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Given the figure below:
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| Photo by Math Principles in Everyday Life |
1. Given AF ≅ AD, and FE ≅ ED. Prove that ΔAFE ≅ ΔAED.
2. Given ABDE is a square and ΔACD is an isosceles triangle. Prove that ΔAED ≅ ΔBDC.
3. Given ABDE is a square. Prove that ΔABD ≅ ΔADE.
4. Given AB ≅ DE, AD bisects ∠BAE and ∠BDE. Prove that ΔBAD ≅ ΔADE.
5. Given AF ≅ DC and DC ≅ ED. Prove that ΔAFD ≅ ΔACD.
Solution:
Consider Case 1:
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| Photo by Math Principles in Everyday Life |
Proof:
1. Statement: AF ≅ AD and FE ≅ ED.
Reason: Given items.
2. Statement: AE ≅ AE.
Reason: Reflexive property of congruence.
Therefore, ΔAFE ≅ ΔAED.
Reason: SSS (Side-Side-Side) Postulate
Consider Case 2:
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| Photo by Math Principles in Everyday Life |
Proof:
1. Statement: ABDE is a square and ΔACD is an isosceles triangle.
Reason: Given items.
2. Statement: AE ≅ BD ≅ AB ≅ ED
Reason: All sides of a square are congruent.
3. Statement: AD ≅ AD
Reason: Reflexive property of congruence.
4. Statement: ∠EDA ≅ ∠BAD
Reason: The alternating interior angles of a two parallel lines that passes a transversal line are congruent. The two opposite sides of a square are parallel.
5. Statement: ∠DAB ≅ ∠BCD
Reason: The base angles of an isosceles triangle are congruent.
6. Statement: AD ≅ DC
Reason: The two sides of an isosceles triangle are congruent.
7. Statement: AB ≅ BC
Reason: The base altitude (BD) of an isosceles triangle bisects the line segment (AC) of a base.
Therefore, ΔAED ≅ ΔBDC.
Reason: SAS (Side-Angle-Side) Postulate
Consider Case 3:
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| Photo by Math Principles in Everyday Life |
Proof:
1. Statement: ABDE is a square.
Reason: Given item.
2. Statement: AE ≅ BD ≅ AB ≅ ED
Reason: All sides of a square are congruent.
3. Statement: AD ≅ AD
Reason: Reflexive property of congruence.
Therefore, ΔABD ≅ ΔAED.
Reason: SSS (Side-Side-Side) Postulate
Consider Case 4:
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| Photo by Math Principles in Everyday Life |
Proof:
1. Statement: AB ≅ ED, ∠EAD ≅ ∠BAD, and ∠BDA ≅ ∠ADE.
Reason: Given items.
2. Statement: AD ≅ AD
Reason: Reflexive property of congruence.
Therefore, ΔBAD ≅ ΔADE.
Reason, ASA (Angle-Side-Angle) Postulate.
Consider Case 5:
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| Photo by Math Principles in Everyday Life |
Proof:
1. Statement: AF ≅ DC and DC ≅ ED.
Reason: Given items.
2. Statement: AB ≅ ED and AE ≅ BD.
Reason: The opposite sides of a rectangle are congruent.
3. Statement: AD ≅ AD
Reason: Reflexive property of congruence.
4. Statement: AE ┴ ED and BD ┴ AB.
Reason: The sides of a rectangle ABDE are perpendicular to each other.
5. Statement: ∠FEA ≅ ∠AED, and ∠ABD ≅ ∠DBC.
Reason: The sum of the supplementary angles is 180. If ∠AED and ∠ABD are 90°, then ∠FEA and ∠DBC msut be 90°.
6. Statement: ΔFEA and ΔDBC are right triangles.
Reason: One of the angles of each triangles is 90°.
7. Statement: FE ≅ BC
Reason: Since ΔFEA and ΔDBC are right triangles, we can use Pythagorean Theorem (c2 = a2 + b2) to solve the other side of a right triangle. If AF ≅ CD and AE ≅ BD, then FE ≅ BC.
8. Statement: FD ≅ AC
Reason: Since FE ≅ BC and ED ≅ AB, then FE + ED ≅ AB + BC.
Therefore, ΔAFD ≅ ΔACD.
Reason: SSS (Side-Side-Side) Postulate