Difference between revisions of "1997 AJHSME Problems/Problem 12"
Talkinaway (talk | contribs) (Created page with "==Problem== <math>\angle 1 + \angle 2 = 180^\circ </math> <math>\angle 3 = \angle 4</math> Find <math>\angle 4.</math> <asy> pair H,I,J,K,L; H = (0,0); I = 10*dir(70); J = I ...") |
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label("$3$",L+dir(162.5),WNW); label("$4$",K+dir(-52.5),SE); | label("$3$",L+dir(162.5),WNW); label("$4$",K+dir(-52.5),SE); | ||
</asy> | </asy> | ||
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<math>\text{(A)}\ 20^\circ \qquad \text{(B)}\ 25^\circ \qquad \text{(C)}\ 30^\circ \qquad \text{(D)}\ 35^\circ \qquad \text{(E)}\ 40^\circ</math> | <math>\text{(A)}\ 20^\circ \qquad \text{(B)}\ 25^\circ \qquad \text{(C)}\ 30^\circ \qquad \text{(D)}\ 35^\circ \qquad \text{(E)}\ 40^\circ</math> | ||
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== See also == | == See also == | ||
| − | {{AJHSME box|year=1997|num-b= | + | {{AJHSME box|year=1997|num-b=11|num-a=13}} |
* [[AJHSME]] | * [[AJHSME]] | ||
* [[AJHSME Problems and Solutions]] | * [[AJHSME Problems and Solutions]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 11:37, 2 May 2021
Problem
Find 
![[asy] pair H,I,J,K,L; H = (0,0); I = 10*dir(70); J = I + 10*dir(290); K = J + 5*dir(110); L = J + 5*dir(0); draw(H--I--J--cycle); draw(K--L--J); draw(arc((0,0),dir(70),(1,0),CW)); label("$70^\circ$",dir(35),NE); draw(arc(I,I+dir(250),I+dir(290),CCW)); label("$40^\circ$",I+1.25*dir(270),S); label("$1$",J+0.25*dir(162.5),NW); label("$2$",J+0.25*dir(17.5),NE); label("$3$",L+dir(162.5),WNW); label("$4$",K+dir(-52.5),SE); [/asy]](http://latex.artofproblemsolving.com/9/f/0/9f0cab5df9bf98e5c9cd3ffe8661ec2772becc8f.png)
Solution
Using the left triangle, we have:
Using the given fact that 
, we have 
.
Finally, using the right triangle, and the fact that , we have:
Thus, the answer is 
See also
| 1997 AJHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 11 |
Followed by Problem 13 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
