Points D, E, and F are the midpoints of sides \overline{BC}, \overline{CA}, and \overline{AB} of \triangle ABC, respectively, and \overline{CZ} is an altitude of the triangle. If \angle BAC = 71^\circ, \angle ABC = 39^\circ, and \angle BCA = 70^\circ, then what is \angle EZD+\angle EFD in degrees?
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To find the sum of angles \angle EZD and \angle EFD, we need to determine the measures of these angles.
Let's start by drawing the triangle \triangle ABC. We are given that CZ is an altitude, so let's draw it as well. It is always helpful to visualize the given information.
\triangle ABC:
A
/ \
CZ/ \B
/ \
/_______\
C BC A
Since D, E, and F are the midpoints of sides BC, CA, and AB respectively, we can mark them on the diagram as well:
\triangle ABC:
A
/ \
CZ/ \B
/ E \
/_______\
C D BC F A
Now, we are given the measures of angles \angle BAC, \angle ABC, and \angle BCA. Let's mark them on the diagram:
\triangle ABC:
A
/ \
CZ/ \B
/ E \
/_______\
C 70° D 39° BC F 71° A
To find \angle EZD, we need to determine the measure of \angle D. Since D is the midpoint of side BC, it divides that side into two equal parts. Therefore, \angle BDC is a right angle, and we know that \angle ABC = 39°.
Using this information, we can find \angle D using the relationship between \angle BCD and \angle ABC in a triangle. In \triangle BCD, we have two angles: \angle BCD and \angle BDC. Since the sum of all angles in a triangle is 180°, we can write the equation:
\angle BCD + \angle BDC + \angle ABC = 180°
Substituting the given angles, we have:
\angle BCD + 90° + 39° = 180°
Simplifying, we find:
\angle BCD = 180° - 90° - 39° = 51°
Since D is the midpoint of side BC, \angle EZD is equal to \angle BCD. Therefore, \angle EZD = 51°.
Now, let's find \angle EFD. Since E is the midpoint of side CA, it divides that side into two equal parts. Therefore, \angle AEC is a right angle, and we know that \angle BCA = 70°.
Using this information, we can find \angle F using the relationship between \angle ACE and \angle BCA in a triangle. In \triangle ACE, we have two angles: \angle ACE and \angle AEC. Since the sum of all angles in a triangle is 180°, we can write the equation:
\angle ACE + \angle AEC + \angle BCA = 180°
Substituting the given angles, we have:
\angle ACE + 90° + 70° = 180°
Simplifying, we find:
\angle ACE = 180° - 90° - 70° = 20°
Since E is the midpoint of side CA, \angle EFD is equal to \angle ACE. Therefore, \angle EFD = 20°.
Now, we can find the sum of angles \angle EZD and \angle EFD:
\angle EZD + \angle EFD = 51° + 20° = 71°.
Therefore, the sum of angles \angle EZD and \angle EFD is 71°.