Sunday, May 18, 2008

Elearn Geometry Problem 5

Geometry Problem, Triangle, Angles

See complete Problem 5 at:

Triangle, Angles, Congruence. Level: High School, SAT Prep, College geometry

Post your solutions or ideas in the comments.


  1. Triangle CDB is isosceles since BD=DC, therefore angle BCD=angle DBC=2x.

    Let point E be a point inside ABD triangle so that AED triangle is congruent to CDB triangle, where AE=ED=BD=DC and as it's given AD=BC. Note that both angles EAB and EAD are equal 2x.
    Angle ADB = 4x=>angle EDB=2x. Triangle EDB is isosceles since ED=BD.
    Angle BAE = angle BAD - angle EAD = 3x - 2x = x.

    Let's reflect point E over AB calling it F. Triangles AFB and AEB are congruent. AF=AE, FB=EB. Angle FAE = 2 * angle BAE = 2x.

    FAE and BDE are congruent isosceles triangles since AF=AE=DE=DB and angle FAE=angle EDB=2x
    =>FE=EB, but FB=EB => triangle FBE is equilateral => angle FBE=60=>AEB=30.

    Angle EBD = (180 - angle EDB) / 2 = 90 - x.
    Angle ABC = 180 - angle BAC - angle BCA = 180 - 5x.
    But angle ABC = angle ABE + angle EBD + angle DBC, or
    180 - 5x = 30 + (90 - x) + 2x=> 180 - 5x = 120 +x => x=10

    1. "Note that both angles EAB and EAD are equal 2x"

      What? How come? If you subtract <EAD=2x from <BAD=3x, the remainder <BAE=x, not 2x.


  3. Video solution (Spanish version) by Eder Contreras and Cristian Baeza at Geometry problem 5. Thanks Eder and Cristian.

  4. Draw an angular bisector of BDA.
    Draw DAE=2x, so that the angular bisector of BDA meets it at E.
    So, AE=ED (EAD=EDA=2x)
    But ∆EAD and ∆BDC are congruent under A.A.S case)
    So, ED=BD
    Now, Draw a Circle having the center D and Radius DC
    Obviously it passes through E, B and C
    Let AC and the circle meet at G
    So, GD=BD (Radius)
    Now connect BG
    H be the point of intersection of BG and DE
    Since ∆BDG is an Isosceles DH is perpendicular to BG
    So, HGD=90-2x
    Let F be the reflex of E over AB
    As AF=AE, FAB=EAB=x, AB=AB
    ∆AFB and ∆ AEB are congruent
    So FB=EB
    Also, ∆AFE and ∆BDE are congruent (BD=AF, BDE=FAE=2x, DE=AE)
    So, BE=FE
    Eventually EB=EF=BF
    So, ∆BFE is an equilateral
    Therefore FBE=600
    Since FE and AB are perpendicular to each other
    Since, Ang.ABC=ABE+EBD+DBC,
    But GBC=90Deg as it is an angle on semicircle (Note that GC is a diameter)
    So, ABG=120+x-90
    Since BGD=BAG+ABG
    So, x=100
    Note: Since ABD = 120-x the proof for the “PROBLEM 4 is inbuilt”

  5. Extend BD to E such that BD = DE
    Verify BCE is a right angle; Angle BDA = 4x; ABD = Pi - 7x
    Now cos 2x = BC / BE = BC / 2BD
    => 2 cos 2x = BC / BD = AD / BD = sin (pi - 7x) / sin3x = sin 7x / sin 3x
    => 2 sin 3x cos 2x = sin 7x
    => sin 5x + sin x = sin 7x
    => sin 7x - sin 5x = sin x
    => 2 cos 6x sin x = sin x
    => cos 6x = 1/2
    => 6x = pi/3
    => x = pi/18 = 10 deg

  6. Dear Antonio - is there a proof using properties of 80-20-80 triangles?


  8. Since BDC is isosceles, m(BDA) = 4x
    Now construct an isosceles triangle AED such that m(EAD) = m(EDA) = 2x
    the traingles AED and BDE are congruent implying AE = ED = DB and M(BDE) = 2*m(BAE)
    => on careful observation the quadrilateral BAED is similar to problem -4
    so m(ABD) = 120-x
    from triangle, ABC, 3x+120-x+2x+2x = 180
    => x = 10

  9. Solution:

  10. Grab a pen and a piece of paper, and illustrate along.

    Firstly, let us calculate some angles:
    Angle BCD = Line AE = line DE = line BD = line CD
    > Triangles ADE and BCD are identical.
    > Angle BAE = x
    > Angle BDE = 2x

    Add point F to line AB and connect it to point D such that line DF is the angle bisector of angle BDE. Now, connect point F to point E.

    We can see that:
    > Angle BDF = angle EDF = x
    > Triangles BDF and EDF are congruent by SAS because:
    · Angles BDF and EDF are equal.
    · Line BD = line DE
    · They share a side.
    > Triangle ADF is isosceles because angle DAF = angle ADF = 3x.
    > Since triangle ADF is isosceles, line AF = line DF.
    > Triangles FAE and EDF are congruent by SAS because:
    · Angle FAE = angle FDE = x
    · Line AE = line DE
    · Line AF = line DF

    Let angle AFE have a value of q such that angle AFE = angle DFE = angle DFB = q.

    Now, we solve the problem:
    > 3q = 180°, angle DFB = q = 180°/3 = 60°
    > Since all three angles in a triangle add up to 180°,
    180° - 7x + x + 60° = 180°
    180° - 6x = 120°
    6x = 60°
    x = 60°/6 = 10°

  11. This comment has been removed by the author.

  12. <BCD=2x

    Consider triangle and by sine law

    Consider triangle BDC and by sine law again

    Equating (1) and (2) as AD=BC & BD=CD