Answer: See the attached image below. I've filled in the table with the proper values.
Row 2: AFB = 36, BFC = 54, EFA = 144
Row 3: AFB = 40, BCF = 50, DFE = 40
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Explanation:
angles AFB and BFC add to 90 degrees since they form a right angle.
In row 1, we see that 48+42 = 90
Angles AFB and DFE are congruent since they are vertical angles. For row 1, we have both of these angles equal to 48 degrees.
Also for row 1, we have angles BFC and CFD add to 42+90 = 132, which is exactly the measure of angle BFD. This angle is congruent to angle EFA because they are vertical angles. So that's how angle EFA is 132.
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Now onto row 2
We'll use the ideas of row 1 to work our way backwards to fill in the missing items.
We're given DFE is 36 degrees. This must mean that AFB is also 36 degrees. These angles are vertical angles.
Then we'll use the idea that (angleAFB)+(angleBFC) = 90 to find that
angle BFC = 90 - (angle AFB) = 90 - 36 = 54
Lastly for this row, we can say,
angle BFD = (angle BFC) + (angle CFD) = 54 + 90 = 144
angle EFA = angle BFD = 144
To summarize the missing items of row 2, we have...
- angle AFB = 36
- angle BFC = 54
- angle EFA = 144
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Now onto row 3. We use the same tricks (more or less) that were used on row 2.
If angle EFA is 140, then so is angle BFD. These are vertical angles.
(angle BFC) + (angle CFD) = angle BFD
(angle BFC) + 90 = 140
angle BFC = 140-90
angle BFC = 50
This in turn lets us say,
(angle AFB) + (angle BFC) = 90
angle AFB = 90 - (angle BFC)
angle AFB = 90 - 50
angle AFB = 40
angle DFE = angle AFB = 40
To summarize the missing pieces we found:
- angle AFB = 40
- angle BFC = 50
- angle DFE = 40
All of the values are filled in the table shown below.
