Question

In quadrilateral ABCD, diagonal BD is the angle bisector of ∠B and ∠D. Find m∠DAC if m∠ADB=20° and m∠DBC=60°.

A) 30°
B) 40°
C) 50°
D) 60°

Answer 1

The measure of angle DAC in quadrilateral ABCD is 120°.

Step-by-step explanation:

To find the measure of angle DAC, we can use the angle bisector theorem. According to the theorem, the ratio of the lengths of the two segments created by the angle bisector is equal to the ratio of the opposite sides. In this case, we can say that AD/DB = AC/BC. Since BD is the angle bisector and m∠ADB = 20°, we have m∠ADB = m∠BDC = 20°.

Let's use this ratio to set up an equation:

AD/DB = AC/BC

AD/BC = AC/DB

Since m∠ADB = m∠BDC = 20°,

AD/BC = AC/DB = 1

Therefore, AD = BC and AC = DB.

Since m∠DBC = 60°, we have m∠CAB = 60° as well.

Since m∠DAC + m∠CAB = 180° (the sum of angles in a quadrilateral),

m∠DAC + 60° = 180°

m∠DAC = 120°

Therefore, the measure of angle DAC is 120°. Answer choice (D) 60° is incorrect.