• Ankita

MemberNov 2, 2011

## Solution to Grand Quiz Question #2

From figure we have,

[1] PR = QR
[2] QS = SR
-----------------------------------------------

In Figure 2, parallel to PQ, we draw SU
Because [1] and PQ || SU, we get
[3] RS =RU
[4] PS = QU
-----------------------------------------------

In Figure 3, we draw line PU.
From [4], and PQ = QP, and angle RPQ = RQP (=80),
the triangle PQS equals to triangle PQU.
So
[5] PU = QS
From [2], PU = SR
From [3], we get
[6] PU = RU
So angle RPU = 20, TPU = 10, UPQ = 60,
[7] SVU = 60
[8] PQ = QV = VP
From [5], [7] and [8], we get
[9] SU = UV = VS
-----------------------------------------------

In Figure 4, we draw UW vertical to PR.
From [6] and angle RPU = PRU, we get triangle PUW equals to triangle RUW. So
[10] PW = RW
-----------------------------------------------

Figure 5
In Figure 5, we draw RX.
From [10] and UW vertical to PR, we get
[11] angle SRX = SPX = 10
[12] RX = PX
From [11], we get
[13] angle URX = 10 = SRX
[13'] angle URX = UPX
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Figure 6

In Figure 6, extend RX to PQ.
From [13], and angle PQR = QPR = 80, we get
line RX vertical to PQ,
[14] line RX is the same as line RV
From [12], [13], [14], we get triangle RTX equals to triangle PVX. So
[15] TX = VX
From [12], [15], we get
[16] RV = PT
From [13'], [16], we get triangle RVU equals to triangle PTU. So
[17] TU = UV
--------------------------------------------------------

Finally
From [9], [17], we get
[18] TU = SU
Because SU || PQ, so
[19] angle RUS = RQP = 80
From [18], [19], we get
[20] angle STU = TSU = 50
So angle x = angle STU - angle PTU
= 50 - (180 - 70 - 60 - 20)
= 50 - 30
= 20
--------------------------------------------------------

This is just one of the solutions.
The fun about geometry is that, the same problem can have different approaches to land up to a solution.

We are expecting some logical approach (may be) different than what I mentioned above. 😀
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Yesterday's question's Coins will not be credited to anyone.

But the solution finding is still open to all.
Please come up with better solutions.

😀
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• MemberNov 2, 2011