Difference between revisions of "1968 AHSME Problems/Problem 6"

Latest revision as of 18:06, 17 July 2024

Problem

Let side $AD$ of convex quadrilateral $ABCD$ be extended through $D$ , and let side $BC$ be extended through $C$ , to meet in point $E.$ Let $S$ be the degree-sum of angles $CDE$ and $DCE$ , and let $S'$ represent the degree-sum of angles $BAD$ and $ABC.$ If $r=S/S'$ , then:

$\text{(A) } r=1 \text{ sometimes, } r>1 \text{ sometimes}\quad\\ \text{(B) } r=1 \text{ sometimes, } r<1 \text{ sometimes}\quad\\ \text{(C) } 0<r<1\quad \text{(D) } r>1\quad \text{(E) } r=1$

Solution

$[asy] draw((-16,0)--(16,-24)); draw((0,24)--(16,-24)); draw((-16,0)--(0,24)); draw((0,-12)--(8,0)); dot((16,-24)); label("E",(16,-24),SE); dot((-16,0)); label("A",(-16,0),W); dot((0,24)); label("B",(0,24),N); dot((0,-12)); label("D",(0,-12),SW); dot((8,0)); label("C",(8,0),NE); [/asy]$

Because $ABCD$ is a convex quadrilateral, the sum of its interior angles is $360^{\circ}$ . Thus, $S'=\measuredangle BAD + \measuredangle ABC=360^{\circ}-(\measuredangle BCD+\measuredangle ADC)$ . Furthermore, because $\angle BCD$ and $\angle ADC$ are supplementary to $\angle DCE$ and $\angle CDE$ , respectively, the four angles sum to $2*180^{\circ}=360^{\circ}$ , so $\measuredangle BCD+\measuredangle ADC=360^{\circ}-(\measuredangle DCE+\measuredangle CDE)=360^{\circ}-S$ . Plussing this expression for $\measuredangle BCD+\measuredangle ADC$ into the first equation, we see that $S'=360^{\circ}-(360^{\circ}-S)=S$ , so $r=\frac{S}{S'}=1$ , which is answer choice $\fbox{E}$ .

1968 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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Difference between revisions of "1968 AHSME Problems/Problem 6"

Latest revision as of 18:06, 17 July 2024

Problem

Solution

See also

@@ Line 10: / Line 10: @@
 == Solution ==
-<math>\fbox{E}</math>
+<asy>
+draw((-16,0)--(16,-24));
+draw((0,24)--(16,-24));
+draw((-16,0)--(0,24));
+draw((0,-12)--(8,0));
+dot((16,-24));
+label("E",(16,-24),SE);
+dot((-16,0));
+label("A",(-16,0),W);
+dot((0,24));
+label("B",(0,24),N);
+dot((0,-12));
+label("D",(0,-12),SW);
+dot((8,0));
+label("C",(8,0),NE);
+</asy>
+Because <math>ABCD</math> is a convex [[quadrilateral]], the sum of its interior angles is <math>360^{\circ}</math>. Thus, <math>S'=\measuredangle BAD + \measuredangle ABC=360^{\circ}-(\measuredangle BCD+\measuredangle ADC)</math>. Furthermore, because <math>\angle BCD</math> and <math>\angle ADC</math> are [[supplementary]] to <math>\angle DCE</math> and <math>\angle CDE</math>, respectively, the four angles sum to <math>2*180^{\circ}=360^{\circ}</math>, so <math>\measuredangle BCD+\measuredangle ADC=360^{\circ}-(\measuredangle DCE+\measuredangle CDE)=360^{\circ}-S</math>. Plussing this expression for <math>\measuredangle BCD+\measuredangle ADC</math> into the first equation, we see that <math>S'=360^{\circ}-(360^{\circ}-S)=S</math>, so <math>r=\frac{S}{S'}=1</math>, which is answer choice <math>\fbox{E}</math>.
 == See also ==
-{{AHSME box|year=1968|num-b=5|num-a=7}}
+{{AHSME 35p box|year=1968|num-b=5|num-a=7}}
 [[Category: Introductory Geometry Problems]]
 {{MAA Notice}}