Question

Question 5(Multiple Choice Worth 6 points)

(05.03 MC)

The figure below shows a quadrilateral ABCD with diagonal BD bisecting angle ADC:

A quadrilateral ABCD is drawn with BD as a diagonal. BD bisects angle ADC and forms two triangles ABD and CBD. AD and CD are equal in length.

Which equation is true?

Angle DCB = angle ADB
Angle DAB = angle DCB
Angle ABD = angle BCD
Angle ABD = angle CDB

Answer 1

2.
`\angle DAB=\angle DCB`

Explanation:

We have been given a diagram of a quadrilateral ABCD with diagonal BD bisecting angle ADC, which means that
`m\angle ADB=m\angle CDB`.

We can see that diagonal BD has divided quadrilateral ABCD into two triangles. We can see that in triangles ADB and CDB side AD equals to side CD and side BD equals to itself, therefore, triangles ADB and CDB are congruent by SAS postulate of congruence.

Since triangle ADB is congruent to triangle CDB, therefore, all sides and angles of triangle ADB will be congruent to corresponding sides and angles of triangle CDB.

Let us see which of our given options represent corresponding angles of both triangles.

1.
`\angle DCB=\angle ADB`

We can see from our diagram that angle DAB corresponds to angle DCB, therefore, 1st equation is not true.

2.
`\angle DAB=\angle DCB`

We can see from our diagram that angle DAB corresponds to angle DCB, therefore, 2nd equation is true.

3.
`\angle ABD=\angle BCD`

We can see from our diagram that angle CBD corresponds to angle ABD, therefore, 3rd equation is not true.

4.
`\angle ABD=\angle CDB`

We can see from our diagram that angle CDB corresponds to angle ADB, therefore, 4th equation is not true.