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

In the figure below, △𝐷𝐡𝐢 is an isosceles triangle, π‘šβˆ πΆπ΅π· = 40Β°, π‘šβˆ π·πΆπΈ = 15Β°, π‘šβˆ π΅π·πΆ = (8π‘₯ + 20)Β°, and π‘šβˆ πΈπ΅πΆ = 11(𝑦 βˆ’ 2)Β°. Find the values of π‘₯ and 𝑦.

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Video Transcript

In the figure below, 𝐷𝐡𝐢 is an isosceles triangle, angle 𝐢𝐡𝐷 has measure 40 degrees, angle 𝐷𝐢𝐸 has measure 15 degrees, angle 𝐡𝐷𝐢 has measure eight π‘₯ plus 20, and angle 𝐸𝐡𝐢 has measure 11 times 𝑦 minus two. Find the values of π‘₯ and 𝑦.

Let’s find π‘₯ first. Since 𝐷𝐡𝐢 is an isosceles triangle, we know that angles 𝐷𝐢𝐡 and 𝐢𝐡𝐷 are equal. Therefore, angle 𝐡𝐷𝐢 has measure 180 minus 40 minus 40, which is 100 degrees. The question tells us that the measure of angle 𝐡𝐷𝐢 is equal to eight π‘₯ plus 20 degrees. Substituting in our value of 100 for the measure of 𝐡𝐷𝐢 and simplifying, we find that π‘₯ equals 10.

Since angles 𝐷𝐡𝐸 and 𝐷𝐢𝐸 are subtended by the same arc, they must be equal. We are told that the measure of 𝐷𝐢𝐸 is 15 degrees. Therefore, the measure of 𝐷𝐡𝐸 is 15 degrees too. Therefore, angle 𝐸𝐡𝐢, which is the sum of angles 𝐢𝐡𝐷 and 𝐷𝐡𝐸, has measure 40 plus 15, which is 55. We are told that the measure of 𝐸𝐡𝐢 is equal to 11 times 𝑦 minus two degrees. We can now substitute in our value of 55 for the measure of 𝐸𝐡𝐢 and simplify, giving us 𝑦 equals seven. The answer to the question is π‘₯ equals 10 and 𝑦 equals seven.

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