[分享]Some basic geometric theorems

Isosceles triangle

Definition: The triangle has two equal sides

Base angles of isosceles triangle

In an isosceles triangle ABC, if AB=AC, then ∠ABC=∠ACB (Base angles of isosceles triangle)

Proof

D is a mid-point of BC, join AD.
AB=AC (Given)
BD=CD (Given)
DA=DA (Common sides)
∴△ABD≡△ACD (SSS)
∴∠ABC=∠ACB (Corresponding angles, congruent triangles)

Side opposite equal angles

If ∠ABC=∠ACB, then AB=AC (Side opposite equal angles)

Proof

∠ABD=∠ACD (Given)
∠BDA=∠CDA=90° (Given)
DA=DA (Common sides)
∴△ABD≡△ACD (AAS)
∴AB=AC (Corresponding sides, congruent triangles)

Equilateral triangle

Definition: All sides of the triangle are equal

Property of equilateral triangle

If ABC is an equilateral triangle, then AB=BC=CA and ∠ABC=∠BCA=∠CAB=60° (Property of equilateral triangle)

Proof
△ABC is an equilateral triangle (Given)
∴The triangle has three equal sides (By definition)
∴AB=BC=CA

∵AB=BC=CA (Prove)
∴∠ABC=∠ACB (Base angles of isosceles triangle)
∴∠BAC=∠BCA (Base angles of isosceles triangle)
∴∠ABC=∠BCA=∠CAB
∵∠ABC+∠BCA+∠CAB=180° (Angle sum of triangle)
∴3∠ABC=180°
∴∠ABC=∠BCA=∠CAB=60°

If AB=BC=CA or ∠ABC=∠BCA=∠CAB=60°, then △ABC is an equilateral triangle

Parallelogram

Definition: A quadrilateral which two pairs of opposite sides are parallel

Opposite sides of parallelogram

If ABCD is a parallelogram, then AB=DC and AD=BC (Opposite sides of parallelogram)

Proof

Join AC.
ABCD is a parallelogram (Given)
∴∠BAC=∠DCA (Alternate angles, AB//DC)
∴AC=CA (Common sides)
∴△BAC≡△DCA (ASA)
∴AB=DC and AD=BC (Corresponding sides, congruent triangles)

Opposite angles of parallelogram

If ABCD is a parallelogram, then ∠BAD=∠BCD and ∠ABC=∠CDA (Opposite angles of parallelogram)

Proof
ABCD is a parallelogram (Given)
∴∠ABC+∠BCD=180° (Interior angles, AB//DC)

Diagonals of parallelogram

If ABCD is a parallelogram, then AB=DC and AD=BC (Diagonals of parallelogram)

Proof
ABCD is a parallelogram (Given)
∴∠BOA=∠DOC (Vertical opposite angles)
∴∠OAB=∠OCD (Alternate angles, AB//DC)
∴AB=CD (Opposite sides of parallelogram)
∴△BOA≡△DOC (AAS)
∴AO=OC及BO=OD (Corresponding sides, congruent triangles)

Opposite sides equal

If AB=DC and AD=BC, then ABCD is a parallelogram (Opposite sides equal)

Proof

Join AC.
AB=CD (Given)
BC=DA (Given)
CA=AC (Common sides)
∴△ABC≡△CDA (SSS)
∴∠CAB=∠ACD (Corresponding angles, congruent triangles)
∴∠BCA=∠DAC (Corresponding angles, congruent triangles)
∴AB//DC and AD//BC (Alternate angles equal)
∴ABCD is a parallelogram

Opposite angles equal

If ∠DAB=∠BCD and ∠ABC=∠CDA, then ABCD is a parallelogram (Opposite angles equal)

Proof
∠DAB=∠BCD (Given)
∠ABC=∠CDA (Given)
∠DAB+∠BCD +∠ABC+∠CDA=(4-2)180° (Angles sum of polygon)
∴2(∠DAB+∠ABC)=360°
∴∠DAB+∠ABC=180°
∴2(∠ABC+∠BCD)=360°
∴∠ABC+∠BCD=180°
∴AB//DC (Interior angles supplementary)
∴ABCD is a parallelogram

Diagonals bisect each other

If AO=OC and BO=OD, then ABCD is a parallelogram (Diagonals bisect each other)

Proof
AO=CO (Given)
∠AOB=∠COD (Vertical opposite angles)
∠AOD=∠COB (Vertical opposite angles)
BO=DO (Given)
∴△AOB≡△COD (SAS)
∴△AOD≡△COB (SAS)
∴AB=CD (Corresponding sides, congruent triangles)
∴BC=DA (Corresponding sides, congruent triangles)
∴ABCD is a parallelogram (Opposite sides equal)

2 sides equal and parallel

If AB=DC and AB//DC, then ABCD is a parallelogram (2 sides equal and parallel)

Proof

Join AC.
BA=DC (Given)
BA//DC (Given)
∴∠BAC=∠DCA (Alternate angles, BA//DC)
AC=CA (Common sides)
∴△BAC≡△DCA (SAS)
∴BC=DA (Corresponding sides, congruent triangles)
∴ABCD is a parallelogram (Opposite sides equal)