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 \(\overset{\bar}{AB},\overset{\bar}{BC,}\) and \(\overset{\bar}{AC},\) we have a triangle, as shown in .

Triangles are classified by their angles and their sides. All angles in an acute triangle measure \(<{90}^{∘}.\) One of the angles in a right triangle measures \({90}^{∘},\) symbolized by □. One angle in an obtuse triangle measures between \({90}^{∘}\) and \({180}^{∘}.\) Sides that have equal length are indicated by the same hash marks. 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 \({180}^{∘}.\) Euclid discovered and proved this property using parallel lines. The completed sketch is shown in .

This is how the proof goes:

Step 1: Start with a straight line \(\overset{↔}{AB}\) and a point \(C\) not on the line.

Finding Measures of Angles Inside a Triangle

Try it.

Find the measure of each angle in the triangle shown (). We know that the sum of the angles must equal \({180}^{∘}.\)

Solution

Step 1: As the sum of the interior angles equals \({180}^{∘},\) we can use algebra to find the measures: \[\begin{array}{lll}x+(x+17)+(3x-62) & = & 180 \\ 5x-45 & = & 180 \\ 5x & = & 225 \\ x & = & \frac{225}{5}=45\end{array}\]

Step 2: Now that we have the value of \(x\), we can substitute 45 into the other two expressions to find the measure of those angles: \[\begin{array}{lll}(x+17) & = & 45+17={62}^{∘} \\ (3x-62) & = & 3(45)-62=135-62={73}^{∘}\end{array}\]

Step 3: Then, \(m∡x={45}^{∘}\), \(m∡(x+17)={62}^{∘}\), and \(m∡(3x-62)={73}^{∘}.\)

Finding Angle Measures

Try it.

Find the measure of angles numbered 1–5 in .

Solution

The \(m∡1={119}^{∘}\) because it is supplementary with the unknown angle of the adjacent triangle. The unknown angle measures \({61}^{∘}.\) The \(m∡2={61}^{∘}\) because of vertical angles. The \(m∡5={59}^{∘}\) because the angle that is supplementary to the \({134}^{∘}\) measures \({46}^{∘}\), and angle 5 is the unknown angle in that triangle. The \(m∡4={59}^{∘}\) by vertical angles. Finally, \(m∡3={60}^{∘},\) as it is the third angle in the triangle with angles measuring \({59}^{∘}\) and \({61}^{∘}.\)

Condensed — the full section is in OpenStax Contemporary Mathematics.

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.

Determining If Triangles Are Congruent

Try it.

In , is the triangle \(\text{ABC}\) congruent to triangle \(\text{DEF}\)?

Solution

Triangle \(\text{ABC}\) is congruent to triangle \(\text{DEF}\). Angles \(A\) and \(C\) are congruent to angles \(D\) and \(F\), which implies that angle \(B\) is congruent to angle \(E\). Side \(\text{AB}\) is congruent to side \(\text{DE}\), and side \(\text{CB}\) is congruent to side \(\text{FE}\), which implies that side \(\text{AC}\) is congruent to side \(\text{DF}\).

The Congruence Theorems

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

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 .

We have that \(\overset{\bar}{DF}≅\overset{\bar}{RT}\), \(\overset{\bar}{EF}≅\overset{\bar}{ST},\) and \(\overset{\bar}{\text{DE}}≅\overset{\bar}{\text{RS}}\text{,}\) then \(\text{\Delta }DEF≅\text{\Delta }RST.\)

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 . We see that \(\overset{\bar}{AB}≅\overset{\bar}{{A}^{'}{B}^{'}}\) and \(\overset{\bar}{BC}≅\overset{\bar}{{B}^{'}{C}^{'}}\), \(m∡B=m∡{B}^{'}\), then \(\text{\Delta }ABC≅\text{\Delta }{A}^{'}{B}^{'}{C}^{'}\).

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 . Notice that \(m∡\text{A}≅m∡\text{F}\), and \(m∡\text{C}≅m∡\text{D}\), \(\overset{\bar}{\text{AC}}≅\overset{\bar}{\text{DF}}\), then \(\text{\Delta }\text{ABC}≅\text{\Delta }\text{DEF}.\)

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 . We see that \(m∡X≅m{X}^{'}\), \(m∡Z≅m∡{Z}^{'}\), and \(\overset{\bar}{XY}≅\overset{\bar}{{X}^{'}{Y}^{'}}\), then \(\text{\Delta }XYZ≅\text{\Delta }{X}^{'}{Y}^{'}{Z}^{'}\).

Identifying Congruence Theorems

Try it.

What congruence theorem is illustrated in ?

Solution

AAS: Two angles and a non-included side in one triangle are congruent to the corresponding angles and side in the second triangle.

Determining the Congruence Theorem

Try it.

What congruence theorem is illustrated in ?

Solution

The SSS theorem.

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 are similar? These two images have the same proportions between elements. Therefore, they are similar.

Determining If Triangles Are Similar

Try it.

Are the two triangles shown in similar?

Solution

Step 1: We will look at the proportions within each triangle. In triangle \(\alpha\) (alpha), the side opposite the \({57}^{∘}\) angle measures 7, and the side opposite the \({33}^{∘}\) angle measures 4. Then, the measures of the corresponding sides in triangle \(\beta\) (beta) measures 3.5 and 2, respectively. We have \[\frac{4}{7}=0.5714\ \frac{2}{3.5}=0.5714.\]

This is the proportion \(\frac{4}{7}=\frac{2}{3.5}\). The scaling factor is 0.5714.

Step 2: Let’s try another correspondence. In triangle \(\alpha\), the hypotenuse measures 8.06 and the side opposite the \({57}^{∘}\) angle measures 7. In triangle \(\beta\), the hypotenuse measure 4.03 and the side opposite the \({57}^{∘}\) angle measures 3.5. We have \[\frac{7}{8.06}=0.8685\ \frac{3.5}{4.03}=0.8685.\]

Step 3: Now, let’s look at the proportions between triangle \(\alpha\) and triangle \(\beta .\) The side measuring 2 in triangle \(\beta\) corresponds to the side measuring 4 in triangle \(\alpha\), the side measuring 3.5 in triangle \(\beta\) corresponds to the side measuring 7 in triangle \(\alpha ,\) and the hypotenuse in triangle \(\beta\) corresponds to the hypotenuse in triangle \(\alpha .\) We have \[\frac{2}{4}=0.5\ \frac{3.5}{7}=0.5\ \frac{4.03}{8.06}=0.5\]

Thus, the corresponding angles are equal and the proportions between each pair of corresponding sides equals 0.5. In other words, the scaling factor is 0.5. Therefore, the triangles are similar.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Key Concepts

• The sum of the interior angles of a triangle equals \({180}^{∘}.\)
• 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.

Намунаи коркардашуда: triangle 3 4 5

Practice (9)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. Find the measure of each angle in the triangle shown (). We know that the sum of the angles must equal \({180}^{∘}.\)

   Ҷавоби ҷавобро нишон диҳед

   Step 1: As the sum of the interior angles equals \({180}^{∘},\) we can use algebra to find the measures: \[\begin{array}{lll}x+(x+17)+(3x-62) & = & 180 \\ 5x-45 & = & 180 \\ 5x & = & 225 \\ x & = & \frac{225}{5}=45\end{array}\]

   Step 2: Now that we have the value of \(x\), we can substitute 45 into the other two expressions to find the measure of those angles: \[\begin{array}{lll}(x+17) & = & 45+17={62}^{∘} \\ (3x-62) & = & 3(45)-62=135-62={73}^{∘}\end{array}\]

   Step 3: Then, \(m∡x={45}^{∘}\), \(m∡(x+17)={62}^{∘}\), and \(m∡(3x-62)={73}^{∘}.\)

2. Find the measure of angles numbered 1–5 in .

   Ҷавоби ҷавобро нишон диҳед

   The \(m∡1={119}^{∘}\) because it is supplementary with the unknown angle of the adjacent triangle. The unknown angle measures \({61}^{∘}.\) The \(m∡2={61}^{∘}\) because of vertical angles. The \(m∡5={59}^{∘}\) because the angle that is supplementary to the \({134}^{∘}\) measures \({46}^{∘}\), and angle 5 is the unknown angle in that triangle. The \(m∡4={59}^{∘}\) by vertical angles. Finally, \(m∡3={60}^{∘},\) as it is the third angle in the triangle with angles measuring \({59}^{∘}\) and \({61}^{∘}.\)

3. In , is the triangle \(\text{ABC}\) congruent to triangle \(\text{DEF}\)?

   Ҷавоби ҷавобро нишон диҳед

   Triangle \(\text{ABC}\) is congruent to triangle \(\text{DEF}\). Angles \(A\) and \(C\) are congruent to angles \(D\) and \(F\), which implies that angle \(B\) is congruent to angle \(E\). Side \(\text{AB}\) is congruent to side \(\text{DE}\), and side \(\text{CB}\) is congruent to side \(\text{FE}\), which implies that side \(\text{AC}\) is congruent to side \(\text{DF}\).

4. What congruence theorem is illustrated in ?

   Ҷавоби ҷавобро нишон диҳед

   AAS: Two angles and a non-included side in one triangle are congruent to the corresponding angles and side in the second triangle.

5. What congruence theorem is illustrated in ?

   Ҷавоби ҷавобро нишон диҳед

   The SSS theorem.

6. Are the two triangles shown in similar?

   Ҷавоби ҷавобро нишон диҳед

   Step 1: We will look at the proportions within each triangle. In triangle \(\alpha\) (alpha), the side opposite the \({57}^{∘}\) angle measures 7, and the side opposite the \({33}^{∘}\) angle measures 4. Then, the measures of the corresponding sides in triangle \(\beta\) (beta) measures 3.5 and 2, respectively. We have \[\frac{4}{7}=0.5714\ \frac{2}{3.5}=0.5714.\]

   This is the proportion \(\frac{4}{7}=\frac{2}{3.5}\). The scaling factor is 0.5714.

   Step 2: Let’s try another correspondence. In triangle \(\alpha\), the hypotenuse measures 8.06 and the side opposite the \({57}^{∘}\) angle measures 7. In triangle \(\beta\), the hypotenuse measure 4.03 and the side opposite the \({57}^{∘}\) angle measures 3.5. We have \[\frac{7}{8.06}=0.8685\ \frac{3.5}{4.03}=0.8685.\]

   Step 3: Now, let’s look at the proportions between triangle \(\alpha\) and triangle \(\beta .\) The side measuring 2 in triangle \(\beta\) corresponds to the side measuring 4 in triangle \(\alpha\), the side measuring 3.5 in triangle \(\beta\) corresponds to the side measuring 7 in triangle \(\alpha ,\) and the hypotenuse in triangle \(\beta\) corresponds to the hypotenuse in triangle \(\alpha .\) We have \[\frac{2}{4}=0.5\ \frac{3.5}{7}=0.5\ \frac{4.03}{8.06}=0.5\]
   

   Thus, the corresponding angles are equal and the proportions between each pair of corresponding sides equals 0.5. In other words, the scaling factor is 0.5. Therefore, the triangles are similar.

7. In , is triangle \(\delta\) (delta) similar to triangle \(\epsilon\) (epsilon)? Find the lengths of sides \(x\) and \(y\) as part of your answer.

   Ҷавоби ҷавобро нишон диҳед

   We can see that all three angles in triangle \(\delta\) are equal to the corresponding angles in triangle \(\epsilon\). That is enough to determine similarity. However, we want to find the values of \(x\) and \(y\) to prove similarity.

   Step 1: We have to do is set up the proportions between the corresponding sides. We have the side that measures 2.375 in triangle \(\delta\) corresponding to the side measuring 1.069 in triangle \(\epsilon\). We have the hypotenuse/side in triangle \(\delta\) measuring 6 corresponding to the hypotenuse/side labeled \(y\) in triangle \(\epsilon\). And, finally, the side labeled \(x\) in triangle \(\delta\) corresponds to the side measuring 2.475 in triangle \(\epsilon .\)

   Each proportion should be equal. We start with the proportion of the shorter sides. Thus \[\frac{1.069}{2.375}=0.45\]

   Step 2: We solve for \(y\) using the first proportion. Set the two ratios equal to each other, cross-multiply, and solve for \(y\). We have: \[\begin{array}{lll}\frac{1.069}{2.375} & = & \frac{y}{6} \\ (6)(1.069) & = & (2.375)(y) \\ 6.414 & = & 2.375y \\ \frac{6.414}{2.375} & = & 2.7=y\end{array}\]

   So, \(y=2.7\).

   Step 3: Checking that length in the proportion factor of 0.45, we have: \[\frac{y}{6}=\frac{2.7}{6}=0.45\]

   Step 4: Solving for \(x\), we will use the same proportion we used to solve for \(y\). We have: \[\begin{array}{lll}\frac{1.069}{2.375} & = & \frac{2.475}{x} \\ 1.069(x) & = & 2.475(2.375) \\ 1.069(x) & = & 5.878 \\ x & = & \frac{5.878}{1.069}=5.5 \\ \frac{2.475}{x} & = & \frac{2.475}{5.5}=0.45\end{array}\]

   Step 5: We test the proportions. We have the following: \[\frac{2.7}{6}=\frac{2.475}{5.5}=\frac{1.069}{2.375}=0.45\]
   

   The proportions are all equal. Therefore, we have proven the property of similarity between triangle \(\delta\) and triangle \(\epsilon .\)

8. A person who is 5 feet tall is standing 50 feet away from the base of a tree (). The tree casts a 57-foot shadow. The person casts a 7-foot shadow. What is the height of the tree?

   Ҷавоби ҷавобро нишон диҳед

   The bigger triangle includes a tree at side \(x\) and the smaller triangle includes the person at the side labeled 5 ft. These two triangles are similar because the smaller triangle fits inside the larger triangle at the smallest angle. It would fit inside the larger triangle at either of the other two angles as well. That all angles are equal is one of the criteria for similar triangles, so we can solve using proportions: \[\begin{array}{lll}\frac{5}{7} & = & \frac{x}{57} \\ 5(57) & = & 7x \\ \frac{285}{7} & = & x=40.7\end{array}\]

   The tree is 40.7 feet tall.

9. At a certain time of day, a radio tower casts a shadow 180 feet long (). At the same time, a 9-foot truck casts a shadow 15 feet long. What is the height of the tower?

   Ҷавоби ҷавобро нишон диҳед

   These are similar triangles and the problem can be solved by using proportions: \[\begin{array}{lll}\frac{x}{180} & = & \frac{9}{15} \\ 9(180) & = & 15x \\ 1620 & = & 15x \\ 108 & = & x\end{array}\]

   The height of the tower is 108 ft.

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

\sin,\ \cos,\ \tan

sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.

\neg,\ \wedge,\ \vee,\ \Rightarrow,\ \Leftrightarrow

not, and, or, implies, iff
Logical connectives.

\varepsilon,\ \delta

epsilon, delta
Small positive tolerances in the definition of a limit.

^\circ

degrees
1/360 of a full turn. 180° = π radians.

\approx

approximately equal
Equal to the precision shown, not exactly.

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

\theta

theta
The usual name for an angle.

\angle ABC,\ \triangle ABC

angle, triangle
The angle at B between BA and BC; the triangle with those vertices.

\parallel,\ \perp,\ \cong,\ \sim

parallel, perpendicular, congruent, similar
Never meet; meet at 90°; identical shape and size; same shape.

How to: Triangles

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.
6. The points where the line segments meet are called the
7. We often refer to sides of a triangle by the angle they are opposite. In other words, side
8. acute

Questions people ask

Why does every triangle have angles adding to 180°?

Draw a line through one vertex parallel to the opposite side: the two alternate angles equal the other two angles of the triangle, and the three angles at the vertex lie on a straight line. That is Euclid's proof, and it is why the fact holds only on a flat plane.

When do I use the law of sines versus the law of cosines?

Cosines when you know three sides, or two sides and the angle between them. Sines when you know an angle and the side opposite it, plus one more piece.

What is the difference between area and perimeter?

Perimeter is the length of the boundary (one dimension, measured in metres); area is the amount of surface inside (two dimensions, square metres). Doubling every side doubles the perimeter but quadruples the area.

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⌨ Type ✎ Draw

Масъалаи математикии шумо

& Ҷудо кардан

x²√÷×π( )=∫d/dxlim[ ]sinlogeˣ

triangle 5 12 13area of triangle 7 8 9triangle 2 3 4triangle 5 7 with angle 60triangle 1 2 5

Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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