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Triangles – Answers
1. D
2. C
3. C
4. D
5. D
6. B
7. C
8. D
9. A
10. B
11. D
12. B
13. B
14. B
15. B
Triangles - Explanations
1.
An angle bisector is a line that passes through an angle, dividing it into two smaller angles of equal measure.
2.
The sum of the measures of the angles of a triangle always equals 180°. Use this fact to find the measure of
F.
180° - 62° - 57° = 61°
The Side-Angle Inequality states that the side opposite the largest angle is the longest side and that the side
opposite the smallest angle is the shortest side.
Since angle G is the smallest angle, side EF is the shortest side.
So, side EF must be shorter than each of the other two sides, but greater than their difference.
Therefore, side EF must be between 1 cm and 15 cm.
3.
The Side-Angle Inequality states that the side opposite the largest angle is the longest side and that the side
opposite the smallest angle is the shortest side.
Since side UW is the longest side,
4.
V is the largest angle.
The Triangle Inequality states that for any triangle, the sum of the lengths of any two sides must be greater than
the third side.
The best way to check if three lengths will make a triangle is to add the two smallest sides and then compare this
sum to the third side.
15 + 25 ? 28
40 > 28
Since the sum of the two smallest sides is greater than the third side, the lengths do make a triangle.
To find the smallest angle, sketch a basic triangle and label the angles and side lengths. (It does not have to be
drawn to scale.)
28
The Side-Angle Inequality states that the side opposite the largest angle is the longest side and that the side
opposite the smallest angle is the shortest side.
Since side HI is the shortest side,
J is the smallest angle.
5.
A perpendicular bisector is a line or segment that passes through the midpoint of a side of a triangle and is
perpendicular to that side.
6.
The Side-Angle Inequality states that the side opposite the largest angle is the longest side and that the side
opposite the smallest angle is the shortest side.
Since side JK is the shortest side, angle L is the smallest angle.
Since side JL is the longest side, angle K is the largest angle.
Therefore, the angles listed from greatest to least are
7.
K,
J,
L.
In the given figure, x is the measure of an angle of GJK. Since the sum of the measures of the angles of a
triangle is 180°, set up an equation and solve for m x.
m
x+m J+m K
m x + 24° + 96°
m x
m x
=
=
=
=
180°
180°
180° - 120°
60°
In the given figure, y is the measure of an angle of FHL. Since the sum of the measures of the angles of a
triangle is 180°, set up an equation and solve for m y.
m
y+m F+m L
m y + 62° + 72°
m y
m y
=
=
=
=
180°
180°
180° - 134°
46°
Since m
x = 60° and m
y = 46°, set up an equation and solve for m
m
8.
x+m y+m
60° + 46° + m
m
m
Since the triangle is an isosceles triangle, m
Therefore, m
z=
z=
z=
z=
C equals m
z.
180°
180°
180° - 106°
74°
A.
C equals 29°.
9.
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle is greater than
the length of the third side.
10.
The sum of the measures of the angles of a triangle always equals 180°. Use this fact to find the measure of
L + M.
180° - 42° = 138°
Since sides LN and MN are congruent,
L and
M are also congruent. Thus, divide 138° by 2.
138° ÷ 2 = 69°
So, the measures of the angles in the triangle are listed below.
N = 42°,
L = 69°,
M = 69°
The Side-Angle Inequality states that the side opposite the largest angle is the longest side and that the side
opposite the smallest angle is the shortest side.
Since angle N is the smallest angle, side LM is the shortest side.
11.
The sum of the measures of the angles of a triangle is 180°.
m
12.
A + 29° + 90° = 180°
m A = 61°
The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the
third side.
So, the length of each side of the triangle must be greater than the difference and less than the sum of the other
two side lengths.
Since 7 yd + 4 yd < 13 yd, a triangle cannot be formed from the three beams.
13.
An equilateral triangle has three sides of equal length and three 60° angles.
Therefore, the length of ST is 5.1 mm.
14.
The sum of the measures of the angles of a triangle always equals 180°. Use this fact to find the measure of
G.
180° - 74° - 33° = 73°
The Side-Angle Inequality states that the side opposite the largest angle is the longest side and that the side
opposite the smallest angle is the shortest side.
Since angle F is the smallest angle, side GH is the shortest side.
Since angle H is the largest angle, side FG is the longest side.
Therefore, the sides listed from longest to shortest are FG, HF, GH.
15.
The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the
third side.
So, the length of the third side, y, must be greater than the difference and less than the sum of the two given
side lengths.
z-x<y< z+x
41.6 cm - 37.6 cm < y < 41.6 cm + 37.6 cm
4 cm < y < 79.2 cm