Congruent triangles

Geometry Level 3

$ABC$ is a triangle in which angle $B$ is twice of $C$ . $D$ is a point on $BC$ such that $AD$ bisects angle $BAC$ and $AB=CD$ . Find angle $BAC$ .

90 70 77 72

1 solution

Alisina Zayeni
Jan 9, 2018

Draw line AE which is parallel to BC and equals with DC. The triangles AEC & ADE are congruent. So ADCE is a parallelogram and EC=AD , AE=DC=AB, ACB=CAE=ABC/2=♧. The angle BAE=BAC+CAE>>BAE=180-2♧. Because of that the triangle BAE is isosceles, angle ABE=AEB=♧>>EBC=♧. By an easy length chasing we will get AC=BE. By congruency of tow triangles BAC & CBE we will get EC=AD=AB. So ABD is an isosceles triangle. By easy angle chasing you can prove that the triangle ABC is an isosceles triangle, too. So 5♧=180>>>>♧=36>>>2♧=72 Note: ♧ is a diverse.

Congruent triangles

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