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Isoceles Triangle Theorem

(A-1) Each angle isoceles triangle theorem _gr_1.gif] is associated with a unique real number between 0 and 180, called its measure and denoted isoceles triangle theorem _gr_2.gif] No angle can have measure 0 nor 180.

Definition A point isoceles triangle theorem _gr_3.gif] is an interior point of isoceles triangle theorem _gr_4.gif] if an only if there exists a segment isoceles triangle theorem _gr_5.gif] containing isoceles triangle theorem _gr_6.gif] with isoceles triangle theorem _gr_7.gif] and isoceles triangle theorem _gr_8.gif] that extends from one side of the angle to the other ( isoceles triangle theorem _gr_9.gif] and isoceles triangle theorem _gr_10.gif] isoceles triangle theorem _gr_11.gif] isoceles triangle theorem _gr_12.gif]).

(A-2) If isoceles triangle theorem _gr_13.gif] lies in the interior of isoceles triangle theorem _gr_14.gif] then isoceles triangle theorem _gr_15.gif] isoceles triangle theorem _gr_16.gif] Conversely, if isoceles triangle theorem _gr_17.gif] isoceles triangle theorem _gr_18.gif] then isoceles triangle theorem _gr_19.gif] is an interior point of isoceles triangle theorem _gr_20.gif]

Definition For any three rays isoceles triangle theorem _gr_21.gif] isoceles triangle theorem _gr_22.gif] and isoceles triangle theorem _gr_23.gif] (having the same endpoint) we say that isoceles triangle theorem _gr_24.gif] lies between rays isoceles triangle theorem _gr_25.gif] and isoceles triangle theorem _gr_26.gif] and we write isoceles triangle theorem _gr_27.gif] if and only if the rays are distinct and isoceles triangle theorem _gr_28.gif]

(A-3) The set of rays isoceles triangle theorem _gr_29.gif] lying on one side of a given line isoceles triangle theorem _gr_30.gif] including ray isoceles triangle theorem _gr_31.gif] may be assigned to the entire set of real numbers isoceles triangle theorem _gr_32.gif] isoceles triangle theorem _gr_33.gif] called coordinates, in such a manner that

    (i) each ray is assigned to a unique coordinate
    
    (ii) no two rays are assigned to the same coordinate
    
    (iii) the coordinate of isoceles triangle theorem _gr_34.gif] is 0
    
     (iv) if rays isoceles triangle theorem _gr_35.gif] and isoceles triangle theorem _gr_36.gif] on isoceles triangle theorem _gr_37.gif] have coordinates isoceles triangle theorem _gr_38.gif] and isoceles triangle theorem _gr_39.gif] then isoceles triangle theorem _gr_40.gif]

Theorem (12) If the rays isoceles triangle theorem _gr_41.gif] isoceles triangle theorem _gr_42.gif] and isoceles triangle theorem _gr_43.gif] have coordinates isoceles triangle theorem _gr_44.gif] isoceles triangle theorem _gr_45.gif] and isoceles triangle theorem _gr_46.gif] relative to some half-plane, then isoceles triangle theorem _gr_47.gif] if and only if either isoceles triangle theorem _gr_48.gif] or isoceles triangle theorem _gr_49.gif]

Definition We say ray isoceles triangle theorem _gr_50.gif] is an angle bisector of angle isoceles triangle theorem _gr_51.gif] when isoceles triangle theorem _gr_52.gif] lies between isoceles triangle theorem _gr_53.gif] and isoceles triangle theorem _gr_54.gif] such that isoceles triangle theorem _gr_55.gif]

Theorem (13) If isoceles triangle theorem _gr_56.gif] there is a unique ray isoceles triangle theorem _gr_57.gif] such that isoceles triangle theorem _gr_58.gif] and isoceles triangle theorem _gr_59.gif]

Theorem (14) The bisector of any angle exists and is unique.

Definition Given isoceles triangle theorem _gr_60.gif] then the two rays isoceles triangle theorem _gr_61.gif] and isoceles triangle theorem _gr_62.gif] are called opposing rays.  

Definition Two angles are said to form a linear pair if and only if they have one side in common and the other two sides are opposite rays.

Definition Any two angles whose angle measure sum to 180 is called a supplementary pair and any two angles whose angle measures sum to 90 is called a complementary pair.

Theorem (15) Angles supplementary (or complementary) to the same angles have the same measure.

    Proof. We will use a direct proof for the theorem with supplementary angles.
    
isoceles triangle theorem _gr_63.gif]

We will use a direct proof for the theorem with complementary angles.
    
isoceles triangle theorem _gr_64.gif]
isoceles triangle theorem _gr_65.gif]

(A-4) A linear pair of angles is supplementary pair.

Definition A right angle is any angle having measure 90. An acute angle is any angle whose measure is less than 90 and an obtuse angle is any angle who measure is greater than 90.

Definition Two distinct lines isoceles triangle theorem _gr_66.gif] and isoceles triangle theorem _gr_67.gif] are called perpendicular lines if and only if they contain the sides of a right angle.

    For convenience, segments are perpendicular if and only if they lie, respectively, on perpendicular lines. Similar terminology applies to segment and ray, two rays, and so.

Theorem (16) If isoceles triangle theorem _gr_68.gif] then isoceles triangle theorem _gr_69.gif] and isoceles triangle theorem _gr_70.gif] are perpendicular at isoceles triangle theorem _gr_71.gif]

isoceles triangle theorem _gr_72.gif]

Two lines isoceles triangle theorem _gr_73.gif] and isoceles triangle theorem _gr_74.gif] are perpendicular at isoceles triangle theorem _gr_75.gif] then isoceles triangle theorem _gr_76.gif]

isoceles triangle theorem _gr_77.gif]
isoceles triangle theorem _gr_78.gif]

Definition A set isoceles triangle theorem _gr_79.gif] in isoceles triangle theorem _gr_80.gif] is called convex provided it has the property that for all points isoceles triangle theorem _gr_81.gif] and isoceles triangle theorem _gr_82.gif] the segment joining isoceles triangle theorem _gr_83.gif] and isoceles triangle theorem _gr_84.gif] lies in isoceles triangle theorem _gr_85.gif] that is, isoceles triangle theorem _gr_86.gif]

(H-1) Let isoceles triangle theorem _gr_87.gif] be any line lying in any plane isoceles triangle theorem _gr_88.gif] The set of all points in isoceles triangle theorem _gr_89.gif] not on isoceles triangle theorem _gr_90.gif] consists of the union of two subsets isoceles triangle theorem _gr_91.gif] and isoceles triangle theorem _gr_92.gif] of isoceles triangle theorem _gr_93.gif] such that

     (i) isoceles triangle theorem _gr_94.gif] and isoceles triangle theorem _gr_95.gif] are convex sets
     
     (ii) isoceles triangle theorem _gr_96.gif] and isoceles triangle theorem _gr_97.gif] have no points in common
     
     (iii) If isoceles triangle theorem _gr_98.gif] lies in isoceles triangle theorem _gr_99.gif] and isoceles triangle theorem _gr_100.gif] lies in isoceles triangle theorem _gr_101.gif] the line isoceles triangle theorem _gr_102.gif] intersects the segment isoceles triangle theorem _gr_103.gif]
     

Definition The two sets isoceles triangle theorem _gr_104.gif] and isoceles triangle theorem _gr_105.gif] in the axiom (H-1) are called the two sides of isoceles triangle theorem _gr_106.gif] or also, half-planes determined by isoceles triangle theorem _gr_107.gif]

Theorem (17) If isoceles triangle theorem _gr_108.gif] then there exists a unique perpendicular to line isoceles triangle theorem _gr_109.gif] at isoceles triangle theorem _gr_110.gif]

    Proof. First we will prove the following statement using the direct method: if isoceles triangle theorem _gr_111.gif] is any line then there is a perpendicular to line isoceles triangle theorem _gr_112.gif] at isoceles triangle theorem _gr_113.gif]

isoceles triangle theorem _gr_114.gif]

Next we will show, using an indirect method, that the perpendicular is unqiue.

isoceles triangle theorem _gr_115.gif]

Thertefore, any perpendicular is unique. isoceles triangle theorem _gr_116.gif]

Definition Two angles having the sides of one opposite the sides of the other are called vertical angles.

Theorem (18) Vertical angles have equal measures.

    Proof. We will prove the statement: for any vertical angles isoceles triangle theorem _gr_117.gif] and isoceles triangle theorem _gr_118.gif] isoceles triangle theorem _gr_119.gif]

isoceles triangle theorem _gr_120.gif]
    
isoceles triangle theorem _gr_121.gif]     

Theorem (19) Bisectors of a linear pair of angles are perpendicular.

Theorem (20) If isoceles triangle theorem _gr_122.gif] and isoceles triangle theorem _gr_123.gif] are any three rays on one side of a line and having the same end point, then either isoceles triangle theorem _gr_124.gif] isoceles triangle theorem _gr_125.gif] or isoceles triangle theorem _gr_126.gif]

Theorem (21) If two angles have a side in common that passes through an interior point of the angle formed by the other two sides, then the other two sides are perpendicular if and only if the given angles are complementary.

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Isoceles Triangle Theorem
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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