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MATH / Triangles ABC and ABD are isosceles with AB=AC=BD, and (BD)¯ intersects (AC)¯ at E. If (BD)¯⊥(AC)¯, then what is the value of ∠C+∠D? [asy] import olympiad; import geometry; size(15…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Triangles ABCABC and ABDABD are isosceles with AB=AC=BDAB=AC=BD, and BD¯\overline{BD} intersects AC¯\overline{AC} at EE. If BD¯⊥AC¯\overline{BD}\perp\overline{AC}, then what is the value of ∠C+∠D\angle C+\angle D? [asy] import olympiad; import geometry; size(150); defaultpen(linewidth(0.8)); draw(origin--(dir(70))--(2*Cos(70),0)--cycle); draw(origin--(dir(20))--(dir(70))--cycle); pair E = intersectionpoint(origin--dir(20),dir(70)--(2*Cos(70),0)); dot("EE",E); pair A = dir(70); pair B = origin; pair C = (2*Cos(70),0); pair D = dir(20); dot("AA",A,N); dot("BB",B,SW); dot("CC",C,SE); dot("DD",D,E); draw(rightanglemark(B,E,A,1.5)); [/asy]
Plain-text mathematical notation (without MathML)
Triangles ABC and ABD are  isosceles with AB=AC=BD, and (BD)¯ intersects (AC)¯ at E. If (BD)¯⊥(AC)¯, then what is the value of ∠C+∠D? [asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8));
draw(origin--(dir(70))--(2*Cos(70),0)--cycle);
draw(origin--(dir(20))--(dir(70))--cycle);
pair E = intersectionpoint(origin--dir(20),dir(70)--(2*Cos(70),0));
dot("E",E);
pair A = dir(70);
pair B = origin;
pair C = (2*Cos(70),0);
pair D = dir(20);
dot("A",A,N);
dot("B",B,SW);
dot("C",C,SE);
dot("D",D,E);
draw(rightanglemark(B,E,A,1.5));
[/asy]
Original LaTeX notation
Triangles $ABC$ and $ABD$ are  isosceles with $AB=AC=BD$, and $\overline{BD}$ intersects $\overline{AC}$ at $E$. If $\overline{BD}\perp\overline{AC}$, then what is the value of $\angle C+\angle D$? [asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8));
draw(origin--(dir(70))--(2*Cos(70),0)--cycle);
draw(origin--(dir(20))--(dir(70))--cycle);
pair E = intersectionpoint(origin--dir(20),dir(70)--(2*Cos(70),0));
dot("$E$",E);
pair A = dir(70);
pair B = origin;
pair C = (2*Cos(70),0);
pair D = dir(20);
dot("$A$",A,N);
dot("$B$",B,SW);
dot("$C$",C,SE);
dot("$D$",D,E);
draw(rightanglemark(B,E,A,1.5));
[/asy]

level

Level 4

type

Geometry

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Source and history

task_id
ba47e3a6-548e-52c0-98c0-a815895cbe38
task_key
geometry--test--ba47e3a6-548e-52c0-98c0-a815895cbe38
task_revision_id
1
config / split
geometry / test

Official source

initial import