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Question Video: Finding Unknown Angle Measures by Using the Properties of Isosceles Triangles Mathematics

In the figure, if 𝐿𝑋 = π‘Œπ‘‹, 𝐿𝑍 = π‘Œπ‘, π‘šβˆ πΏπ‘Œπ‘‹ = 30Β°, π‘šβˆ π‘πΏπ‘Œ = 55Β°, and 𝑀 is the midpoint of line segment πΏπ‘Œ, find π‘šβˆ π‘€π‘πΏ and π‘šβˆ πΏπ‘€π‘‹.

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Video Transcript

In the following figure, if 𝐿𝑋 equals π‘Œπ‘‹, 𝐿𝑍 equals π‘Œπ‘, the measure of angle πΏπ‘Œπ‘‹ equals 30 degrees, the measure of angle π‘πΏπ‘Œ equals 55 degrees, and 𝑀 is the midpoint of line segment πΏπ‘Œ, find the measure of angle 𝑀𝑍𝐿 and the measure of angle 𝐿𝑀𝑋.

Let’s begin this question by noting that we have three pairs of congruent line segments. Firstly, we have that line segments 𝐿𝑋 and π‘Œπ‘‹ are congruent. Then, 𝐿𝑍 and π‘Œπ‘ are congruent. And finally we can see on the diagram that line segments 𝐿𝑀 and π‘Œπ‘€ are congruent. That’s because the point 𝑀 is the midpoint of the line segment πΏπ‘Œ.

Now, the first two pairs of congruent sides will give us some information about these triangles. We can say that triangle πΏπ‘Œπ‘ is isosceles. And so is triangle πΏπ‘Œπ‘‹ as both these triangles have two sides congruent. Now, let’s consider the angle measures that we need to calculate. The first is the measure of angle 𝑀𝑍𝐿, which means that we need to add in a line segment. Here is the line segment 𝑀𝑍. So angle 𝑀𝑍𝐿 will be here on the left side of the diagram.

Notice that we could describe this interior angle at 𝑍, that is, the angle πΏπ‘π‘Œ, to be the vertex angle of triangle πΏπ‘Œπ‘, because this is an isosceles triangle. And in an isosceles triangle, the angle which is not congruent to any other is called the vertex angle. And this information might help us recall one of the isosceles triangle theorems. This theorem states that the median of an isosceles triangle from the vertex angle is a perpendicular bisector of the base and bisects the vertex angle.

Let’s remind ourselves of what a median is. It is a line segment from any vertex of a triangle to the midpoint of the opposite side. And that’s what we have here. Line segment 𝑀𝑍 is a line from the vertex angle of triangle πΏπ‘Œπ‘ to the midpoint of the opposite side. The theorem then allows us to note that this median is a perpendicular bisector of the base πΏπ‘Œ. Notice that we already knew that the point 𝑀 is the midpoint of line segment πΏπ‘Œ. But now we know that the line segment 𝑀𝑍 bisects it at right angles.

By noticing that this angle is part of the smaller triangle 𝐿𝑀𝑍, then we can find its measure by recalling that the interior angle measures in a triangle sum to 180 degrees. So we have the measure of angle 𝑀𝑍𝐿 plus 55 degrees plus 90 degrees equals 180 degrees. Then, simplifying and subtracting 145 degrees from both sides, we have that the measure of angle 𝑀𝑍𝐿 is 35 degrees. And that’s the first part of the question answered.

Next, we need to find the measure of angle 𝐿𝑀𝑋. And so we need to draw in another line segment. Here is the line segment 𝑀𝑋. And we can add in the angle 𝐿𝑀𝑋 whose measure we need to calculate. At this point, we can’t simply assume that the larger line segment 𝑋𝑍 is a straight line which passes through 𝑀. So let’s see instead what we can determine.

We can note that line segment 𝑀𝑋 is a median of triangle πΏπ‘‹π‘Œ. And as we previously noted, triangle πΏπ‘‹π‘Œ is an isosceles triangle. Applying the same theorem as before, then this median from the vertex angle of the isosceles triangle must be a perpendicular bisector of the base. Therefore, the measure of angle 𝐿𝑀𝑋 is 90 degrees.

We can give the answers for both the required angle measures. The measure of angle 𝑀𝑍𝐿 is 35 degrees, and the measure of angle 𝐿𝑀𝑋 is 90 degrees.

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