12 multiple-choice questions, progressively harder.
In the triangle shown, which segment has length aaa?
Solution
Correct answer: C
Side aaa is opposite the vertex AAA, so its endpoints are BBB and CCC.
Which expression gives twice the area KKK of triangle ABCABCABC using the sides that form angle BBB?
Correct answer: A
The sides that meet at BBB are BABABA and BCBCBC. Their lengths are ccc and aaa, respectively.
In an area proof, divide 2K=absinC2K=ab\sin C2K=absinC by the product abcabcabc. What is the simplified right side?
Correct answer: D
The factors aaa and bbb cancel from the numerator and denominator. The opposite length ccc stays in the denominator.
The angles at BBB and CCC and the length of ACACAC are given. Which arrangement is this?
Correct answer: B
The side between the known angles would be BCBCBC. The supplied side has endpoint AAA, whose angle was missing, so these are angle-angle-side data.
Two interior angles of triangle ABCABCABC are 41∘41^\circ41∘ at AAA and 79∘79^\circ79∘ at BBB. What is the angle at CCC?
Subtract both known angles from the triangle angle sum.
In triangle ABCABCABC, AAA is 30∘30^\circ30∘, CCC is 60∘60^\circ60∘, and aaa is 7 units. What is ccc?
Multiply the known length by the sine of the requested angle and divide by the sine of the known opposite angle.
One angle strictly between 0∘0^\circ0∘ and 180∘180^\circ180∘ with a certain sine is 28∘28^\circ28∘. What is the other angle with that sine?
Supplementary angles share the same sine.
An SSA calculation gives sinB=21/20\sin B=21/20sinB=21/20. What can you conclude?
The ratio is greater than 1, the largest sine of a triangle angle. No candidate angle exists.
In triangle ABCABCABC, the known angle is A=30∘A=30^\circA=30∘. Side bbb is 18 units long and side aaa is 8 units long. How many nondegenerate triangles are possible?
The shortest distance from CCC to the ray is its perpendicular height.
The radius 8 is less than 9, so the circle falls short.
The known angle is A=30∘A=30^\circA=30∘. Side bbb is 22 units long and side aaa is 11 units long. Which statement is true?
The perpendicular height is 11 units.
The radius equals that height. The only intersection is the foot of the perpendicular, which gives a right angle at BBB.
Angle AAA is 42∘42^\circ42∘ and the lengths aaa and bbb are equal. Which angle BBB gives a valid triangle?
Equal opposite side lengths give the sine equation with candidates 42∘42^\circ42∘ and 138∘138^\circ138∘. The supplement uses the whole angle sum with AAA, leaving zero for CCC.
The known angle AAA is right. Its opposite side aaa is 21 cm and the given adjacent side bbb is 20 cm. How many valid triangles fit?
The circle radius 21 exceeds the distance 20 from its center to the starting point of the ray. For a right angle at AAA, that distance increases strictly along the ray, giving one crossing.
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