How do I solve this math question?---**Chart/Diagram Description:**
* **Type:** Geometric figure.
* **Main Elements:**
* **Shape:** A parallelogram labeled ABCD. Points are arranged such that AB is parallel to DC and AD is parallel to BC.
* **Points:** Vertices A, B, C, D. A point F is on side BC.
* **Lines:** Sides AB, BC, CD, DA of the parallelogram. Diagonal AC. Line segment AF connecting vertex A to point F on BC.
* **Angles:**
* Angle between side AD and diagonal AC is labeled 27 degrees (∠DAC = 27°).
* Angle between line segment AF and diagonal AC is labeled d degrees (∠FAC = d°).
* Angle between line segment AF and side BC is labeled 34 degrees (∠AFB = 34°).
* **Labels and Annotations:** Vertices A, B, C, D. Point F. Angle measures 27°, d°, 34°. Note: Figure not drawn to scale.
* **Relative Position and Direction:** A is approximately bottom-left, B top-left, C top-right, D bottom-right. F is on the segment BC. AC is a diagonal from A to C. AF is a line segment from A to F.
**Other Relevant Text:**
Note: Figure not drawn to scale.
**Question Stem:**
The figure, $ABCD$, is a parallelogram. What is the value of $d$?
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In this problem, we have a parallelogram ABCD. Point F is located on side BC. We're given that angle DAC is 27 degrees, angle AFB is 34 degrees, and angle FAC is d degrees. Our task is to find the value of d. Let's start by understanding the properties of a parallelogram and how they relate to the angles in this figure.
Let's start by using the properties of a parallelogram. In a parallelogram, opposite sides are parallel. When a transversal line intersects two parallel lines, alternate interior angles are equal. Here, diagonal AC is a transversal that intersects the parallel lines AD and BC. Therefore, angle DAC equals angle ACB. Since we're given that angle DAC is 27 degrees, we can conclude that angle ACB is also 27 degrees.
Now, let's find angle AFC. Since point F lies on the straight line BC, angles AFB and AFC form a straight line, which means they sum to 180 degrees. We're given that angle AFB is 34 degrees. Therefore, angle AFC equals 180 degrees minus 34 degrees, which is 146 degrees. This is an important angle that we'll use in the next step to find the value of d.
Now we can find the value of d by using the triangle angle sum property. Let's focus on triangle AFC. In any triangle, the sum of all interior angles is 180 degrees. In triangle AFC, we have angle FAC, which is d degrees, angle AFC, which we found to be 146 degrees, and angle ACF, which is the same as angle ACB, which is 27 degrees. Setting up the equation: d degrees plus 146 degrees plus 27 degrees equals 180 degrees. Solving for d, we get d equals 180 degrees minus 146 degrees minus 27 degrees, which gives us d equals 7 degrees.
Let's summarize the steps we took to find the value of d. First, we identified that ABCD is a parallelogram, which means AD is parallel to BC. Using the property of alternate interior angles, we found that angle ACB equals 27 degrees. Next, we recognized that since F lies on line BC, angles AFB and AFC form a straight line, which gave us angle AFC equals 146 degrees. Then, we applied the triangle angle sum property to triangle AFC: d degrees plus 146 degrees plus 27 degrees equals 180 degrees. Solving this equation, we determined that d equals 7 degrees, which is our final answer.