10.3 Triangles

Learning Objectives

After completing this section, you should be able to:

1. Identify triangles by their sides.
2. Identify triangles by their angles.
3. Determine if triangles are congruent.
4. Determine if triangles are similar.
5. Find the missing side of similar triangles.

Identifying Triangles

Joining any three noncollinear points with line segments produces a triangle. For example, given points $A$, $B$, and $C$, connected by the line segments $\overline{AB},\overline{BC,}$ and $\overline{AC},$ we have a triangle, as shown in Figure 10.43.

Triangles are classified by their angles and their sides. All angles in an acute triangle measure $<{90}^{\circ }.$ One of the angles in a right triangle measures ${90}^{\circ },$ symbolized by □. One angle in an obtuse triangle measures between ${90}^{\circ }$ and ${180}^{\circ }.$ Sides that have equal length are indicated by the same hash marks. Figure 10.44 illustrates the shapes of the basic triangles, their names, and their properties.

A few other facts to remember as we move forward:

• The points where the line segments meet are called the vertices (plural for vertex).
• We often refer to sides of a triangle by the angle they are opposite. In other words, side $a$ is opposite angle $A$, side $b$ is opposite angle $B$, and side $c$ is opposite angle $C$.

We want to add a special note about right triangles here, as they are referred to more than any other triangle. The side opposite the right angle is its longest side and is called the hypotenuse, and the sides adjacent to the right angle are called the legs.

One of the most important properties of triangles is that the sum of the interior angles equals ${180}^{\circ }.$ Euclid discovered and proved this property using parallel lines. The completed sketch is shown in Figure 10.45.

This is how the proof goes:

Step 1: Start with a straight line $\overleftrightarrow{AB}$ and a point $C$ not on the line.

Step 2: Draw a line through point $C$ parallel to the line $\overleftrightarrow{AB}$.

Step 3: Construct two transversals (a line crossing the parallel lines), one angled to the right and one angled to the left, to intersect the parallel lines.

Step 4: Because of the property that alternate interior angles inside parallel lines are equal, we have that

$$m\measuredangle 2=m\sphericalangle 1\text{and}m\measuredangle 3=m\measuredangle 4.$$

Step 5: Notice that $m\measuredangle 2+m\measuredangle 5+m\measuredangle 4={180}^{\circ }$ by the straight angle property.

Step 6: Therefore, by substitution, $m\measuredangle 1=m\measuredangle 2,$ and $m\measuredangle 3=m\measuredangle 4,$ we have that

$$m\measuredangle 1+m\measuredangle 3+m\sphericalangle 5={180}^{\circ }\text{.}$$

Therefore, the sum of the interior angles of a $\text{triangle}={180}^{\circ }.$

Congruence

If two triangles have equal angles and their sides lengths are equal, the triangles are congruent. In other words, if you can pick up one triangle and place it on top of the other triangle and they coincide, even if you have to rotate one, they are congruent.

The Congruence Theorems

The following theorems are tools you can use to prove that two triangles are congruent. We use the symbol $\cong$ to define congruence. For example, $\text{Δ}ABC\cong \text{Δ}DEF$.

Side-Side-Side (SSS). If three sides of one triangle are equal to the corresponding sides of the second triangle, then the triangles are congruent. See Figure 10.49.

We have that $\overline{DF}\cong \overline{RT}$, $\overline{EF}\cong \overline{ST},$ and $\overline{\mathit{DE}}\cong \overline{\mathit{RS}}\text{,}$ then $\text{Δ}DEF\cong \text{Δ}RST.$

Side-Angle-Side (SAS). If two sides of a triangle and the angle between them are equal to the corresponding two sides and included angle of the second triangle, then the triangles are congruent. See Figure 10.50. We see that $\overline{AB}\cong \overline{A^{\prime }{B}^{\prime }}$ and $\overline{BC}\cong \overline{B^{\prime }{C}^{\prime }}$, $m\measuredangle B=m\measuredangle B^{\prime }$, then $\text{Δ}ABC\cong \text{Δ}{A}^{\prime }{B}^{\prime }{C}^{\prime }$.

Angle-Side-Angle (ASA). If two angles and the side between them in one triangle are congruent to the two corresponding angles and the side between them in a second triangle, then the two triangles are congruent. See Figure 10.51. Notice that $m\measuredangle \text{A}\cong m\measuredangle \text{F}$, and $m\measuredangle \text{C}\cong m\measuredangle \text{D}$, $\overline{\mathit{AC}}\cong \overline{\mathit{DF}}$, then $\text{Δ}\mathit{ABC}\cong \text{Δ}\mathit{DEF}.$

Angle-Angle-Side (AAS). If two angles and a nonincluded side of one triangle are congruent to two angles and the nonincluded corresponding side of a second triangle, then the triangles are congruent.

See Figure 10.52. We see that $m\measuredangle X\cong m{X}^{\prime }$, $m\measuredangle Z\cong m\measuredangle Z^{\prime }$, and $\overline{XY}\cong \overline{X^{\prime }{Y}^{\prime }}$, then $\text{Δ}XYZ\cong \text{Δ}{X}^{\prime }{Y}^{\prime }{Z}^{\prime }$.

Similarity

If two triangles have the same angle measurements and are the same shape but differ in size, the two triangles are similar. The lengths of the sides of one triangle will be proportional to the corresponding sides of the second triangle. Note that a single fraction $\frac{a}{b}$ is called a ratio, but two fractions equal to each other is called a proportion, such as $\frac{a}{b}=\frac{c}{d}.$

This rule of similarity applies to all shapes as well as triangles. Another way to view similarity is by applying a scaling factor, which is the ratio of corresponding measurements between an object or representation of the object, to an image that produces the second, similar image.

For example, why are the two images in Figure 10.55 are similar? These two images have the same proportions between elements. Therefore, they are similar.

Key Terms

• acute
• obtuse
• isosceles
• equilateral
• hypotenuse
• congruence
• similarity
• scaling factor

Key Concepts

• The sum of the interior angles of a triangle equals ${180}^{\circ }.$
• Two triangles are congruent when the corresponding angles have the same measure and the corresponding side lengths are equal.
• The congruence theorems include the following: SAS, two sides and the included angle of one triangle are congruent to the same in a second triangle; ASA, two angles and the included side of one triangle are congruent to the same in a second triangle; SSS, all three side lengths of one triangle are congruent to the same in a second triangle; AAS, two angles and the non-included side of one triangle are congruent to the same in a second triangle.
• Two shapes are similar when the proportions between corresponding angles, sides or features of two shapes are equal, regardless of size.