Geometry Isosceles and Equilateral Triangles

Isosceles and Equilateral Triangles

∠DEF = 6x + 4, m ∠DFE = 3x + 2, ―DE ≅ ―DF , find x.

• ∆ DEF is an isosceles triangle.
• ∠DEF = m ∠DFE
• 6x + 4 = 3x + 2

x = − [2/3].

Determine whether the following statement is true or false.
If two sides of a triangle are congruent, then there must be a pair of congruent angles in this triangle.

  

True.

Determine whether the following statement is true or false.
All the angles in an isosceles triangle are 60 degree.

  

False.

ABC ≅ ACB, ―AC = 16 − x, ―AB = 3 + 4x, find x.

• ∆ ABC is an isosceles triangle.
• ―AC = ―AB
• 16 − x = 3 + 4x

x = [13/5].

Fill in the blank in the statement with always, sometimes, or never.
Isosceles angles are _____ equilateral angles.

  

Sometimes.

∆ ABC is an Equilateral triangle, AD ⊥ BC, find m ∠BAD.

• ∠ABC = 60
• ∠ADB = 90
• ∠ABC + m ∠ABD + m ∠BAD = 180
• ∠BAD = 180 − m ∠ABC − m ∠ABD

∠BAD = 180 − 60 − 90 = 30

Determine whether the following statement is true or false.
If all the three sides of one triangle are congruent, then the triangle is an isosceles triangle.

  

True.

Given: AD||BC, E is the midpoint of AD, ―BE ≅ ―CE
Prove: ∆ ABE ≅ ∆ DCE

• Statements ; Reasons
• AD||BC ; Given
• ∠AEB ≅ ∠CBE, ∠DEC ≅ ∠BCE ; alternate interior angles
• ―BE ≅ ―CE ; Given
• ∠CBE ≅ ∠BCE ; Isosceles ∆ theorem thearom
• ∠AEB ≅ ∠DEC ; Transitive prop of ≅∠s
• E is the midpoint of AD ; Given
• ―AE ≅ ―DE ; Definition of midpoint
• ―BE ≅ ―CE ; Given
• ∆ ABE ≅ ∆ DCE ; SAS post

Statements; Reasons
AD||BC ; Given
∠AEB ≅ ∠CBE, ∠DEC ≅ ∠BCE; alternate interior angles
―BE ≅ ―CE ; Given
∠CBE ≅ ∠BCE ; Isosceles ∆ theorem
∠AEB ≅ ∠DEC ; Transitive prop of ≅∠s
E is the midpoint of AD ; Given
―AE ≅ ―DE ; Definition of midpoint
―BE ≅ ―CE ; Given
∆ ABE ≅ ∆ DCE ; SAS post

Given: ―AC ≅ ―AB , ―BC ≅ ―BD ,

Prove: BAC ≅ CBD

• Statements ; Reasons
• ―AC ≅ ―AB ; Given
• ∠ABC ≅ ∠C ; Isosceles ∆ theorem thearom
• ∠ABC = m∠C ; definition of ≅ angles
• ∠BAC + m ∠ABC + m∠C = 180 ; triangle angles sum theorem
• ∠BAC + m∠C + m∠C = 180 ; transitive prop ( = )
• ∠BAC = 180 - 2 m∠C ; substraction prop ( = )
• ―BC ≅ ―BD ; Given
• BDC ≅ C ; Isosceles ∆ theorem
• ∠BDC = m∠C ; definition of ≅ angles
• ∠CBD + m ∠BDC + m∠C = 180 ; triangle angles sum theorem
• ∠CBD + m∠C + m∠C = 180 ; transitive prop ( = )
• ∠CBD = 180 - 2 m∠C ; substraction prop ( = )
• ∠BAC = m∠CBD ; transitive prop ( = )
• BAC ≅ CBD ; definition of congruent angles.

Statements ; Reasons
―AC ≅ ―AB ; Given
∠ABC ≅ ∠C ; Isosceles ∆ theorem
∠ABC = m∠C ; definition of ≅ angles
∠BAC + m ∠ABC + m∠C = 180 ; triangle angles sum theorem
∠BAC + m∠C + m∠C = 180 ; transitive prop ( = )
∠BAC = 180 - 2 m∠C ; substraction prop ( = )
―BC ≅ ―BD ; Given
BDC ≅ C ; Isosceles ∆ theorem
∠BDC = m∠C ; definition of ≅ angles
∠CBD + m ∠BDC + m∠C = 180 ; triangle angles sum theorem
∠CBD + m∠C + m∠C = 180 ; transitive prop ( = )
∠CBD = 180 - 2 m∠C ; substraction prop ( = )
∠BAC = m∠CBD ; transitive prop ( = )
BAC ≅ CBD ; definition of congruent angles.

―AB ≅ ―AC , ―AD ⊥―BC , ∠BAC = 70^o, find m ∠BAD.

• ∠B ≅ ∠C
• ―AD ⊥―BC
• ∠ADB = m ∠ADC = 90
• ―AD ≅ ―AD
• ∆ ABD ≅ ∆ ACD
• ∠BAD ≅ ∠CAD
• ∠BAD = m∠CAD

∠BAD = [1/2]m ∠BAC = 35^o

Isosceles and Equilateral Triangles

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

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• Intro 0:00
• Isosceles Triangle Theorem 0:07
• Isosceles Triangle Theorem
• Isosceles Triangle Theorem 2:26
• Example: Using the Isosceles Triangle Theorem
• Isosceles Triangle Theorem Converse 3:29
• Isosceles Triangle Theorem Converse
• Equilateral Triangle Theorem Corollaries 4:30
• Equilateral Triangle Theorem Corollary 1
• Equilateral Triangle Theorem Corollary 2
• Extra Example 1: Find the Value of x 7:08
• Extra Example 2: Find the Value of x 10:04
• Extra Example 3: Proof 14:04
• Extra Example 4: Proof 22:41