College Geometry Online, eLearning

Master Geometry Angles with Theorems and Problems - Part 1: Essential Concepts for Solving Angle Problems

Angle IndexIn geometry, an angle is formed when two lines or rays meet at a common point. The measure of an angle is typically expressed in degrees, although other units such as radians may also be used.
Here are some important concepts related to angles in geometry:

  1. Acute angle: An angle that measures less than 90 degrees.

  2. Right angle: An angle that measures exactly 90 degrees.

  3. Obtuse angle: An angle that measures between 90 and 180 degrees.

  4. Straight angle: An angle that measures exactly 180 degrees.

  5. Reflex angle: An angle that measures between 180 and 360 degrees.

  6. Complementary angles: Two angles that add up to 90 degrees.

  7. Supplementary angles: Two angles that add up to 180 degrees.

  8. Vertical angles: A pair of opposite angles formed by the intersection of two lines. Vertical angles are equal in measure.

  9. Adjacent angles: Two angles that share a common vertex and a common side. The sum of adjacent angles is equal to the measure of the larger angle formed by their non-common sides.

  10. Congruent angles: Two angles that have the same measure.

Understanding these concepts is essential in solving geometry problems involving angles.



Euclid's Elements 23 definitions

Euclid's Elements Book I, 23 Definitions. One-page visual illustration. Angles and more.
Euclid's Elements Book. Index.

Point, Theorems and Problems, Index

Points, Theorems and Problems - Index.

Acute Angle, Slideshow

Acute Angle in the Real World - Slideshow.

Wordflex Point

Meanings and syntactic of 'ANGLE'.
Created using Wordflex Touch Dictionary for iPad in association with Oxford University Press.

Obtuse Angle Index

Obtuse Angle. Theorems and Problems.

Angles 30, 60, 120 Index

Angles: 30, 60, 120 degrees.

Angle 45, 90, 135, Theorems and Problems

Angles: 45, 135 degrees.

Angle Bisector Index

Angle Bisector.

Isogonal Lines

Isogonal Lines.
Theorems and Problems.

Billiards and Geometry Index

Billiards and Geometry - Index
Angles of Incidence and Reflection.

Bisect an angle

Book I, Proposition 9: Bisect a given angle

Incenter, Incircle, Index

Incenter. Index.
Angle bisectors.

Excenter, Excircle, Index

Excenter. Index.
Interior and Exterior Angle bisectors.

Stone of twelve angles, Cuzco

Inca Stone of Twelve Angles, Cuzco. Tunnel effect.

Angle Bisector Theorem

Euclid's Elements Book VI, Proposition 3: Angle Bisector Theorem

Morley's Theorem

Animated illustration of the celebrated Morley's angle trisector theorem 

Morley's Triangle & Center: Trisection of angles.

Blanchet Theorem. Altitude, Angle Bisector.

Isogonic-Jacobi Theorem: Using TracenPoche Dynamic Geometry Software

Mascheroni Construction with compass alone. Midpoint of a segment. Equal angles.

Interactive Simson Line

Interactive Simson Line. Dynamic Geometry.
Step-by-Step construction, Manipulation, and animation.
Vertical angles.

Interactive Geometry

Triangle: Incircle, Perpendicular, Angle Bisector. Dynamic Geometry.
Step-by-Step construction, Manipulation, and animation.

Geometry Problem 1561

Geometry Problem 1561: Triangle ABC, Circumcenter O, Orthocenter H, Parallel Line, and Angle Secrets.

Geometry Problem 1560

Geometry Problem 1560: Trapezoid ABCD: Unlocking Angle Secrets at G and H.

Geometry Problem 1559

Geometry Problem 1559: Proving BC Bisects Segment DE. This geometry problem challenges students at high school and college levels.

Geometry Problem 1558

Geometry Problem 1558: The midpoints of segments connecting corresponding vertices of equilateral triangles form an equilateral triangle.

Geometry Problem 1557

Geometry Problem 1557: Calculate the Angle DHG in Right Triangle ABC. The problem involves: equilateral, isosceles, midpoint and congruence

Geometry Problem 1556

Geometry Problem 1556: Right Triangle ABC and Inscribed Circle. The problem involves circle, chords, tangent, perpendicular lines, and congruence

Geometry Problem 1555

Geometry Problem 1555: Find Length of DE. The problem involves chords, tangent, circles, and intersections of line

Geometry Problem 1554

Geometry Problem 1554: Finding the Length of Side AB in Triangle ABC

Geometry Problem 1553

Geometry Problem 1553: Solving for OC in Triangle ABC with Unique Angle Bisectors

Geometry Problem 1552

Geometry Problem 1552: Exploring Angle C in Triangle ABC with Given Angle A and Side Lengths

Geometry Problem 1551

Geometry Problem 1551: Unraveling Angle Relations in Cyclic Quadrilaterals: Solving for Angle GEJ: A High School Level Investigation

Geometry Problem 1550

Geometry Problem 1550: Solving for Segment BD: An Angle Puzzle in Right Triangle ABC: A High School Level Investigation

Geometry Problem 1535

Geometry Problem 1535: Crack the Code: Inscribed Circle in Square - Angle Challenge! Solve the Mystery.

Geometry Problem 1534

Geometry Problem 1534: High School Brainteaser: Tangent Circles, Common External Tangent, and Angle Conundrums!.

Geometry Problem 1533

Geometry Problem 1533: Discovering Relationships between Angles and Lines in an Exterior Right Triangle of a Square - A High School Challenge.

Geometry Problem 1532

Geometry Problem 1532: Crack the Code of Geometry Problem 1532: How to Find the Angle in a Square with a Tangent Semicircle! - A High School Challenge.

Geometry Problem 1531

Geometry Problem 1531: Discover How to Calculate the Length of a Chord in a Circle with Diameter Intersection and an Angle between the Diameter and Chord - A High School Challenge.

Geometry Problem 1529

Geometry Problem 1529: Unlock the Mystery of Triangles: Solving for the Missing Angle with 100-50-30 Degree Angles and Cevian Lengths - A High School Challenge.

Geometry Problem 1528

Geometry Problem 1528: Cracking the Circle Code: Unveiling the Tangent and Angle of an Inscribed Circle within a 90-Degree Circular Sector.

Geometry Problem 1520

Geometry Problem 1520 and a Thematic Poem.
Discovering Distances in a Rectangle with an Exterior Point: A Geometry Challenge.

Geometry Problem 1518

Geometry Problem 1518 and a Thematic Poem.
Boost Your Geometry Skills: Solve for the Number of Sides in an Equiangular Polygon with an Interior Point and Bisected Angle.

Geometry Problem 1508

Geometry Problem 1508.
Understanding Triangle Geometry: Isosceles, Equilateral, Quadrilateral, Angular. Difficulty Level: High School.

Geometry Problem 1507

Geometry Problem 1507.
Understanding Triangle Geometry: Double Angles, Altitudes, and Measurements. Difficulty Level: High School.

Geometry Problem 1506

Geometry Problem 1506.
Parallel Rays, Angles, Corresponding, Alternate.

Geometry Problem 1505

Geometry Problem 1505.
Parallel Rays, Angles, and Bisector, Corresponding, Alternate, Conjugate.

Geometry Problem 1504

Geometry Problem 1504.
Squares, Equilateral Triangle, Parallel, Angle.

Geometry Problem 1491

Geometry Problem 1491.
Cyclic Quadrilateral, Diagonal, Incircle, Angle, Measurement.

Geometry Problem 1490

Geometry Problem 1490.
Triangle, Cevian, Incircle, Tangent, Measurement.

Problem 1442

Geometry Problem 1442.
Triangle, Altitudes, Orthocenter, Perpendicular, Congruent Angles, Similarity, Measurement, Art, Poster, Typography, iPad Apps.

Problem 1431

Geometry Problem 1431.
Triangle, Circumcircle, Parallel Chord, Apollonius, LLC, Congruent Angles, Poster, Conformal Mapping.

Problem 1429

Geometry Problem 1429.
Triangle, Median, 15-30 degree, Angle, Congruence.

Problem 1423

Geometry Problem 1423.
Triangle, Area, Three Equal Angles, Circumradius, Measurement.

Problem 1418

Geometry Problem 1418.
Triangle, Circumcircle, Incenter, Side, Sum of Segments, Tangent, Parallel, Angle.

Problem 1399

Geometry Problem 1399.
Triangle, Circumcircle, Perpendiculars, Angles, Congruence, Cyclic Quadrilateral.

Geometry problem 1364

Geometry Problem 1364.
Square, Inscribed Circle, Area, Angle Bisector, Perpendicular, 45 Degrees, Sketch, iPad Apps.

Geometry problem 1363

Geometry Problem 1363.
Square, Angle Bisector, Perpendicular, 45 Degrees, Angle, Sketch, iPad Apps.

Geometry problem 1361

Geometry Problem 1361.
Triangle, Two Nine-Point Circles, Feuerbach's Circle, Euler's Circle, Congruent Angles, Sketch, iPad Apps.

Geometry problem 1359

Geometry Problem 1359.
Triangle, Circumcircle, Congruent Angles, Midpoints.

Geometry problem 1347

Geometry Problem 1347.
Triangle, Angle Bisector, Congruence, Circumcircle, Tangent, Equilateral Triangle.

Geometry problem 1341

Problem 1341.
Geometry Problem 1341: Isosceles Triangle, 80-20-80 Degrees, Circumcenter, Angle Bisector.

Geometry problem 1338

Problem 1338.
Four Squares, Diagonals, Angle, 45 Degrees, Areas.

Geometry problem 1336

Problem 1336.
Right Triangle, Cevian, Double Angle, Perpendicular, 90 Degrees.

Geometry problem 1329

Geometry Problem 1329.
Two Squares Side by Side, Parallel, Perpendicular, 90 Degrees, Congruence.

Geometry problem 1328

Geometry Problem 1328.
Two Squares Side by Side, Perpendicular, 90 Degrees, Congruence.

Geometry problem 1327

Geometry Problem 1327.
Two Squares Side by Side, Perpendicular, 90 Degrees, Angle.

Geometry problem 1326

Geometry Problem 1326.
Triangle, Cevian, Incenters, Sum of Angles, 270 Degrees.

Geometry problem 1324

Geometry Problem 1324.
Quadrilateral, Diagonal, 45 Degrees, Angle Bisector, Isosceles Triangle, Congruence.

Geometry problem 1323

Geometry Problem 1323.
Square, Angle Trisector, Diagonal, Regular Dodecagon, Area, 30-60 degrees.

Geometry problem 1322

Geometry Problem 1322.
Triangle, Angle Bisector, Circumcircle, Chord, Secant, Sum of two Angles.

Geometry problem 1299

Geometry Problem 1299.
Triangles, Interior Point, Angles, 30 degrees.

Geometry problem 1298

Geometry Problem 1298.
Arbelos, Semicircles, Diameters, Circle, Incircle, Tangent, Angle Bisector, Perpendicular, Midpoint.

Geometry problem 1297

Geometry Problem 1297.
Arbelos, Semicircles, Diameters, Circle, Incircle, Tangent, Angle Bisector.

Geometry problem 1295

Geometry Problem 1295.
Right Triangle, Incenter, Incircle, Excenter, Excircle, Congruence, Angle.

Geometry problem 1288

Geometry Problem 1288.
Triangle, 30-50-100 Degrees, Area, Inradius, Metric Relations, Measurement.

Geometry problem 1287

Geometry Problem 1287.
Triangle, 40-100-40 Degrees, Congruence, Area, Metric Relations, Measurement.

Geometry problem 1286

Geometry Problem 1286.
Triangle, Quadrilateral, 60, 135, 105 Degree Angles, Congruence, Measurement.

Geometry problem 1285

Geometry Problem 1285.
Triangle, Quadrilateral, 60, 150, 90 Degree Angles, Congruence, Measurement.

Geometry problem 1281

Geometry Problem 1281.
Triangle, Circle, Diameter, Tangent, Altitude, Congruent Angles.

Geometry problem 1274

Geometry Problem 1274.
Isosceles Triangle, 80-20-80 Degrees, Area, Inradius, Circumradius, Angle Bisector, Metric Relations, Measurement.

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Last updated: Feb 18, 2024