Question Video: Completing a Proof About Triangles | Nagwa Question Video: Completing a Proof About Triangles | Nagwa

Question Video: Completing a Proof About Triangles Mathematics • First Year of Preparatory School

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In the following triangle π΄π΅πΆ, if πβ πΆ = πβ πΆπ΄π· = 43Β° and πβ π΅ = πβ π΅π΄π·, find πβ π΅π΄πΆ.

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Video Transcript

In the following triangle π΄π΅πΆ, if the measure of angle πΆ equals the measure of angle πΆπ΄π· equals 43 degrees and the measure of angle π΅ equals the measure of angle π΅π΄π·, find the measure of angle π΅π΄πΆ.

We can start this question by identifying the two pairs of congruent angle measures. We have that the measure of angle πΆ is equal to the measure of angle πΆπ΄π·, and those are both 43 degrees. We also have that the measure of angle π΅ is equal to the measure of angle π΅π΄π·, although we arenβt given an exact measurement for those. We can then identify that the angle that we wish to calculate is that of the measure of angle π΅π΄πΆ, which occurs at the vertex π΄ in the larger triangle π΄π΅πΆ.

A property that we can apply in this question is that the sum of the measures of the interior angles in a triangle is 180 degrees. So then, if we consider the large triangle π΄π΅πΆ, we can say that the measure of angle π΅π΄πΆ plus the measure of angle π΅ plus the measure of angle πΆ is equal to 180 degrees. From the diagram then, we can observe that the measure of angle π΅π΄πΆ actually consists of an angle of 43 degrees and the measure of angle π΅π΄π·.

Then, we are given in the question that the measure of angle π΅ is equal to the measure of angle π΅π΄π·. Adding in the measure of angle πΆ, which is 43 degrees, we can add the left-hand side, and it will be equal to 180 degrees. We can then simplify this by adding the two 43 degrees, which is 86 degrees. And we know that there will be two lots of the measure of angle π΅π΄π·. Subtracting 86 degrees from both sides, we have that two times the measure of angle π΅π΄π· is equal to 94 degrees. Finally, dividing through by two, we have that the measure of angle π΅π΄π· is 47 degrees. Now that we know the measure of this angle, we can calculate the measure of angle π΅π΄πΆ. It will be equal to 43 degrees plus 47 degrees, which gives us a final answer of 90 degrees.

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