All About Triangles

Recall from the first section, Fundamental Geometry Concepts, that polygons are flat, enclosed 2D shapes with straight sides. At least three sides are needed to create a polygon. You probably already know what a triangle is, but a triangle is formally defined as a polygon with three sides and three angles. “Tri” means three and “angle” means angle, so a triangle has three angles.

The symbol for a triangle is a triangle: \(\triangle\)

To name a triangle, use the triangle symbol followed by three of the vertices (that’s plural for vertex). The example triangle shown is named \(\triangle KEV\). I hope it’s not pretentious to name a triangle after myself. 

An opposite angle is an angle directly across from a side. An opposite side is the side directly across from an angle. In the sample triangle KEV, the angle K is opposite the line segment \(\overline{EV}\). \(\angle E\) is opposite \(\overline{KV}\). \(\overline{KV}\) is opposite \(\angle E\).

As described above, an isosceles triangle is a triangle with two congruent sides and two congruent angles.

The side that is different is called the base of the triangle. The two congruent sides are called legs of the triangle.

The angles opposite (across from) the legs are called the base angles. They’re the angles touching the base.

Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite them are also congruent.

Converse of the Isosceles Triangle Theorem

If two angles of a triangle are congruent, then the sides opposite them are also congruent.

Interior angles are the angles on the inside of the triangle.

Triangle Sum Theorem

The triangle sum theorem says that the interior angles of a triangle add up to 180°. This is true regardless of the triangle’s classification.

Exterior Angles

Exterior angles are the angles on the outside of the triangle.

Note that the exterior angles are measured from the extension of the lines making up the side, inwards on the shortest angle towards the next side of the triangle. It’s always going to be less than 180°.

The measure of the exterior angle of a triangle is equal to the sum of the two nonadjacent interior angles.

As was mentioned in chapter 1, congruent polygons are polygons with the same shape and size. So congruent triangles are triangles with the same shape and size.

Corresponding angles are angles in the same relative position on each congruent triangle. Corresponding sides are sides in the same relative position.

In these example triangles, \(\angle A\) corresponds to \(\angle F\), \(\angle B\) corresponds to \(\angle E\), and \(\angle C\) corresponds to \(\angle D\).

\(\overline{AB}\) corresponds to \(\overline{FE}\), \(\overline{BC}\) corresponds to \(\overline{ED}\), and \(\overline{AC}\) corresponds to \(\overline{DF}\).

The Side-Side-Side (SSS) Congruence Postulate says that if three sides of one triangle are congruent to three sides of another, then the triangles are congruent.

Since congruent triangles have corresponding congruent angles, if the corresponding sides are all congruent, we can also say that the corresponding angles are all congruent.

In this example, an isosceles triangle has a perpendicular bisector going through the base. This splits the triangle into two congruent triangles. In this case, the segment acting as a bisector is congruent to itself in both triangles. Line segments are congruent to themselves.

Since \(\overline{GI}\cong\overline{GI}\), \(\overline{GH}\cong\overline{GJ}\), and \(\overline{HI}\cong\overline{IJ}\), the triangles satisfy the Side-Side-Side Congruence Postulate, and the triangles are congruent.

The included angle is the angle between two sides of a triangle. The Side-Angle-Side Congruence Postulate (SAS) says if two sides and the included angle of one triangle are congruent to two sides and the included angle of another, then the triangles are congruent.

Just like an included angle is between two sides, an included side is a side between two angles.

The Angle-Side-Angle Congruence Postulate says if two angles and the included side of one triangle are congruent to two angles and the included side of another, then the two triangles are congruent.

This postulate is very similar to the ASA postulate. It requires two angles and one side to be congruent. The difference is that the Angle-Angle-Side Congruence Theorem says if two angles and a nonincluded side of one triangle are congruent to the angles and nonincluded side of another, then the triangles are congruent.

The Hypotenuse-Leg (HL) Theorem says if the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent.

Remember, they must be right triangles! If they’re both right triangles, we already know there are two congruent 90 degree angles. So this is kind of like the SAS congruence postulate. The difference is the side-angle-side congruence postulate requires the two sides to be adjacent to the included angle. In this case, we’re using an adjacent and opposite side length to the right angle.

1. Determine the measure of \(\angle{1}\)

Solution:

Since \(m\angle{ABC}=40°\) and this is an isosceles triangle, we know that \(m\angle{ACB}\) is also 40°. \(\angle{ACB}\) is supplementary to \(\angle{1}\), so \(m\angle{ACB}+m\angle{1}=180°\).

This means that the measure of angle 1 must equal 140°, as \(180°-40°=140°\).

2. Are these two triangles congruent? If so, how do you know?

Solution:

These two triangles are congruent because of the Side-Side-Side congruence postulate. Each side of the first triangle has a corresponding congruent side to the second. They must be congruent triangles.

3. Determine the length of the side length x.

Solution:

First, determine if the triangles are congruent. Since they have the same interior angle measure between two corresponding congruent sides, the triangles are congruent by the Side-Angle-Side congruence postulate. 

Since the triangles are congruent, and the side with the x corresponds to the 50m side of the first triangle, \(x=50m\).

4. These two triangles are not drawn to scale. Can you prove that they’re congruent? Explain.

Solution:

Although these triangles look like they could be congruent, we do not have enough information to prove that they are. They’re not drawn to scale, so measuring it manually with a ruler wouldn’t count.

The figure must show that the hypotenuse of each triangle is congruent to the other to satisfy the Hypotenuse-Leg Theorem. The longer legs would have to be congruent to satisfy the Side-Angle-Side congruence postulate.

These triangles might be congruent, but they also might not be. We cannot prove it.

5. These triangles are not drawn to scale. Given \(\triangle{ABC}\cong\triangle{DEF}\), which side length of the second triangle corresponds to \(\overline{BC}\)?

Solution:

Since \(\overline{BC}\) is opposite \(\angle{BAC}\) in the first triangle, we know the corresponding side must be opposite the 40° \(\angle{DEF}\) on the second triangle. Therefore, \(\overline{DF}\) corresponds to \(\overline{BC}\). Since they’re congruent triangles, we can also say that \(\overline{DF}\cong\overline{BC}\).

6. Determine the measure of \(\angle{C}\) in the triangle below.

Solution:

The triangle sum theorem says the interior angles of a triangle add up to 180°. We know \(m\angle{A}=49°\) and \(m\angle{B}=90°\) from the figure. \(m\angle{A}+m\angle{B}+m\angle{C}=180°\). So \(m\angle{C}=180°-90°-49°\) which works out to 41°. So \(m\angle{C}=41°\).

The circumcenter theorem says that the circumcenter of a triangle is equidistant from the vertices.

The Perpendicular Bisector Theorem says any point on the perpendicular bisector of a line segment will be equidistant (the same distance) from each of the line segment’s endpoints.

In the example, \(\overrightarrow{FA}\) is a perpendicular bisector to \(\overline{BC}\). Point A is equidistant from points B and C, point D is equidistant from points A and C, and point E is also equidistant from points A and C. Any other point you could make on \(\overrightarrow{FA}\) would also be equidistant from points B and C.

Recall that angle bisectors split angles evenly into two congruent parts. The incenter of a triangle is the point where all the angle bisectors meet.

A circle whose circumference touches each side of a triangle one time forms an incircle.

The incenter theorem says that the incenter is equidistant to each side of the triangle.

If you connect perpendicular line segments from the sides of the triangle to the incenter (the radius of the incircle), they will all be congruent.

A median of a triangle is a line between a vertex and the midpoint of the opposite side. The three medians of a triangle intersect at a point called the centroid.

The centroid theorem says that the centroid is located \(\frac{2}{3}\) the distance from each vertex to opposite side’s midpoint.

The perpendicular line drawn from any vertex to the opposite side is the altitude of the triangle from that vertex. This differs from the perpendicular bisector, which comes from the midpoint of a side.

Since triangles have three vertices, there are three altitudes for every triangle.

The point at which all the altitudes meet is known as the orthocenter of the triangle. Note that for isosceles triangles, like the example shown, both the orthocenter and the circumcenter are at the same point.