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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 0° 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. 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 0A,B90 then A and B are complementary if A+B=90.
  2. Two angles between 0 and 180 are supplementary if their sum equals 180. In other words, if 0A,B180 then A and B are supplementary if A+B=180.
  3. Two angles between 0 and 360 are conjugate (or explementary) if their sum equals 360. In other words, if 0A,B360 then A and B 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 BAC as angle A.

Solution:

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

35+B+20=180B=1803520B=125.

For the right triangle DEF,E=53 and F=90, and we know that the two acute angles D and E are complementary, so

D+E=90D=9053D=37.

For triangle XYZ, 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, and Z:

α+3α+α=1805α=180α=36X=36,Y=3×36=108,Z=36

Example 1.2: Thales' Theorem

Thales' Theorem states that if A,B, and C are (distinct) points on a circle such that the line segment ¯AB is a diameter of the circle, then the angle ACB is a right angle (see Figure 1.1.3(a)). In other words, the 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 ¯OC, as in Figure 1.1.3(b). Let α=BAC and β=ABC. Since ¯AB is a diameter of the circle, ¯OA and ¯OC have the same length (namely, the circle’s radius). This means that OAC 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), 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 ABC is similar to CBD, by proportionality of corresponding sides we see that

¯AB is to ¯CB (hypotenuses) as ¯BC is to ¯BD (vertical legs)ca = adcd = a2 .

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

bcd = cbb2 = c2  cd = c2  a2a2 + b2 = c2 .QED

Note: The symbols and denote perpendicularity and similarity, respectively. For example, in the above proof we had ¯CD¯AB and ABCCBDACD.

Example 1.3

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

1.3 example.png

Solution:

For triangle ABC, the Pythagorean Theorem says that

a2 + 42 = 52a2 = 25  16 = 9a = 3 .

For triangle DEF, the Pythagorean Theorem says that

e2 + 12 = 22e2 = 4  1 = 3e = 3 .

For triangle XYZ, the Pythagorean Theorem says that

12 + 12 = z2z2 = 2z = 2 .

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

h2 + 82 = 172h2 = 289  64 = 225h = 15 ft .


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 via source content that was edited to the style and standards of the LibreTexts platform.

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