure of an Arc Mathematics • Third Year of Preparatory School

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In the following figure, π΄π΅πΆ is an equilateral triangle inscribed in a circle. Segment π΄π΅ and segment π·πΈ are two chords of the circle, and line πΉπΊ is a tangent to the circle at πΉ. If segment π·πΈ β₯ segment π΄π΅ β₯ line πΉπΊ, and the measure of arc πΆπ· = 79Β°, find the measure of arc πΈπ΅. Find the measure of arc π·πΉ.

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### Video Transcript

In the following figure, π΄π΅πΆ is an equilateral triangle inscribed in a circle. Segment π΄π΅ and segment π·πΈ are two chords of the circle, and line πΉπΊ is a tangent to the circle at πΉ. If segment π·πΈ is parallel to segment π΄π΅, which is parallel to line πΉπΊ, and the measure of arc πΆπ· is 79 degrees, find the measure of πΈπ΅. Find the measure of π·πΉ.

The first thing we can do is add in the information we know into our circle. π΄π΅πΆ is an equilateral triangle. Therefore, each of their interior angles measures 60 degrees. We also need to note that π·πΈ is parallel to π΄π΅, which is parallel to πΉπΊ. Because we know that these lines are parallel, we can also say the arcs between them will be equal in measure. This means the measure of arc π·π΄ will be equal to the measure of arc πΈπ΅. And the measure of arc π΄πΉ will be equal to the measure of arc πΉπ΅. We know the first two chords will be equal in measure by the measure of arcs between parallel chords theorem and the second set, arc π΄πΉ and arc πΉπ΅, by the measure of arcs between a parallel chord and tangent theorem. We know that arc πΆπ· equals 79 degrees.

The first arc we need to find the measure of is πΈπ΅. To do that, letβs consider the inscribed angle πΆπ΅π΄. Using this inscribed angle, we can identify the measure of arc π΄πΆ. The arc π΄πΆ is two times the measure of this inscribed angle, which we know is 60 degrees. Therefore, the arc π΄πΆ measures 120 degrees. We know that the larger arc π΄πΆ is made up of the two smaller arcs π΄π· and π·πΆ. Therefore, we can say that 120 degrees equals the measure of arc π΄π· plus 79 degrees. Subtracting 79 degrees from both sides, we find the measure of arc π΄π· to be 41 degrees. And weβve already shown that the measure of arc πΈπ΅ will be equal to the measure of arc π΄π· by the measure of arcs between parallel chords theorem. This makes the measure of arc πΈπ΅ equal to 41 degrees.

Our next unknown arc measure that weβre looking for is π·πΉ. To do that, letβs first look at the inscribed angle π΄πΆπ΅. The inscribed angle π΄πΆπ΅ subtends the arc π΄π΅. Therefore, we can say the measure of arc π΄π΅ equals two times the measure of angle π΄πΆπ΅, which again will be 120 degrees. Since the measure of arc π΄π΅ equals the measure of arc π΄πΉ plus the measure of arc πΉπ΅ and since arc πΉπ΅ and arc π΄πΉ will be equal in measure, we can substitute two times the measure of arc π΄πΉ in this equation. 120 degrees equals two times the measure of arc π΄πΉ. The measure of arc π΄πΉ equals 60 degrees, which of course means that arc πΉπ΅ also measures 60 degrees.

Back to our unknown arc, thatβs π·πΉ, the measure of arc π·πΉ equals the measure of arc π·π΄ plus the measure of arc π΄πΉ, which is 41 degrees plus 60 degrees, 101 degrees.

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