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:

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.

Index

Euclid's Elements Book I, 23 Definitions. One-page 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.

Isogonic-Jacobi Theorem: Using TracenPoche Dynamic Geometry Software

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

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

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

Geometry Problem 1584: Proving a Ratio Involving Incircle and Angle Bisector
Unveiling a Potentially Unexplored Geometric Relationshp.

Geometry Problem 1583: Prove Lines CK and BL are Parallel.

Geometry Problem 1582: Prove That Angles AFD and AEF Are Equal in This Secant Problem.

Geometry Problem 1581: Prove the Angle Bisection in a Cyclic Quadrilateral.

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 7-24-25 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 Right-angled 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!.
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Geometry Problem 1533: Discovering Relationships between Angles and Lines in an Exterior Right Triangle of a Square - A High School Challenge.
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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.
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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.
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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.
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Geometry Problem 1528: Cracking the Circle Code: Unveiling the Tangent and Angle of an Inscribed Circle within a 90-Degree Circular Sector.
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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, 15-30 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 Nine-Point 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, 80-20-80 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, 30-60 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, 30-50-100 Degrees, Area, Inradius, Metric Relations, Measurement.

Geometry Problem 1287.
Triangle, 40-100-40 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, 80-20-80 Degrees, Area, Inradius, Circumradius, Angle Bisector, Metric Relations, Measurement.