Points D, E, and F are the midpoints of sides BC, CA, and AB of ABC, respectively, and CZ is an altitude of the triangle. If <BAC=71, <ABC=39, and <BCA=70, then what is <EZD+<EFD in degrees?
Please help I tried making a picture but its not helping....please help!
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To solve this problem, you can use the properties of triangles and the angles formed by points D, E, F, and Z.
Step 1: Draw a diagram of triangle ABC and label the given information. Label the midpoints D, E, and F on sides BC, CA, and AB, respectively. Also, label the altitude CZ.
Step 2: Since D, E, and F are midpoints, you can use the midpoint theorem to find the lengths of DE, EF, and FD. The midpoint theorem states that the line segment joining two midpoints of a triangle is parallel to the third side and half its length.
Step 3: Label the points of intersection between CZ and DE as X and CZ and EF as Y. Notice that CZ is an altitude, so XZ is perpendicular to DE and YZ is perpendicular to EF.
Step 4: Since XZ is perpendicular to DE, triangle DXZ is a right triangle. Similarly, YZ is perpendicular to EF, so triangle EYZ is a right triangle.
Step 5: Use triangle DXZ and triangle EYZ to find the measures of angles <EZD and <EFD. In triangle DXZ, <EZD is the complement of <XZD (since <XZD + <EZD = 90 degrees). Similarly, in triangle EYZ, <EFD is the complement of <YZE (since <EZD + <EFD = 90 degrees).
Step 6: To find <XZD and <YZE, use the angle sums in triangles DXZ and EYZ. In triangle DXZ, <XZD + <DXZ + <ZDX = 180 degrees. Since <DXZ and <ZDX are both right angles, <XZD = 180 - 90 - 90 = 0 degrees. Similarly, in triangle EYZ, <YZE = 0 degrees.
Step 7: Recall that <EZD is the complement of <XZD, so <EZD = 90 degrees. Similarly, <EFD is the complement of <YZE, so <EFD = 90 degrees.
Step 8: Therefore, <EZD + <EFD = 90 + 90 = 180 degrees.
Hence, <EZD + <EFD is equal to 180 degrees.