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Given: In ∆ABC, D is the midpoint of AC; DF bisects ∠BDC and ED bisects ∠ADB.
Prove: EF ║ AC
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| Photo by Math Principles in Everyday Life |
Solution:
Consider the given figure and we will label further as follows
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| Photo by Math Principles in Everyday Life |
Proof:
1. Statement: D is the midpoint of line AC.
Reason: Given item
2. Statement: ED bisects ∠ADB and DF bisects BDC.
Reason: Given items
3. Statement: ∠ADE ≅ ∠EDB and ∠CDF ≅ ∠FDB.
Reason: Two bisected angles are congruent to each other.
4. Statement: ∠ADE + ∠EDB + ∠BDF + ∠FDC = 180°
Reason: The sum of the angles of a straight line is always equal to 180°.
5. Statement: ∠EDB and ∠BDF are complementary angles.
Reason: From the equation of Statement #4,
∠ADE + ∠EDB + ∠BDF + ∠FDC = 180°
but ∠ADE ≅ ∠EDB and ∠CDF ≅ ∠FDB
then the above equation becomes
2∠EDB + 2∠BDF = 180°
2(∠EDB + ∠BDF) = 180°
∠EDB + ∠BDF = 90°
6. Statement: ∆EDF is a right triangle.
Reason: ∠EDF is equal to 90°. ∠EDB and ∠BDF are complementary angles.
7. Statement: EF is the hypotenuse of a rt∆EDF.
Reason: EF is the opposite side of a right angle of a right triangle. EF is the longest side of a right triangle.
8. Statement: EO ≅ OF
Reason: If BD is a line bisector of line AC, D is the midpoint of line AC, and D is also a vertex of a rt∆EDF which is a right angle, then line EF which is the hypotenuse of a right triangle will be bisected into EO and OF.
9. Statement: ∆EDO and ∆DOF are isosceles triangles.
Reason: If BD is a line bisector that passes through the midpoint of the hypotenuse and a vertex of a right triangle which is the opposite angle of the hypotenuse, then BD will bisect a right triangle into two isosceles triangles.
10. Statement: EO ≅ OD and OD ≅ OF.
Reason: The two sides of an isosceles triangle are congruent.
11. Statement: ∠OED ≅ ∠ODE and ∠ODF ≅ ∠OFD.
Reason: The two base angles of an isosceles triangles are congruent.
12. Statement: If ∠OED ≅ ∠ODE and ∠ODE ≅ ∠ADE, then ∠OED ≅ ∠ADE.
Reason: Transitive property of congruence.
13. Statement: If ∠OFD ≅ ∠ODF and ∠ODF ≅ ∠CDF, then ∠OFD ≅ ∠CDF.
Reason: Transitive property of congruence.
14. Statement: EF ║ AC
Reason: If the alternating interior angles of a transversal line are congruent, then the two lines that are adjacent to the interior angles are parallel to each other. In the first case, line ED is the transversal line and ∠FED and ∠ADE are the alternating interior angles while in the second case, line DF is the transversal line and ∠EFD and ∠FDC are the alternating interior angles.