How many properties of parallelograms are there?
1 answer:
Answer: The answer is six.
Step-by-step explanation: There are six important properties of a parallelogram. Please see the attached figure. The properties are as follows :
(i) The opposite sides of a parallelogram are parallel and congruent, i.e., PQ║RS, QR║PS, PQ ≅ RS and QR ≅ PS.
(ii) The opposite angles of a parallelogram are congruent, i.e., ∠P = ∠R and ∠Q = ∠S.
(iii) The diagonals of a parallelogram bisect each other, i.e., OP = OR and OQ = OS.
(iv) Consecutive angles of a parallelogram are supplementary. For example, ∠P + ∠Q = 180°.
(v) If one angles of a parallelogram is 90°, then all the angles are 90°.
(vi) Each diagonal of a parallelogram separates it into two congruent triangles. For example, ΔPQR ≅ ΔSQR.
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Answer: The triangles are congruent by the HL theorem
HL = hypotenuse leg
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Explanation:
Through the segment addition postulate, we can say:
WZ = WT+TZ
YT = TZ+ZY
Notice how both WZ and YT have TZ as part of their segments.
Since WZ and YT are the same length, we can then say,
WZ = YT
WT+TZ = TZ+ZY
WT = ZY
I used substitution for the second step.
The last step is where I subtracted TZ from both sides, and they canceled out.
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We have the following info:
- WT = ZY (just found)
- AT = UZ (given)
- angle AWT = angle UYZ (given)
Normally, the conclusion from here would be "not enough info" because this is a SSA case. Note how the angles are not between the congruent sides. So we wouldn't use SAS.
However, despite us having SSA, it doesn't mean we can't keep going. The reason why is because we have two right triangles. So we can use the HL theorem instead. Statement 1 above refers to the L of HL, while statement 2 is the H.
HL = hypotenuse leg
Again, HL only works for right triangles. It's the only time we can use SSA to prove right triangles are congruent; otherwise, SSA is not a valid congruence theorem.
