Question

In quadrilateral ABCD, where AB and diagonal BD are perpendicular, AB=BD=BC, and M is the midpoint of CD, find the measure of the acute angle formed by diagonal AC and segment BM.

a. 45 degrees
b. 60 degrees
c. 30 degrees
d. 90 degrees

Answer 1

The measure of the acute angle formed by diagonal AC and segment BM is 90 degrees.

Step-by-step explanation:

Given that AB and diagonal BD are perpendicular, AB=BD=BC, and M is the midpoint of CD, we can determine the measure of the acute angle formed by diagonal AC and segment BM.

Since AB=BC, triangle ABC is an isosceles triangle. Therefore, angles BAC and BCA are congruent.

Since M is the midpoint of CD, we have MC=MD. Thus, triangle MBC is an isosceles triangle, and angles BMC and BCM are congruent.

Since angles BCA and BCM are congruent, and angles BAC and BMC are congruent, it follows that the acute angle formed by diagonal AC and segment BM is 90 degrees.