In a triangle ABC, AC=BC and angle ACB=40. The vertices A,B and C lie on the circumference of a circle. D is a point outside this circle such that DC is perpendicular to BC, BC=DC and the points B and D are on the same side of the line AC. DA cuts the circumference of the circle at E. Find the size of angle DCE.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

We are given `bar(AC) cong bar(BC)` with `Delta ABC` inscribed in a circle. `bar(DC)` is drawn on the same side of `bar(AC)` as B such that `bar(DC) _|_ bar(BC)`and `bar(DC) cong bar(BC)` . Then `bar(DA)` intersects the circle at E; we are to find the measure of `/_DCE` :

`Delta ABC` is isosceles with vertex angle `40^@` so the base angles `/_A cong /_B = 70^@` .

`Delta ACD` is isosceles with vertex angle `130^@` (`/_DCB=90^@` plus `40^@` from `/_ACB` ) so `m/_CAD=m/_CDA=25^@` .

Then `m/_DAB=45^@` (70-25=45)

The measure of arc EBA is 170. (The measure of an inscribed angle is half the measure of its inscribed arc.) The measure of arc EB=90 and the measure of arc BA is 80.

Then `m/_ECB=45^@` (half of arc EB)

`m/_ACD=130^@;m/_ACB=40^@;m/_BCE=45^@` which implies that `m/_DCE=45^@`

----------------------------------------------------------------

`m/_DCE=45^@`

----------------------------------------------------------------

Approved by eNotes Editorial Team

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial