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Mathematics LibreTexts

1.1: Angles

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Recall the following definitions from elementary geometry:

  1. An angle is acute if it is between and 90°.
  2. An angle is a right angle if it equals 90°.
  3. An angle is obtuse if it is between 90° and 180°.
  4. An angle is a straight angle if it equals 180°.
1.1.1.png
Figure 1.1.1 Types of angles

In elementary geometry, angles are always considered to be positive and not larger than 360^\circ . For now we will only consider such angles. The following definitions will be used throughout the text:

  1. Two acute angles are complementary if their sum equals 90^◦. In other words, if 0^◦ ≤ ∠ A , ∠B ≤ 90^◦ \text{ then }∠ A \text{ and }∠B are complementary if ∠ A +∠B = 90^◦.
  2. Two angles between 0^◦ \text{ and }180^◦ are supplementary if their sum equals 180^◦. In other words, if 0^◦ ≤ ∠ A , ∠B ≤ 180^◦ \text{ then }∠ A \text{ and }∠B are supplementary if ∠ A +∠B = 180^◦.
  3. Two angles between 0^◦ \text{ and }360^◦ are conjugate (or explementary) if their sum equals 360^◦. In other words, if 0^◦ ≤ ∠ A , ∠B ≤ 360^◦ \text{ then }∠ A \text{ and }∠B\text{ are conjugate if }∠ A+∠B = 360^◦.
1.1.2.png
Figure 1.1.2 Types of pairs of angles

Instead of using the angle notation ∠ A to denote an angle, we will sometimes use just a capital letter by itself (e.g. A, B, C) or a lowercase variable name (e.g. x, y, t). It is also common to use letters (either uppercase or lowercase) from the Greek alphabet, shown in the table below, to represent angles:

Table 1.1 The Greek alphabet

1.1 Table.png

In elementary geometry you learned that the sum of the angles in a triangle equals 180^◦, and that an isosceles triangle is a triangle with two sides of equal length. Recall that in a right triangle one of the angles is a right angle. Thus, in a right triangle one of the angles is 90^◦ and the other two angles are acute angles whose sum is 90^◦ (i.e. the other two angles are complementary angles).

Example 1.1

For each triangle below, determine the unknown angle(s):

1.1 Example.png

Note: We will sometimes refer to the angles of a triangle by their vertex points. For example, in the first triangle above we will simply refer to the angle \angle\,BAC as angle A.

Solution:

For triangle \triangle\,ABC, A = 35^\circ and C = 20^\circ, and we know that A + B + C = 180^\circ, so

\nonumber 35^◦ + B + 20^◦ = 180^◦ ⇒ B = 180^◦ − 35^◦ − 20^◦ ⇒ \fbox{\(B = 125^◦\)} . \nonumber

For the right triangle △DEF,\, E = 53^◦ \text{ and }F = 90^◦, and we know that the two acute angles D and E are complementary, so

\nonumber D + E = 90^◦ ⇒ D = 90^◦ − 53^◦ ⇒ \fbox{\(D = 37^◦\)} . \nonumber

For triangle △ XY Z, the angles are in terms of an unknown number α, but we do know that X +Y + Z = 180^◦, which we can use to solve for α and then use that to solve for X, Y, \text{ and }Z:

\nonumber α + 3α + α = 180^◦ ⇒ 5α = 180^◦ ⇒ α = 36^◦ ⇒ \fbox{\(X = 36^◦ ,\, Y = 3×36^◦ = 108^◦ ,\, Z = 36^◦\)} \nonumber

Example 1.2: Thales' Theorem

Thales' Theorem states that if A, \, B,\text{ and }C are (distinct) points on a circle such that the line segment \overline{AB} is a diameter of the circle, then the angle \angle\,ACB is a right angle (see Figure 1.1.3(a)). In other words, the triangle \triangle\,ABC is a right triangle.

1.1.3.png
Figure 1.1.3 Thales’ Theorem: ∠ ACB = 90^◦

To prove this, let O be the center of the circle and draw the line segment \overline{OC}, as in Figure 1.1.3(b). Let α = ∠BAC \text{ and }β = ∠ ABC. Since \overline{AB} is a diameter of the circle, \overline{OA} \text{ and }\overline{OC} have the same length (namely, the circle’s radius). This means that △OAC \text{ is an isosceles triangle, and so }∠OCA = ∠OAC = α. Likewise, △OBC is an isosceles triangle and ∠OCB = ∠OBC = β. So we see that ∠ ACB = α+β. And since the angles of △ ABC must add up to 180^◦, we see that 180^◦ = α+(α+β)+β = 2 (α+β), \text{ so }α+β = 90^◦. Thus, ∠ ACB = 90^◦. QED

1.1.4.png
Figure 1.1.4

By knowing the lengths of two sides of a right triangle, the length of the third side can be determined by using the Pythagorean Theorem:

Theorem 1.1. Pythagorean Theorem

The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of its legs.

1.1.5.png
Figure 1.1.5 Similar triangles △ ABC, △CBD, △ ACD

Recall that triangles are similar if their corresponding angles are equal, and that similarity implies that corresponding sides are proportional. Thus, since \triangle\,ABC is similar to \triangle\,CBD , by proportionality of corresponding sides we see that

\nonumber \overline{AB}~\text{is to}~\overline{CB}~\text{(hypotenuses)}\text{ as } \overline{BC}~\text{is to}~\overline{BD}~\text{(vertical legs)} \quad\Rightarrow\quad \frac{c}{a} ~=~ \frac{a}{d} \quad\Rightarrow\quad cd ~=~ a^2 ~. \nonumber

Since \triangle\,ABC is similar to \triangle\,ACD , comparing horizontal legs and hypotenuses gives

\nonumber \frac{b}{c-d} ~=~ \frac{c}{b} \quad\Rightarrow\quad b^2 ~=~ c^2 ~-~ cd ~=~ c ^2 ~-~ a^2 \quad\Rightarrow\quad a^2 ~+~ b^2 ~=~ c^2 ~. \textbf{QED} \nonumber

Note: The symbols \perp and \sim denote perpendicularity and similarity, respectively. For example, in the above proof we had \,\overline{CD} \perp \overline{AB}\, and \,\triangle\,ABC \sim \triangle\,CBD \sim \triangle\,ACD .

Example 1.3

For each right triangle below, determine the length of the unknown side:

1.3 example.png

Solution:

For triangle \triangle\,ABC , the Pythagorean Theorem says that

\nonumber a^2 ~+~ 4^2 ~=~ 5^2 \quad\Rightarrow\quad a^2 ~=~ 25 ~-~ 16 ~=~ 9 \quad\Rightarrow\quad \fbox{\(a ~=~ 3\)} ~. \nonumber

For triangle \triangle\,DEF , the Pythagorean Theorem says that

\nonumber e^2 ~+~ 1^2 ~=~ 2^2 \quad\Rightarrow\quad e^2 ~=~ 4 ~-~ 1 ~=~ 3 \quad\Rightarrow\quad \fbox{$e ~=~ \sqrt{3}$} ~. \nonumber

For triangle \triangle\,XYZ , the Pythagorean Theorem says that

\nonumber 1^2 ~+~ 1^2 ~=~ z^2 \quad\Rightarrow\quad z^2 ~=~ 2 \quad\Rightarrow\quad \fbox{$z ~=~ \sqrt{2}$} ~. \nonumber

Example 1.4

A 17 ft ladder leaning against a wall has its foot 8 ft from the base of the wall. At what height is the top of the ladder touching the wall?

1.4. example.png

Solution

Let h be the height at which the ladder touches the wall. We can assume that the ground makes a right angle with the wall, as in the picture on the right. Then we see that the ladder, ground, and wall form a right triangle with a hypotenuse of length 17 ft (the length of the ladder) and legs with lengths 8 ft and h ft. So by the Pythagorean Theorem, we have

\nonumber h^2 ~+~ 8^2 ~=~ 17^2 \quad\Rightarrow\quad h^2 ~=~ 289 ~-~ 64 ~=~ 225 \quad\Rightarrow\quad \fbox{$h ~=~ 15 ~\text{ft}$} ~. \nonumber


This page titled 1.1: Angles is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Michael Corral.

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