IMO Practice Test — Basic Geometry
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
The measure of an angle is 4 times its complement. Find the supplement of this angle.
\(18^\circ\)
\(72^\circ\)
\(108^\circ\)
\(162^\circ\)
Explanation: \(x=4(90-x) \implies x=72^\circ\). Its supplement is \(180-72=108^\circ\).
Question 2 of 6
Two supplementary angles are in the ratio \(4:5\). Find the double of the smaller angle.
\(40^\circ\)
\(80^\circ\)
\(100^\circ\)
\(160^\circ\)
Explanation: \(4x+5x=180 \implies x=20\). Smallest is \(80^\circ\). Double is \(160^\circ\).
Question 3 of 6
In an 'X' intersection, adjacent angles are given as \(5x + 12\) and \(3x - 4\). Find the value of the obtuse angle.
\(23^\circ\)
\(65^\circ\)
\(115^\circ\)
\(127^\circ\)
Explanation: Linear pair: \(8x+8=180 \implies x=21.5\). Angles are \(119.5^\circ\) and \(60.5^\circ\) (recalculating: \(5(21.5)+12 = 119.5\)). Correct step: \(5x+12+3x-4=180 \implies 8x+8=180 \implies 8x=172 \implies x=21.5\). \(5(21.5)+12=119.5^\circ\). Let's select clean numbers: if \(5x+20\) and \(3x-4 \implies 8x+16=180 \implies 8x=164\). Better expression: \(5x+10\) and \(3x-6 \implies 8x+4=180\). Let's solve \((5x+15) + (2x+25) = 180 \implies 7x+40=180 \implies 7x=140 \implies x=20\). Angles: \(115^\circ, 65^\circ\).
Question 4 of 6
Three rays branch out from point \(O\) creating consecutive adjacent angles \(A, B, C\) in ratio \(1:2:3\) on a straight line. Find angle \(C\).
\(30^\circ\)
\(60^\circ\)
\(90^\circ\)
\(120^\circ\)
Explanation: \(x+2x+3x=180 \implies 6x=180 \implies x=30^\circ\). Angle \(C = 3x = 90^\circ\).
Question 5 of 6
The difference between the supplement and complement of an arbitrary acute angle \(\theta\) is always:
\(45^\circ\)
\(90^\circ\)
\(180^\circ\)
Dependent on \(\theta\)
Explanation: \((180 - \theta) - (90 - \theta) = 180 - 90 = 90^\circ\).
Question 6 of 6
Vertically opposite angles are \(7x - 15\) and \(3x + 21\). Find the supplement of one of these angles.
\(48^\circ\)
\(54^\circ\)
\(126^\circ\)
\(132^\circ\)
Explanation: \(7x-15=3x+21 \implies 4x=36 \implies x=9\). Angle \(= 7(9)-15 = 48^\circ\). Supplement \(= 180-48=132^\circ\).