12 multiple-choice questions, progressively harder.
Triangle ABC has angle A of 46 degrees and angle B of 59 degrees. Which side is longest?
Solution
Correct answer: C
Find the missing angle first.
C is the largest angle, so its opposite side AB is longest.
An existing triangle ABC has AB of 12 units, BC of 9 units, and CA of 15 units. Order its angles from smallest to largest.
Correct answer: A
The sides, from shortest to longest, are BC, AB, and AC.
Their opposite angles are A, C, and B, respectively.
An existing triangle ABC has angle B larger than angle C. Its lengths satisfy
Which restriction follows?
Correct answer: D
AC faces the larger angle, so
Subtract x and 2 to obtain
Divide by 2:
An existing triangle ABC has angle B larger than angle C. Side AC is 7 units long and AB has length 3x−23x-23x−2. Which is the sharp upper bound from the angle comparison?
Correct answer: B
AC must exceed AB, since it faces the larger angle.
Add 2 and divide by 3:
Equality is excluded because the opposite angles are unequal.
In triangle ABC, BC is extended through C to D. The exterior angle ACD is 118 degrees. The remote angles are
Which side is longest?
The remote angles add to the exterior angle.
The interior angles A, B, and C are 75, 43, and 62 degrees, respectively. A is largest, so BC is longest.
Triangle ABC is right at B. Its acute angles satisfy
Which leg is shorter?
The acute angles sum to 90 degrees.
Angle A is 50 degrees and angle C is 40 degrees. AB faces the smaller angle C, so AB is the shorter leg.
Triangle ABC has equal sides AB and AC. The angle between them, at A, is 100 degrees. Which side is longest?
The base angles at B and C are equal.
Angle C is also 40 degrees, so the largest angle is A. Its opposite side BC is longest.
P is outside a line, H is its perpendicular foot, and Q is a different point on the line. The lengths are
Which inequality is equivalent to PQ being longer than PH?
The slanted segment PQ is the hypotenuse.
Subtract 4, then divide by 2:
Equality would make a slanted segment as short as the perpendicular.
Three points A, B, and C lie on a line, and P is outside the line. Their distances from P are 12, 9, and 7 units, respectively. What does this establish?
The given distances put C ahead of A and B.
They do not compare C with every other point on the line. A closer perpendicular foot may lie somewhere else.
In triangle ABC, BC is extended through C to D. The measures are
Which is the sharp lower bound on x forced by the exterior angle inequality?
The exterior angle exceeds remote angle A.
Subtract x and 25:
At 15, the other remote angle would have zero measure.
In triangle ABC, BC is longer than AC. In triangle BCD, BC is longer than BD. Which conclusion about AC and BD follows?
Both AC and BD are shorter than BC.
Two lengths can both be below a third without having a determined order relative to each other.
Point P lies on line ℓ\ellℓ. What is its shortest distance to the line?
The point P itself is available as an endpoint on the line.
Its distance from itself is zero. Every different point on the line has positive distance from P, so the minimum is zero.
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