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

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

Right angle: An angle that measures
exactly 90 degrees.

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

Straight angle: An angle that measures exactly 180
degrees.

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

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

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

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

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 noncommon sides.

Congruent angles: Two angles that have the same measure.
Understanding
these concepts is essential in solving geometry problems involving angles.


Index


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

Points, Theorems and Problems  Index.
Vertex. 

Acute Angle in the Real World  Slideshow. 

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

Obtuse Angle. Theorems and Problems. 

Angles: 30, 60, 120 degrees. 

Angles: 45, 135 degrees. 

Angle Bisector. 

Isogonal Lines.
Theorems and Problems. 

Billiards and Geometry 
Index
Angles of Incidence and Reflection. 

Book I, Proposition 9: Bisect a given angle


Incenter. Index.
Angle bisectors.


Excenter. Index.
Interior and Exterior Angle bisectors.


Inca
Stone of Twelve Angles, Cuzco.
Tunnel effect. 

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

Animated
illustration of the celebrated Morley's angle trisector theorem 

Morley's Triangle & Center: Trisection of angles. 

Blanchet Theorem.
Altitude, Angle Bisector.


IsogonicJacobi
Theorem: Using TracenPoche Dynamic
Geometry Software 

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

Interactive Simson
Line. Dynamic Geometry.
StepbyStep construction, Manipulation, and animation.
Vertical angles. 

Triangle: Incircle, Perpendicular, Angle Bisector.
Dynamic Geometry.
StepbyStep construction, Manipulation, and animation. 

Geometry Problem 1577:
Prove that in triangle ABC, segment AB equals the sum of segments BD and CD, with given angles and congruent segments. 

Geometry Problem 1576:
Congruency of Segments in Triangle ABC with Angles 30 and 20 Degrees and an Interior Cevian 

Geometry Problem 1575:
Prove an Angle Bisector in a Triangle Involving an Altitude, Midpoint, and Excircle 

Geometry Problem 1573:
Two circles are externally tangent at point C. Given specific secants and tangents, find the angle at point K 

Geometry Problem 1572:
Calculate Angle C with Geometric Methods Using Sides AB=17, BC=25, and Angle A=45 Degrees 

Geometry Problem 1571:
Prove that the intermediate angle in a Pythagorean Triple 72425 measures 74 degrees. 

Geometry Problem 1570:
Calculating BE in Overlapping Squares: A High School and College Geometry Challenge. 

Geometry Problem 1569:
Prove a Relationship: EG Equals the Incircle Diameter in Triangle ABC with Square CDEF. 

Geometry Problem 1568:
Concyclicity of Points B, D , H, J in a Triangle ABC with an Incircle and a Tangent Circle. 

Geometry Problem 1566:
Demonstrate AC Equals AE + CD in an Equilateral Triangle ABC Involving Cevians and 60 Degrees.. 

Geometry Problem 1565:
Find the Length of BF in Triangle ABC Involving Median, Perpendicular, Midpoint, and Congruence. 

Geometry Problem 1564:
Find the Area of Quadrilateral BGDJ in a Right Triangle involving the Altitude, Angle bisectors, and Midpoints. 

Geometry Problem 1563:
Perpendicularity in a Right Triangle involving the Altitude, Angle bisectors, and Midpoints. 

Geometry Problem 1562:
Proof of Collinearity in a Rightangled Triangle involving the Altitude, Angle bisectors, and Midpoint. 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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: 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: Unlock the Mystery of Triangles: Solving for the Missing Angle with 1005030 Degree Angles and Cevian Lengths  A High School Challenge.
. 

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

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

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

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

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

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

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

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

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

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

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

Geometry Problem 1429. Triangle, Median, 1530 degree, Angle, Congruence. 

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

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

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

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

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

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

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

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

Problem 1341.
Geometry Problem 1341: Isosceles Triangle, 802080 Degrees, Circumcenter, Angle Bisector.


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


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


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

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

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

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

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

Geometry Problem 1323.
Square, Angle Trisector, Diagonal, Regular Dodecagon, Area, 3060 degrees. 

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

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

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

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

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

Geometry Problem 1288.
Triangle, 3050100 Degrees, Area, Inradius, Metric Relations, Measurement. 

Geometry Problem 1287.
Triangle, 4010040 Degrees, Congruence, Area, Metric Relations, Measurement. 

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

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

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

Geometry Problem 1274.
Isosceles Triangle, 802080 Degrees, Area, Inradius, Circumradius, Angle Bisector, Metric Relations, Measurement. 