# Question Video: Proving a Geometric Statement Using the Corollaries of the Isosceles Triangle Theorems Mathematics

In the figure below, line segment π΄πΆ β© line segment π΅π· = π, ππ· = ππΆ, line segment π΄π΅ β₯ line segment πΆπ·, and πβ ππΆπ· = 40Β°. Draw line segment ππΈ β line segment πΆπ· and cut line segment πΆπ· at πΈ, then draw line segment ππΉ β line segment π΄π΅ and cut line segment π΄π΅ at πΉ. Find πβ πΆππΈ and πβ π΅ππΉ.

04:35

### Video Transcript

In the figure below, the intersection of line segment π΄πΆ and line segment π΅π· equals π. ππ· equals ππΆ. Line segment π΄π΅ and line segment πΆπ· are parallel. And the measure of angle ππΆπ· equals 40 degrees. Draw line segment ππΈ perpendicular to line segment πΆπ·, and cut line segment πΆπ· at πΈ. Then, draw line segment ππΉ perpendicular to line segment π΄π΅, and cut line segment π΄π΅ at πΉ. Find the measure of angle πΆππΈ and the measure of angle π΅ππΉ.

Letβs begin by considering the information that we are given, with the first piece of information being that the two line segments π΄πΆ and π΅π· intersect at the point π. Next, we have two congruent line segments, since ππ· is equal to ππΆ. Then, we are given that the line segments π΄π΅ and πΆπ· are parallel. And the measure of angle ππΆπ· is 40 degrees.

Now we need to draw two new line segments, with the first being the line segment ππΈ, which must be perpendicular to line segment πΆπ· and with line segment ππΈ cutting line segment πΆπ· at point πΈ, which would look like this on the figure. Similarly, we can draw the second new line segment of ππΉ perpendicular to line segment π΄π΅, which cuts line segment π΄π΅ at the point πΉ. We will need to determine the measures of angles πΆππΈ and π΅ππΉ.

One way that we can find the measure of the first angle πΆππΈ is by noticing that we have triangle πΆππΈ and we know two of the angles within it. By recalling that the interior angle measures in a triangle sum to 180 degrees, we can write that 40 degrees plus the right angle of 90 degrees plus the measure of angle πΆππΈ equals 180 degrees. And simplifying the left-hand side and then subtracting 130 degrees from both sides, we have that the measure of angle πΆππΈ is 50 degrees. So, thatβs the first angle measure calculated.

Next, letβs consider how we can calculate the measure of angle π΅ππΉ. To do this, we can take a closer look at triangle πΆππ·. Given that this triangle has two congruent line segments, that means that by definition, triangle πΆππ· is an isosceles triangle. We also drew a perpendicular line from the vertex angle of this isosceles triangle to the base.

We can recall that one of the corollaries of the isosceles triangle theorems states that the straight line that passes through the vertex angle of an isosceles triangle and is perpendicular to the base bisects the base and the vertex angle. The important part for this question is that the line segment ππΈ that we drew bisects the vertex angle. The vertex angle would be the angle πΆππ·. So both angles πΆππΈ and π·ππΈ would have the same measure. They are both 50 degrees.

But we still need to find the measure of angle π΅ππΉ, which is in a different triangle. However, we can observe that the angles π·ππΈ and π΅ππΉ are vertically opposite angles. Therefore, they are both equal in measure. They are both 50 degrees. We can then give the answers for both the required angle measures. The measure of angle πΆππΈ is 50 degrees, and the measure of angle π΅ππΉ is 50 degrees.