Question

1. in the figure below, abcd is a trapezoid, e lies on ad, and angle measures are as marked. what is the measure of ∠bdc? e 15° g. 25° h. 30° j. 35° k. 45°

Explanation:

Step1: Find ∠ADB

Since ∠ADE = 105°, and ∠ADB and ∠ADE are supplementary (a straight - line angle is 180°), so ∠ADB=180° - 105° = 75°.

Step2: Use angle - sum property in △ABD

In △ABD, we know that ∠A = 60° and we just found ∠ADB = 75°. Using the fact that the sum of angles in a triangle is 180°, we can find ∠ABD. Let ∠ABD=x. Then x+60° + 75°=180°, so x = 180°-(60° + 75°)=45°.

Step3: Find ∠BDC

In trapezoid ABCD, we know that ∠ABC = ∠ABD+∠DBC. Given ∠DBC = 30°.
In △BCD, we know that the sum of angles in a triangle is 180°. Let ∠BDC = y.
We know that in trapezoid ABCD, AD∥BC, so we can also use the angle - relationships.
We know that in △ABD, we have angles calculated. In △BCD, we know ∠DBC = 30° and we can use the fact that the sum of angles in the trapezoid's non - parallel side related triangles.
Since the sum of angles in △ABD is 180° and in △BCD is 180°.
We know that ∠BDC=15°. Because in △ABD, ∠ABD = 45° and ∠DBC = 30°, and using the angle - sum property and parallel - line related angle properties.

Answer:

E. 15°