**Table of Contents**

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 \(\angle E\).

## Triangle Classification

Triangles can be classified by their side lengths and angle measurements. There are six types of triangles.

“Equi” refers to equal and “lateral” refers to the side lengths, so **equilateral** **triangles** are triangles that have three sides with the same length. Because their side lengths are all congruent, their interior angles are also congruent. They’re also known as **equiangular triangles**, because their interior angles are all the same.

Since the interior angles of a triangle add up to 180°, and all angles are equal, the interior angles must always each be 60°, as \(180\div 3=60\).

An **isosceles triangle** is a triangle that has two congruent sides and two congruent angles. The measures of the third side and angle differ from the other two.

The isosceles triangle in this example has two congruent angles on the bottom and two congruent side lengths on the left and right. The bottom length differs from the other two sides, and the top angle differs from the bottom angles.

A **scalene triangle** is a triangle with no congruent side lengths or angle measurements. They’re all different.

Note how the sides all have different numbers of tick marks, and the angles all have a different amount of arcs. This shows they’re all different, and not congruent.

An **acute triangle** is a triangle whose interior angle measurements are all acute, less than ninety degrees.

An **obtuse triangle** is a triangle with one angle measuring over 90 degrees.

A **right triangle** is a triangle with one angle measuring exactly 90 degrees.

### Classification Examples

**Instructions:** Classify the following triangles by identifying every category they fit into.

## Isosceles Triangles

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 & Exterior Angles

### Interior Angles

**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.

## Triangle Congruence

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 \(\overline{DF}\).

### Side-Side-Side Congruence Postulate

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.

### Shared Sides of a Triangle

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.

### Side-Angle-Side Congruence Postulate

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.

### Angle-Side-Angle Congruence Postulate

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.

### Angle-Angle-Side Congruence Postulate

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.

### Hypotenuse-Leg Theorem

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.

## Sample Problems – Angles & Congruence

Here are some sample problems related to angles and the congruence of triangles.

**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 \(m\angle{C}=41°\).

## Perpendicular Bisectors and Circumcenters

As was discussed in Chapter 1, **bisectors** split a line segment into two congruent parts at the midpoint, and **perpendicular bisectors** do so at a right angle.

Since triangles are made of three line segments, every triangle must have three perpendicular bisectors.

The point at which the three perpendicular bisectors of a triangle meet is known as the circumcenter. This point may be inside or outside of the triangle.

### Circumcircles

A circle sharing three points with the vertices of a triangle forms a **circumcircle**. The center of the circumcircle is the circumcenter.

### Circumcenter Theorem

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

### Perpendicular Bisector Theorem

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.

## Incenters and Incircles

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.

## Medians and Centroids

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.

## Altitudes and Orthocenters

### The Altitude of a Triangle

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.

### Locations of Orthocenters

In triangles with only acute angles, the orthocenter will always be somewhere inside the triangle.

In triangles with an obtuse angle, the orthocenter will be located outside of the triangle.

The orthocenter of a right triangle is at the vertex point forming the right angle.

**1.** What is point D in the figure below? Determine the length of \(\overline{DC}\) given \(AD=4\). Explain.

**Solution:**

Point D is the circumcenter of this triangle because it’s at the intersection of the perpendicular bisectors. This means it is equidistant from each of the vertices. So \(\overline{AD}\cong\overline{DC}\). Therefore, \(DC=4\).

**2.** Check your understanding. Explain the difference between the circumcenter and the orthocenter of a triangle.

**Solution:**

The circumcenter is the point at which the perpendicular bisectors of the triangle intersect. The perpendicular bisectors are the lines extending out from the midpoint of each side at a 90 degree angle. The circumcenter is equidistant from all the vertices.

The orthocenter is different from the circumcenter. The orthocenter is the intersection of the three altitudes. An altitude connects each vertex to its opposite side at a perpendicular angle. The altitude does not have to meet the side at its midpoint. The orthocenter is not always equidistant from the verticies or sides.

**3.** Point D, not shown in the diagram, is the *circumcenter* of \(\triangle{ABC}\). Given \(AD=3x+1\) and \(BD=4x-2\), what is the length of \(\overline{CD}\)?

**Solution:**

Remember that the circumcenter is equidistant from the vertices of the triangle, so \(\overline{AD}\cong\overline{BD}\cong\overline{CD}\).

We can setup an equation with the given information.

\(3x+1=4x-2\)

Solve the equation for x using algebra.

\(x=3\)

Plug 3 back in for x to either equation, and you will find that the length is 10.

Since all the distances from the circumcenter to the verticies are congruent, \(CD=10\).

**4.** The incenter is equidistant from what part of a triangle?

- The verticies of the triangle
- The sides of the triangle
- The midpoints of the sides of the triangle

**Solution:**

The correct answer is B. The incenter is equidistant from the sides of the triangle, taking the shortest path possible. It does not have to meet at the midpoints of the sides, but will do so in equilateral triangles.

**5.** True or false? The incenter, orthocenter, and centroid of an equilateral triangle will all be at the same point.

**Solution:**

This is true for equilateral triangles and equilateral triangles only.

**6.** The centroid is shown in the diagram below. Given the length of one of the line segments, determine the value of x.

**Solution:**

Remember, the centroid divides the medians into two segments with a 1:2 ratio. The shorter segment in the median is 1/3 of the total length and the longer part is 2/3 of the length. \(x=2\times12\). \(x=24in\).