Interior and Exterior Angles of a Triangle

Moira is a mosaic master and has just taken on a big space to help the "beautify the city" initiative. She is fascinated by the mosaics of ancient Egypt that are made of colorful tiles, stones, or glass. Moira can easily see how to make a mosaic from Equilateral triangles or isosceles triangles but what about scalene triangles, or any triangle for that matter? To find out, we'll need to explore interior and exterior angles of a triangle. Let's start with a scalene triangle, and extend one side. These angles are known as interior angles because they're on the inside. This angle is called an exterior angle because it is outside of the shape but still shares a side. Exterior angles are adjacent, meaning next to, their respective interior angles. In fact, if we extend all the sides of the triangle we create six exterior angles. But what about these angles? They are outside the triangle, but they're not exterior angles, because they aren't adjacent to interior angles. Each of these angles are called vertical to their opposite interior angles. How do exterior angles relate to their adjacent interior angles? To find out, let's look at one vertex of the triangle and label the angle measures. We'll say this interior angle measures 'x.' So, what is the measure of this opposite angle? It is also 'x' because this angle is a vertical angle to this interior angle. If we say this exterior angle measures 'y', then what is the measurement of this angle opposite of it? It is also 'y' because this angle is vertical to the exterior angle. These four angles actually form a complete circle. Therefore, if we sum x, plus x, plus y, plus y, we get 360 degrees. Simplifying we notice that we get 2x plus 2y equals 360. Actually we could even go further. Do you notice that all the numbers are even? So, we could divide everything by 2. This tells us 'x' plus 'y' is 180 degrees. That means that an exterior angle is always supplementary to its adjacent interior angle. This makes sense, because this is a straight angle. An angle whose rays form a straight line and straight angles always measure 180 degrees. Now, Moira has sketched a triangle with an interior angle of 110 degrees. So, what's the measure of this exterior angle, that we can call 'y'? We know these angles are supplementary, so 110, plus 'y,' is 180 degrees. Isolating the 'y,' tells us that the measure of the exterior angle is 70 degrees. Exterior angles and their adjacent interior angles always sum to 180 degrees, but let's explore other angle relationships in this diagram. From the perspective of this exterior angle, these two angles are called remote interior angles. The remote interior angles are inside the triangle and are the angles far away, also known as remote, from the exterior angle. Let's explore their relationship by first recalling what the sum of all 3 interior angles of a triangle is. The angle sum theorem tells us that the sum of the interior angles of a triangle is always 180 degrees. Looking at the tile Moira made based on her earlier sketch, we see the 110 degree angle plus these two remote interior angles sum to be 180 degrees. We can see this by breaking off the two remote interior angles and suming them all visually by placing them next to each other. Notice that the two remote interior angles sum to the measure of the exterior angle! We can use this fact that remote interior angles sum to be the mesaure of the exterior angles of a triangle, to solve for unknowns. Suppose we want to find 'x' in THIS diagram. We are given an exterior angle of 100 degrees and two remote interior angles whose measures are 55 degrees, and 'x'. Recall, the exterior angle is the sum of the two remote interior angles so 100 equals 55 plus 'x'. Subtraction gives us 45 degrees for 'x'. These special relationships between exterior angles and interior angles help Moira create her mosaic. But before she unveals her work of art, let's review the angle relationships of interior and exterior angles of a triangle. First, recall what the angles are that you're looking at. The angles on the inside are called Interior angles. The sum of the interior angles of a triangle is always 180 degrees. The exterior angle is the angle between any side of a shape, and a line extended from the next side. The sum of an exterior angle and its adjacent interior angle is also 180 degrees. Looking at this exterior angle we can see that the remote interior angles are the two angles inside the triangle that a farther away. The measure of an exterior angle is equal to the sum of the two remote interior angles. So. Now! The moment we've all be waiting for. Moira unveils her effort to beautify the city! I guess Moira was already making the city beautiful by just being herself! Now they are getting a bit more Moira than they bargained for!