Little problem to keep sharp

**fireballs619** · October 21st 2010, 05:49 PM

Came across this problem today, and thought it was pretty nice. Most of the kids in my class solved it in about 1-2 minutes, how quickly can you solve it?

$Little problem to keep sharp-math-problem1.png$

I drew it in paint, so pardon sloppiness (I don't want to mess with latex today).

Given:
m||n;
All lines are coplanar;

Find x

Spoiler:

**Also sprach Zarathustra** · October 21st 2010, 06:09 PM

I think that something is missing... are the two points make a perpendicular to m (n) ? is it matter? (asking myself)

**skeeter** · October 21st 2010, 06:37 PM

nothing missing ... sketch an auxiliary line parallel to m and n thru the vertex of angle x

**Soroban** · October 22nd 2010, 08:24 AM

Hello, fireballs619!

Given: $PQ \parallel RS,\;\angle PAB = 115^o,\;\angle RCB = 135^o$

Find: $x \:=\:\angle ABC$

Code:

            A
    P * - - * - - - - - - - - * Q
          115 *
                *
                  *
                    * 
                      *
                      x * B
                       *
                      *
                     *
                    *
               135 *
    R * - - - - - * - - - - * S
                  C

Extend $CB$ to intersect $PQ$ at $\,D.$

Code:

            A                 D
    P * - - * - - - - - - - - * - Q
          115 * 65        45 /
                *           /
                  *        /
                    *  70 / 
                      *  /
                      x * B
                       *
                      *
                     *
                    *
               135 * 45
    R * - - - - - * - - - - * S
                  C

$\angle PAB = 115^o \quad\Rightarrow\quad \angle QAB = 65^o$

$\angle RCB = 135^o \quad\Rightarrow\quad \angle BCS = 45^o = \angle ADB$

$\text{Hence: }\:\angle ABD \:=\:180^o - 65^o - 45^o \;=\;70^o$

$\text{Therefore: }\:x \;=\;\angle ABC \;=\;180^o - 70^o \;=\;110^o$

**Unknown008** · October 22nd 2010, 08:36 AM

I just make use of the parallel lines and the time to grab my calculator, it takes less than 20 seconds.

So, $x = \alpha + \beta = (180 - 115) + (180 - 135) = 110^o$

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