Question

In Fig., AB is the diameter of the circle such that ∠DAB = 40°. Find ∠DCA.

Answer 1

Given:

AB is the diameter of the circle.

∠DAB = 40°.

We must find ∠DCA.

Since AB is a diameter,

∠ADB = 90°

Step 1: Find ∠ABD

In triangle ADB:

∠ADB = 90°

∠DAB = 40°

∠ABD = 180° − (90° + 40°)

∠ABD = 50°

Step 2: Use that D and C lie on the same circle

∠ABD and ∠ACD stand on the same chord AD.

∠ABD = ∠ACD = 50°

Step 3: Use triangle ACD to find ∠DCA     ... [Now ∠DCA = ∠ACD]

∠DCA = 50°