12 multiple-choice questions, progressively harder.
The two triangles have
Also,
The included angle at A is larger. The third sides satisfy
Which necessary restriction follows?
Solution
Correct answer: B
The hinge theorem makes BC longer than EF.
Subtract 3x, then add 3.
Divide by positive 4.
The third side BC is shorter than EF. The included angles, in degrees, are
Which necessary restriction follows from their order?
Correct answer: D
The converse hinge theorem makes the first included angle smaller.
Subtract 8, then divide by positive 4.
Triangles ABC and ADC share AC and have
Angle BAC measures 46 degrees; angle DAC measures 71 degrees. Which relation follows?
Correct answer: A
The marked arms match, and AC supplies the common second arm.
Triangle ADC has the larger included angle, so its third side is longer.
Two triangles have matching arms of lengths 6 and 10 units. The second third side is 11 units. The first has the larger included angle and third side x. Which is the full possible range of x?
Correct answer: C
The hinge theorem gives the lower comparison.
The triangle inequality bounds the first third side below the sum of its arms.
The difference bound is 4 and is already satisfied. Every length in the remaining interval forms a triangle and has the larger opening by the converse.
Their included angles are equal. The third sides are
What is u?
Equal included angles complete SAS congruence, so the third sides match.
Subtract 3u and subtract 1.
Divide by 5.
The proposed matching arms have lengths
The second pair AC and DF both measure 8 units. The included angle at A is larger. Which value and third-side comparison are justified when both pairs match?
v=9v=9v=9
BC<EFBC<EFBC<EF
BC>EFBC>EFBC>EF
v=3v=3v=3
v=15v=15v=15
BC=EFBC=EFBC=EF
Match the first pair before using the angle comparison.
Subtract v and subtract 3.
Both side pairs now match, so the larger included angle at A gives
Triangles RST and UVW have
Angles R and U measure 52 and 68 degrees. Which comparison is correct?
The equal pairs meet at R and U, so those are the included angles.
The opening at U is larger. Its opposite side VW is longer.
In a hinge construction, P is between B and C and
The lengths BP and PC are 5 and 3 units. B, P, and X are noncollinear. Which conclusion must hold?
The triangle inequality applies to the nondegenerate triangle BPX.
Use PX equal to PC, whose length is 3.
Thus
The third side BC is longer than EF. The included angle at A measures (3z+12)∘(3z+12)^\circ(3z+12)∘, and the included angle at D measures 72∘72^\circ72∘. What is the full range permitted by the angle order and nondegeneracy?
The converse gives the included-angle order.
Solving gives
The first angle must be less than 180 degrees.
This gives z less than 56. Positivity is already satisfied.
Three triangles each have corresponding arms of lengths 8 and 5. Their included angles are 40, 90, and 65 degrees, and their third sides are a, b, and c, respectively. Which order follows?
a<ca<ca<c
c<bc<bc<b
a<ba<ba<b
b<cb<cb<c
c<ac<ac<a
The arm pairs match for every pair of triangles. Order the openings: 40 degrees, then 65 degrees, then 90 degrees.
The hinge theorem gives the same order for the third sides.
Two triangles have the same corresponding arms of lengths 10 and 6 units. Their third sides measure 13 and 9 units. What can you conclude about the included angle opposite 13?
Both side triples form triangles, and the two arm pairs match.
The converse hinge theorem compares the included angles opposite the unequal third sides. It makes no comparison of every angle in the triangles.
Triangles ABD and ACD share AD and have
Ray AD lies inside angle BAC, whose measure is 110 degrees. Angle DAC measures 40 degrees. Compare BD and CD.
The first included angle is the remaining part of angle BAC.
It therefore measures 70 degrees, larger than angle DAC.
The matched arms and shared side let us apply the hinge theorem.
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