Angles

Angles and their measurements

An angle is the “opening” between two rays of the same origin. To measure it, we transport the question to an arc: the angle measures the same as the arc it determines in a circle centered at its vertex. The story The The Elements of Euclid (c. 300 BC) defines the plane angle as “the inclination, relative to each other, of two lines in a plane that meet and are not in a straight line”. The definition seems simple, but the Greeks discussed for centuries what is, after all, an angle: a quantity, which can be added and compared? a quality, like a form? or a relation between two lines? The person who tells this dispute is Proclus, in the 5th century, in his commentary on Euclid. The solution we adopted — measuring the angle by the arc it determines in a circle centered on the vertex — leaves the philosophical question aside and transforms the angle into a number.

October 22, 2026 Read

Angles: turns and corners

When you open a door, turn a doorknob, or turn right around a corner, you are making a turn. The size of this rotation is what we call angle. How much did you spin? One full turn: you turn until you look at the same place again. Half turn: you turn your back to where you were looking. Quarter turn: you turn to the side, like a corner. This rotation forms a right angle. Mathematicians divide the entire revolution into 360 equal parts, called degrees (°). So half turn has 180°, and a quarter turn has 90°.

October 8, 2026 Read

Exercises from the chapter 3 — Angles

Exercises and resolutions The calculator below does, step by step, the calculations with degrees, minutes and seconds for exercises 29 to 33. Try solving it first and use it to check. Enter the angles and choose the operation. It shows each step: 60″ becomes 1′ and 60′ becomes 1°. Ative o JavaScript para ver esta figura interativa. 29 Simplify the following measurements: a) $30^{\circ}70^{\prime}$ · b) $45^{\circ}150^{\prime}$ · c) $65^{\circ}39^{\prime}123^{\prime\prime}$ · d) $110^{\circ}58^{\prime}300^{\prime\prime}$ · e) $30^{\circ}56^{\prime}240^{\prime\prime}$ Ver respostaa) $31^{\circ}10^{\prime}$ · b) $47^{\circ}30^{\prime}$ · c) $65^{\circ}41^{\prime}3^{\prime\prime}$ · d) $111^{\circ}3^{\prime}$ · e) $31^{\circ}$

October 2, 2026 Read

Right, acute and obtuse angles — measurement

Why does a complete turn have 360 degrees, and not 100? The answer lies in Mesopotamian astronomy. Who discovered The names straight, acute and obtuse come from Euclid, who defines them in The Elements (definitions I.10 to I.12): straight is the angle equal to its supplementary adjacent; acute, smaller than the straight; obtuse, the biggest. The measurement in degrees has another origin, Babylonian astronomy, as explained in the curiosity “Why 360 degrees?”, below. The words minute and second come from the Latin translations of Claudius Ptolemy (2nd century AD).

October 2, 2026 Read

Congruence and comparison

Copying an angle and dividing it in half are classic constructions. Dividing it into three equal parts, however, has challenged mathematicians for more than two thousand years. Who discovered The two constructions are in The Elements of Euclid (c. 300 BC): the proposition I.9 divides an angle in half (a angle bisector) and the I.23 copies a given angle onto another straight line. The trisection of the angle with a ruler and compass — dividing any angle into three equal parts — was proven impossible in 1837 by the Frenchman Pierre Wantzel, then just over 20 years old.

October 2, 2026 Read

Angle: definitions

Two rays with the same origin form an angle. And the first theorem about angles even has a named discoverer. Who discovered According to Proclus (5th century AD), who cited the History of Geometry by Eudemus of Rhodes, it was Thales of Miletus (c. 624–546 BC) who first realized that angles opposite the vertex are equal; Euclid only later demonstrated it “scientifically”, in proposition I.15. The definition of angle as “inclination of two lines” is in The Elements of Euclid (definition I.8).

October 2, 2026 Read

Introduction: convexity and half-planes

A straight line divides the plane into two sides. A figure is convex when it has no “indentations”. The two ideas go together. Who discovered The first to carefully define a convex figure was Archimedes of Syracuse (c. 287–212 BC), in the work On the sphere and the cylinder, where he uses curves “concave to the same side” to compare perimeters. The separation of the plane by a straight line was formalized by Moritz Pasch in 1882: his axiom says that a straight line that enters a triangle on one side must exit on the other. At the end of the 19th century, Hermann Minkowski made convexity an area of mathematics.

October 2, 2026 Read

Angles and Circumferences

Learning about angles in circles is fundamental. This concept was extensively addressed by Euclid and the Greeks in general, such as Pythagoras and Archimedes. Bow Definition: Let $A$ and $B$ be points on a circle in a plane. The straight line $\vec{AB}$ (arc) separates the plane into two half-planes, each containing a part of the circle. Central Angle Definition: Let $A$ and $B$ be points of a circle on a center plane $C$. The central angle $\angle ACB$ determined by $A$ and $B$ or determined by the chord $AB$.

September 30, 2022 Read

Angles in basic planes

Angles are key pieces for the development of geometry. It is present in our daily lives such as stairs, building cranes, constructions such as the Leaning Tower of Pisa, among others. In Euclid’s book Elements, this concept was well worked on, even decades before Christ. Examples of angle applications Next, we present some applications of angles. Building Crane Tower of Pisa Ladder A widely used example is the clock, with the movement of the hour and minute hands, the angles are formed and the time is defined.

August 28, 2022 Read

Property of Angles in Planes

In the previous post we talked about several important definitions for angles in flat figures. Now, we deepen the concepts through the properties of angles for applications in flat figures. angle bisector of an angle Definition: the angle bisector of an angle is a ray, which starts from the vertex of the angle and divides it into two congruent angles ($\cong$). angle bisector produces two congruent angles. Ampliar figura para ler os detalhes From the figure, we notice that

August 28, 2022 Read