Given: and bisect each other. Prove: Quadrilateral ABCD is a parallelogram. Proof: Statement Reason 1. and bisect each other. gi
ven 2. AE = EC BE = ED definition of bisection 3. m∠AEB = m∠CED 4. ΔABE ≅ ΔCDE SAS criterion 5. ∠ACD ≅ ∠CAB Corresponding angles of congruent triangles are congruent. 6. converse of Alternate Interior Angles Theorem 7. m∠BEC = m∠AED Vertical Angles Theorem 8. ΔBEC ΔDEA SAS criterion for congruence 9. DBC ≅ BDA Corresponding angles of congruent triangles are congruent. 10. converse of Alternate Interior Angles Theorem 11. Quadrilateral ABCD is a parallelogram. definition of a parallelogram 1 What is the reason for step 3 of this proof? A. Alternate Interior Angles Theorem B. Corresponding angles in congruent triangles are congruent. C. For parallel lines cut by a transversal, corresponding angles are congruent. D. Vertical Angles Theorem E.
2 answers:
Answer:
To Prove: Quadrilateral ABCD is a parallelogram.
Proof: In Δ ABE and ΔCDE
1. AE = EC and BE = ED [ Diagonals bisect each other]
2.∠ AEB = ∠ CED [ vertically opposite angles]
Δ ABE ≅ ΔCDE---------- [SAS]
∠ ACD ≅ ∠CAB [Corresponding angles of congruent triangles are congruent⇒This statement is untrue ∴ these are alternate interior angles not corresponding angles.]
6. The converse of alternate interior interior angle theorem states that if two parallel lines are cut by a transversal then alternate interior angles are equal.
7. In ΔBEC and ΔAED
∠BEC = ∠AED [ Vertical Angles Theorem ]
AE = EC and BE = ED [ Diagonals bisect each other]
⇒ ΔBEC≅ ΔDEA [ SAS criterion for congruence]
9. DBC ≅ BDA [ Corresponding angles of congruent triangles are congruent⇒This statement is untrue ∴ these are alternate interior angles not corresponding angles.]
As pair of triangles are congruent ∵ quadrilateral ABCD is a parallelogram.
Step 3 is m∠AEB = m∠CED
These pair of angles are vertically opposite angles of ΔAEB and ΔCED.
Option [D. Vertical Angles Theorem] is correct.
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Answer:
Step-by-step explanation:
we know that
The absolute value function has two solutions
Observing the graph
the solutions are
and
First solution (case positive)
assume the symbol of the first solution and then compare the results
Second solution (case negative)
Multiply by -1 both sides
substitute the value of b and compare the results
-------> is correct
In Δ ABC, ∠A=120°, AB=AC=1
To draw a circumscribed circle Draw perpendicular bisectors of any of two sides.The point where these bisectors meet is the center of the circle.Mark the center as O.
Then join OA, OB, and OC.
Taking any one OA,OB and OC as radius draw the circumcircle.
Now, from O Draw OM⊥AB and ON⊥AC.
As chord AB and AC are equal,So OM and ON will also be equal.
The reason being that equal chords are equidistant from the center.
AM=MB=1/2 and AN=NC=1/2 [ perpendicular from the center to the chord bisects the chord.]
In Δ OMA and ΔONA
OM=ON [proved above]
OA is common.
MA=NA=1/2 [proved above]
ΔOMA≅ ONA [SSS]
∴ ∠OAN =∠OAM=60° [ CPCT]
In Δ OAN
OA=1
∴ OA=OB=OC=1, which is the radius of given Circumscribed circle.
