Steps and solution---**Question:**
6. If the difference of measure of angle ABD and measure of angle DBC is 10°, what is the degree measure of angle ABD?
**Options:**
A) 20
B) 25
C) 30
D) 50
**Diagram Description:**
* **Type:** Geometric figure illustrating angles.
* **Elements:**
* Point B is the vertex of the angles.
* Ray BA points upwards.
* Ray BC points to the right.
* Ray BD is between ray BA and ray BC.
* Angle ABC is indicated as a right angle (90°) by a square symbol at vertex B.
* Points A, D, and C are labeled on the rays.
* Arrows indicate that the lines are rays extending from B.
* **Relationships:** Angles ABD and DBC are adjacent angles. Angle ABC is the sum of angles ABD and DBC. Since angle ABC is a right angle, measure of angle ABD + measure of angle DBC = 90°.
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Let's analyze this angle problem step by step. We have a right angle ABC measuring 90 degrees. Ray BD divides this right angle into two smaller angles: angle ABD and angle DBC. We're told that the difference between these two angles is 10 degrees, and we need to find the measure of angle ABD.
Now let's set up our variables. We'll call the measure of angle ABD as x, and the measure of angle DBC as y. From the problem, we know two key facts: first, the difference between x and y is 10 degrees, so x minus y equals 10. Second, since angle ABC is a right angle, x plus y equals 90 degrees.
Now let's solve this system of equations. We have x minus y equals 10, and x plus y equals 90. To eliminate y, we add both equations together. This gives us 2x equals 100. Dividing both sides by 2, we get x equals 50. Therefore, the measure of angle ABD is 50 degrees.
Let's verify our answer. If angle ABD is 50 degrees, then angle DBC must be 90 minus 50, which equals 40 degrees. Now let's check: the difference between 50 degrees and 40 degrees is indeed 10 degrees, which matches our given condition. Therefore, the answer is D, 50 degrees.
To summarize our problem-solving approach: we identified the given information, defined variables for the unknown angles, set up equations based on the constraints, solved the system using elimination, and verified our solution. The key concepts we used were that adjacent angles in a right angle sum to 90 degrees, and we applied the elimination method to solve our system of linear equations. Our final answer is D, 50 degrees.