Grades 7-8 Video Solutions 2024-2024_7-8

Video Transcription

Problem number 27. Three angles alpha, beta, and gamma are marked on graph paper as shown. What is the value of alpha plus beta plus gamma?  So the first thing we can do is look at this diagram larger and notice that a lot of these lines are parallel.  In particular, we will notice that these two values occur, angle beta and angle gamma. Let's prove this.  For example, let's look at the following red line. It is a transversal of the two blue lines. That is, it is a line that intersects two parallel lines.  In particular, we can show that using these two parallel blue lines, the two angles beta and beta are equal to each other  because this red line intersects them both and since they are parallel, these two angles will be equal.  Similarly, by the same logic, we can prove that these two angles marked gamma are equal because these two blue lines are parallel and this red line intersects both of them.  So that means that this gamma and the other gamma are equal. This is a consequence of the alternate interior angle theorem.  Now let's look at this green triangle that's highlighted here. We notice that it has angles alpha, beta plus gamma, and 90 degrees.  This means that we know in any triangle, the sum of the angles equals 180 degrees. That means that alpha plus beta plus gamma plus 90 degrees is equal to 180 degrees.  If we subtract 90 degrees from both sides, we get alpha plus beta plus gamma equals 90 degrees. The question asks us, what is the value of alpha plus beta plus gamma?  The answer is D, 90 degrees.

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