Final answer:
To solve this problem, use the properties of triangles and angles. Find CE using the Law of Sines and find angle ZBCE using angle sum in a triangle. Also find ED using angle sum in a triangle. Finally, find ZABE using the Law of Sines and find the area of triangle ABD using the formula for the area of a triangle.
Step-by-step explanation:
To solve this problem, we will use the properties of triangles and angles. Let's start by finding CE:
In triangle BDE, we have angle BDE + angle EBD + angle BED = 180°.
Since we know that angle BDA = 25°, we can calculate angle BDE as 180° - 25° - 20° = 135°.
In triangle CBE, we can use the Law of Sines to find CE:
sin(135°)/6 = sin(20°)/CE.
Solving for CE, we find CE ≈ 5.78 cm.
Now, let's find angle ZBCE:
Using the fact that the sum of the angles in a triangle is 180°, we have:
angle ZBCE = 180° - 20° - 45° = 115°.
For part (c), we can use the fact that the sum of the angles in a triangle is 180° to find angle BDE:
angle BDE = 180° - 135° - 20° = 25°.
Finally, for part (d), we can use the Law of Sines in triangle ABE to find angle ZABE:
sin(25°)/7 = sin(ZABE)/6.
Solving for ZABE, we find ZABE ≈ 55.3°.
To find the area of triangle ABD, we can use the formula:
Area = (1/2) * AB * BD * sin(BDA).
Plugging in the values, we get:
Area = (1/2) * 7 * 8 * sin(25°) ≈ 18.0 cm².
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