In the following figure, if 𝐿𝑋
equals 𝑌𝑋, 𝐿𝑍 equals 𝑌𝑍, the measure of angle 𝐿𝑌𝑋 equals 30 degrees, the
measure of angle 𝑍𝐿𝑌 equals 55 degrees, and 𝑀 is the midpoint of line segment
𝐿𝑌, 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 𝑌𝑋 are congruent. Then, 𝐿𝑍 and 𝑌𝑍 are
congruent. And finally we can see on the
diagram that line segments 𝐿𝑀 and 𝑌𝑀 are congruent. That’s because the point 𝑀 is the
midpoint of the line segment 𝐿𝑌.
Now, the first two pairs of
congruent sides will give us some information about these triangles. We can say that triangle 𝐿𝑌𝑍 is
isosceles. And so is triangle 𝐿𝑌𝑋 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 𝐿𝑍𝑌, to be the vertex angle of triangle
𝐿𝑌𝑍, 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 𝐿𝑌𝑍 to the midpoint of the opposite side. The theorem then allows us to note
that this median is a perpendicular bisector of the base 𝐿𝑌. Notice that we already knew that
the point 𝑀 is the midpoint of line segment 𝐿𝑌. 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
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
We can note that line segment 𝑀𝑋
is a median of triangle 𝐿𝑋𝑌. And as we previously noted,
triangle 𝐿𝑋𝑌 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.