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complete the following proof.
prove: bd= ca
In mathematics, the Pythagorean theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It is expressed as:
c^2 = a^2 + b^2
where c represents the hypotenuse of the right triangle.
We calculate as follows:
16^2 = a^2 + 9^2
a = 5√7
1. ∠1 and ∠3 are vertical angles. 1. Given
2. ∠1 and ∠2 form a linear pair. VAT
∠2 and ∠3 form a linear pair. 2. Definition of linear pair
3. ∠1 and ∠2 are supplementary. Substitution Property
∠2 and ∠3 are supplementary. SAME SIDE INTERIOR ANGLES THEOREM
3. VERTICAL ANGLES THEOREM
4. m∠1 + m∠2 = 180˚ SAME SIDE INTERIOR ANGLES THEOREM
m∠2 + m∠3 = 180˚ SAME SIDE INTERIOR ANGLES THEOREM
5. m∠1 + m∠2 = m∠2 + m∠3
6. m∠1 = m∠3 opposite angles
7. ∠1 ≅ ∠3 7. Given
C.Form a pair of alternate interior angles which are congruent
In a parallelogram the opposite pairs of sides are equal and parallel.
And, If interior angles made on two lines by the same transversal are equal then these two lines are parallel to each other.
Since, Given: ABCD is a quadrilateral where sides AB and DC are equal and parallel.
We have to prove that: ABCD is a parallelogram.
Proof: Side AB║ DC ⇒ ∠ABD ≅ ∠ CDB.
AB≅ DC ( given)
DB≅DB ( Reflexive)
Therefore, ΔABD ≅ Δ BCD (by SAS postulate)
By CPCTC, ΔDBC ≅ Δ ADB
AD ≅ BC
∠ DBC and ∠ ADB form a pair of alternate interior angles which are congruent.
Therefore, AD is parallel and equal to BC.
Quadrilateral ABCD is a parallelogram because its opposite sides are equal and parallel.
Option B is true
We are given that three right triangles .Triangles ABD,CAD and CBA are similar.
When two triangles are similar then, the corresponding sidea are in equal proportion.
We have AB=5 units
AC= 7 units
Triangles ABD is similar triangle CBA
When triangle CAD and CBA are similar
In traingle CBA, uisng pythagoras theorem
Hence, option B is true.
ACD by SSS
DCA by SSS
It is given in the question that
BD is the median. So it divides the opposite sides in two equal parts .
Therefore in triangles BAD and BCD,
AB and AC are congruent because of isosceles triangle.
AD and CD are congruent because of the median BD.
And BD and BD are congruent .
So the two triangles are congruent by SSS and the correct option is the first option .
AB and BC are congruent
DB and FB are congruent
Angle EBD congruent to angle EFB
Angle ABD congruent to angle EBD
Alternate angle theorem
Angle FBC congruent to angle EFB
Alternate angle theorem
thats way to much