Card 0 of 496
In isosceles triangle ABC, the measure of angle A is 50 degrees. Which is NOT a possible measure for angle B?
If angle A is one of the base angles, then the other base angle must measure 50 degrees. Since 50 + 50 + x = 180 means x = 80, the vertex angle must measure 80 degrees.
If angle A is the vertex angle, the two base angles must be equal. Since 50 + x + x = 180 means x = 65, the two base angles must measure 65 degrees.
The only number given that is not possible is 95 degrees.
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Let the three angles of a triangle measure ,
, and
.
Which of the following expressions is equal to ?
The sum of the measures of the angles of a triangle is , so simplify and solve for
in the equation:
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The angles of a triangle measure ,
, and
. Give
in terms of
.
The sum of the measures of three angles of a triangle is , so we can set up the equation:
We can simplify and solve for :
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Which of the following is true about a triangle with two angles that measure each?
The measures of the angles of a triangle total , so if two angles measure
and we call
the measure of the third, then
This makes the triangle obtuse.
Also, since the triangle has two congruent angles (the angles), the triangle is also isosceles.
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You are given two triangles, and
.
,
is an acute angle, and
is a right angle.
Which quantity is greater?
(a)
(b)
We invoke the SAS Inequality Theorem, which states that, given two triangles and
, with
,
( the included angles), then
- that is, the side opposite the greater angle has the greater length. Since
is an acute angle, and
is a right angle, we have just this situation. This makes (b) the greater.
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Refer to the above figure. Which is the greater quantity?
(a)
(b)
The Triangle Exterior-Angle Theorem states that the measure of an exterior angle is equal to the sum of its remote interior angles. Therefore,
,
making the quantities equal.
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Note: Figure NOT drawn to scale.
Refer to the above figure. Which is the greater quantity?
(a)
(b)
(a) The measures of the angles of a linear pair total 180, so:
(b) The Triangle Exterior-Angle Theorem states that the measure of an exterior angle is equal to the sum of its remote interior angles. Therefore, .
Therefore (a) is the greater quantity.
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Note: Figure NOT drawn to scale.
Refer to the above figure. Which is the greater quantity?
(a)
(b)
The two angles at bottom are marked as congruent. Each of these two angles forms a linear pair with a angle, so it is supplementary to that angle, making its measure
. Therefore, the other marked angle also measures
.
The sum of the measures of the interior angles of a triangle is , so:
The quantities are equal.
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is equilateral;
is isosceles
Which is the greater quantity?
(a)
(b)
is equilateral, so
.
In , we are given that
.
Since the triangles have two pair of congruent sides, the third side with the greater length is opposite the angle of greater measure. Therefore,
.
Since is an angle of an equilateral triangle, its measure is
, so
.
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Which is the greater quantity?
(a)
(b)
Corresponding angles of similar triangles are congruent, so, since , it follows that
By similarity, and
, and we are given that
, so
Also,
,
and .
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Refer to the above figure. Which is the greater quantity?
(a)
(b)
Extend as seen in the figure below:
The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles; specifically,
,
and
However, , so, by substitution,
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Given: .
. Which is the greater quantity?
(a)
(b)
Below is the referenced triangle along with , an equilateral triangle with sides of length 10:
As an angle of an equilateral triangle, has measure
. Applying the Side-Side-Side Inequality Theorem, since
,
, and
, it follows that
, so
.
Also, since , by the Isosceles Triangle Theorem,
. Since
, and the sum of the measures of the angles of a triangle is
, it follows that
Substituting and solving:
.
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Consider and
with
.
Which is the greater quantity?
(a)
(b)
, so, by the Side-Side-Side Principle, since there are three pairs of congruent corresponding sides between the triangles, we can say they are congruent - that is,
.
Corresponding angles of congruent sides are congruent, so .
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Given and
with
Which is the greater quantity?
(a)
(b)
Examine the diagram below, in which two triangles matching the given descriptions have been superimposed.
Note that and
. The two question marks need to be replaced by
and
. No matter how you place these two points,
. However, with one replacement,
; with the other replacement,
. Therefore, the information given is insufficient to answer the question.
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Which is the greater quantity?
(a)
(b)
, so by definition, the sides are in proportion. Therefore,
.
Substitute:
, so (a) is greater.
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Which is the greater quantity?
(a)
(b)
, so by definition, the sides are in proportion.
(a)
Substitute and solve for :
(b)
Substitute and solve for :
The two are equal.
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Triangle B has a height that is twice that of Triangle A and a base that is one-half that of Triangle A. Which is the greater quantity?
(a) The area of Triangle A
(b) The area of Triangle B
Let and
be the base and height of Triangle A. Then the base and height of Triangle B are
and
, respectively.
(a) The area of Triangle A is .
(b) The area of Triangle B is .
Therefore, (a) and (b) are equal.
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Two triangles on the coordinate plane have a vertex at the origin and a vertex at , where
.
Triangle A has its third vertex at .
Triangle B has its third vertex at .
Which is the greater quantity?
(a) The area of Triangle A
(b) The area of Triangle B
(a) Triangle A has as its base the horizontal segment connecting and
, the length of which is 10. Its (vertical) altitude is the segment from
to this horizontal segment, which is part of the
-axis; its height is therefore the
-coordinate of this point, or
.
The area of Triangle A is therefore
(b) Triangle B has as its base the vertical segment connecting and
, the length of which is 10. Its (horizontal) altitude is the segment from
to this vertical segment, which is part of the
-axis; its height is therefore the
-coordinate of this point, or
.
The area of Triangle B is therefore
, so
. (b), the area of Triangle B, is greater.
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A triangle has sides 30, 40, and 80. Give its area.
By the Triangle Inequality Theorem, the sum of the lengths of the two shorter sides of a triangle must exceed the length of its longest side. However,
;
Therefore, this triangle cannot exist, and the correct answer is "none of the other responses is correct".
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The above depicts Square ;
, and
are the midpoints of
,
, and
, respectively. Which is the greater quantity?
(a) The area of
(b) The area of
For the sake of simplicity, assume that the square has sidelength 2; this reasoning is independent of the actual sidelength.
Since ,
, and
are the midpoints of their respective sides,
, as shown in this diagram.
The area of , it being a right triangle, is half the product of the lengths of its legs:
The area of is half the product of the length of a base and the height. Using
as the base, and
as an altitude:
The two triangles have the same area.
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