Question Video: Using Inscribed Angles Subtended by the Same Arc to Find Missing Angles | Nagwa Question Video: Using Inscribed Angles Subtended by the Same Arc to Find Missing Angles | Nagwa

Question Video: Using Inscribed Angles Subtended by the Same Arc to Find Missing Angles Mathematics • Third Year of Preparatory School

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In the figure below, β³π·π΅πΆ is an isosceles triangle, πβ πΆπ΅π· = 40Β°, πβ π·πΆπΈ = 15Β°, πβ π΅π·πΆ = (8π₯ + 20)Β°, and πβ πΈπ΅πΆ = 11(π¦ β 2)Β°. Find the values of π₯ and π¦.

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Video Transcript

In the figure below, π·π΅πΆ is an isosceles triangle, angle πΆπ΅π· has measure 40 degrees, angle π·πΆπΈ has measure 15 degrees, angle π΅π·πΆ has measure eight π₯ plus 20, and angle πΈπ΅πΆ has measure 11 times π¦ minus two. Find the values of π₯ and π¦.

Letβs find π₯ first. Since π·π΅πΆ is an isosceles triangle, we know that angles π·πΆπ΅ and πΆπ΅π· are equal. Therefore, angle π΅π·πΆ has measure 180 minus 40 minus 40, which is 100 degrees. The question tells us that the measure of angle π΅π·πΆ is equal to eight π₯ plus 20 degrees. Substituting in our value of 100 for the measure of π΅π·πΆ and simplifying, we find that π₯ equals 10.

Since angles π·π΅πΈ and π·πΆπΈ are subtended by the same arc, they must be equal. We are told that the measure of π·πΆπΈ is 15 degrees. Therefore, the measure of π·π΅πΈ is 15 degrees too. Therefore, angle πΈπ΅πΆ, which is the sum of angles πΆπ΅π· and π·π΅πΈ, has measure 40 plus 15, which is 55. We are told that the measure of πΈπ΅πΆ is equal to 11 times π¦ minus two degrees. We can now substitute in our value of 55 for the measure of πΈπ΅πΆ and simplify, giving us π¦ equals seven. The answer to the question is π₯ equals 10 and π¦ equals seven.

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