12 multiple-choice questions, progressively harder.
Triangles KLM and RST have
They also have
Which remaining sides are opposite the included angles K and R, respectively?
Solution
Correct answer: C
The matched sides meet at K and R. The side opposite K does not touch K, so it is LM. The side opposite R is ST.
LM and ST are the third sides whose lengths a hinge comparison would order.
The two triangles have
Also,
The included angles at A and D measure 35 and 75 degrees, respectively. Which comparison follows?
Correct answer: A
Both side pairs match. The larger included angle is at D.
The hinge theorem puts the longer third side opposite D.
Two triangles have matching arms of lengths 5 and 8 units. Their third sides measure 7 and 10 units. Which included angle is larger?
Correct answer: D
The two corresponding arm pairs match.
The converse hinge theorem matches the longer third side to the larger included angle.
Their included angles are equal. What follows about BC and EF?
Correct answer: B
The two side pairs and included angles match. SAS makes the triangles congruent.
Corresponding third sides therefore match.
You know
and that angle BAC is larger than angle EDF. Which additional equality supplies the missing hypothesis for a hinge comparison of BC and EF?
The included angles are at A and D. The sides surrounding them are AB with AC, and DE with DF.
AB already matches DE; the missing pair is AC with DF.
The given angles are ABC and DEF. Are those the included angles for the matched pairs?
The matched sides meet at A in the first triangle and D in the second.
The given angles have vertices B and E, so their comparison cannot be inserted directly into the hinge theorem.
Triangles ABC and DEF have
Angle BAC measures 38 degrees, and angle FDE measures 71 degrees. Which third side is longer?
The correspondence is reversed along the second triangle: AB matches DF, and AC matches DE. These pairs surround angles BAC and FDE.
The larger included angle is at D. The hinge theorem therefore gives
Angle BAC is smaller than angle EDF. What follows?
The side pairs match. Apply the hinge theorem with triangle DEF first.
Its larger opening faces the longer third side.
In a hinge proof, the copied lengths satisfy
The side AP is common, and AP bisects angle CAX. Which criterion proves triangles ACP and AXP congruent?
AC and AX supply one equal pair, while AP matches itself.
The bisector supplies equal angles between those pairs. Those are the SAS hypotheses.
Which proposed included angle cannot belong to a nondegenerate triangle?
Every triangle angle is positive and less than 180 degrees.
An opening of 180 degrees places the two arms on opposite rays, producing a flat arrangement.
Two triangles have the same arm lengths of 3 and 7 units. Their included angles are 40 and 70 degrees. What does the hinge theorem determine?
The arm pairs match, and the opening of 70 degrees is larger.
Its opposite third side is longer. The hinge theorem gives a strict order; it does not compute the lengths.
Two triangles satisfy all hinge hypotheses. One drawing is rotated so its included angle points downward. How does the comparison change?
A rotation preserves every side length and angle measure.
The corresponding pairs and the included-angle order remain the same, so the hinge conclusion remains the same.
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