Triangles - GED Math
Card 0 of 565
Refer to the above diagram. 
is equilateral; 
. How many of the following statements must be true?
I) 
bisects 
II) 
bisects 
III) 
Refer to the above diagram.
I)
II)
III)
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Since corresponding parts of congruent triangles are congruent, it follows that 
. Therefore, 
, by definition, bisects 
, and the first statement is true. A bisector of an angle of an equilateral triangle is also the perpendicular bisector of the opposite side, so the other two statements are immediate consequences.
Since corresponding parts of congruent triangles are congruent, it follows that
Refer to the above diagram. Which of the following expressions gives the length of 
?
Refer to the above diagram. Which of the following expressions gives the length of
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By the Pythagorean Theorem,
By the Pythagorean Theorem,
A right triangle has hypotenuse with length 20 and a leg of length 9. The length of the other leg is:
A right triangle has hypotenuse with length 20 and a leg of length 9. The length of the other leg is:
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By the Pythagorean Theorem, if we let 
be the length of the hypotenuse, or longest side, of a right triangle, and 
and 
be the lengths of the legs, the relation is
Set 
and 
, and solve for 
:
Square the numbers - that is, multiply them by themselves:
Subtract 81 from both sides to isolate 
:
To find out what integers 
falls between, it is necessary to find the perfect square integers that flank 319. We can see by trial and error that
,
so
The length of the second leg thus falls between 17 and 18.
By the Pythagorean Theorem, if we let
Set
Square the numbers - that is, multiply them by themselves:
Subtract 81 from both sides to isolate
To find out what integers
so
The length of the second leg thus falls between 17 and 18.
Which of the following can be the measures of the three angles of an acute isosceles triangle?
Which of the following can be the measures of the three angles of an acute isosceles triangle?
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For the triangle to be acute, all three angles must measure less than 
. We can eliminate 
and 
for this reason.
In an isosceles triangle, at least two angles are congruent, so we can eliminate 
.
The degree measures of the three angles of a triangle must total 180, so, since 
, we can eliminate 
.
is correct.
For the triangle to be acute, all three angles must measure less than
In an isosceles triangle, at least two angles are congruent, so we can eliminate
The degree measures of the three angles of a triangle must total 180, so, since
Which of the following describes a triangle with sides of length 9 feet, 3 yards, and 90 inches?
Which of the following describes a triangle with sides of length 9 feet, 3 yards, and 90 inches?
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One yard is equal to three feet, and one foot is equal to twelve inches. Therefore, 9 feet is equal to 
inches, and 3 yards is equal to 
inches. The triangle has sides of measure 90 inches, 108 inches, and 108 inches. Exactly two sides are of equal measure, so it is isosceles but not equilateral.
One yard is equal to three feet, and one foot is equal to twelve inches. Therefore, 9 feet is equal to
An exterior angle of an isosceles triangle measures 
. What is the least measure of any of the three angles of the triangle?
An exterior angle of an isosceles triangle measures
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The triangle has an exterior angle of 
, so it has an interior angle of 
.
Since this is an obtuse angle, its other two angles must be acute. By the Isosceles Triangle Theorem, an isosceles triangle must have two congruent angles - the acute angles are those. Since the sum of their measures is the same as their remote exterior angle - 
- each has measure 
.This is the correct response.
The triangle has an exterior angle of
Since this is an obtuse angle, its other two angles must be acute. By the Isosceles Triangle Theorem, an isosceles triangle must have two congruent angles - the acute angles are those. Since the sum of their measures is the same as their remote exterior angle -
Note: Figure NOT drawn to scale.
Refer to the above triangle. Evaluate 
.
Note: Figure NOT drawn to scale.
Refer to the above triangle. Evaluate
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The degree measures of a triangle total 
, so
The degree measures of a triangle total
Note: Figure NOT drawn to scale.
Refer to the above figure. Evaluate 
.
Note: Figure NOT drawn to scale.
Refer to the above figure. Evaluate
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The degree measures of the interior angles of a triangle total 
, so, if we let 
be the measure of the unmarked angle, then
Three angles with measures 
together form a straight angle, so
The degree measures of the interior angles of a triangle total
Three angles with measures
Figure drawn to scale.
Refer to the above diagram.
Which of the following is a valid description of 
?
Figure drawn to scale.
Refer to the above diagram.
Which of the following is a valid description of
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One of the angles of 
- namely, 
- can be seen to be an obtuse angle, as it is wider than a right angle. This makes 
, by definition, an obtuse triangle.
One of the angles of
Refer to the above diagram.
Which of the following is a valid description of 
?
Refer to the above diagram.
Which of the following is a valid description of
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One of the angles of 
- namely, 
- is marked as a right angle. This makes 
, by definition, a right triangle.
One of the angles of
Which of the following follows from the fact that 
?
Which of the following follows from the fact that
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A congruency statement about two triangles implies nothing about the relationship between two angles of one of the triangles, so 
is not correct.
Also, letters in the same position between the two triangles refer to corresponding - and subsequently, congruent - angles. Therefore, implies that:
Of the given choices, only 
is a consequence. It is the correct response.
A congruency statement about two triangles implies nothing about the relationship between two angles of one of the triangles, so
Also, letters in the same position between the two triangles refer to corresponding - and subsequently, congruent - angles. Therefore,
Of the given choices, only
An exterior angle of an isosceles triangle measures 
. What is the greatest measure of any of the three angles of the triangle?
An exterior angle of an isosceles triangle measures
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The triangle has an exterior angle of 
, so it has an interior angle of 
.
Since this is an obtuse angle, its other two angles must be acute. Therefore, this angle is the one of greatest measure.
The triangle has an exterior angle of
Since this is an obtuse angle, its other two angles must be acute. Therefore, this angle is the one of greatest measure.
An exterior angle of an isosceles triangle measures 
. What is the greatest measure of any of the three angles of the triangle?
An exterior angle of an isosceles triangle measures
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The triangle has an exterior angle of 
, so it has an interior angle of 
. By the Isosceles Triangle Theorem, an isosceles triangle must have two congruent angles; there are two possible scenarios that fit this criterion:
Case 1: Two angles have measure 
. The third angle will have measure
.
will be the greatest of the angle measures.
Case 2: One angle has measure 
and the others are congruent. Their common measure will be
.
will be the greatest of the angle measures.
The given information is therefore inconclusive.
The triangle has an exterior angle of
Case 1: Two angles have measure
Case 2: One angle has measure
The given information is therefore inconclusive.
An exterior angle of an isosceles triangle measures 
. What is the least measure of any of the three angles of the triangle?
An exterior angle of an isosceles triangle measures
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The triangle has an exterior angle of 
, so it has an interior angle of 
. By the Isosceles Triangle Theorem, an isosceles triangle must have two congruent angles; there are two possible scenarios that fit this criterion:
Case 1: Two angles have measure 
. The third angle will have measure
.
will be the least of the angle measures.
Case 2: One angle has measure 
and the others are congruent. Their common measure will be
.
will be the least of the angle measures.
In both cases, the least of the degree measures of the angles will be 
.
The triangle has an exterior angle of
Case 1: Two angles have measure
Case 2: One angle has measure
In both cases, the least of the degree measures of the angles will be
Which of the following could give the measures of the three angles of a triangle?
Which of the following could give the measures of the three angles of a triangle?
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The degree measures of a triangle must total 
, so we find the sum of the degree measures in each choice.
:
:
:
:
This is the correct choice.
The degree measures of a triangle must total
This is the correct choice.
Which of the following cannot be true of equilateral triangle 
?
Which of the following cannot be true of equilateral triangle
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Every equilateral triangle has three congruent angles - all of which have measure 
- as a result of the Isosceles Triangle Theorem. Therefore, the statements:
is an acute triangle
immediately follow.
However, if 
and 
are complementary, then the sum of their measures is 
; since each angle measures 
, then their measures add up to 
. This is the correct choice.
Every equilateral triangle has three congruent angles - all of which have measure
immediately follow.
However, if
Which of the following could be the measures of two angles of a scalene triangle?
Which of the following could be the measures of two angles of a scalene triangle?
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By the Isosceles Triangle Theorem, angles opposite congruent sides of a triangle are congruent. Therefore, a scalene triangle, having three noncongrent sides, must have three noncongruent angles.
can be eliminated immediately.
For the other three choices, we need to find the measure of the third angle using the fact that the degree measures of the angles of a triangle must total 
.
:
The third angle measure is 
.
This triangle has two 
angles and can be eliminated.
:
The third angle measure is 
.
This triangle has two 
angles and can be eliminated.
:
The third angle measure is 
.
This triangle has three angles of different measure, making it scalene. This is the correct choice.
By the Isosceles Triangle Theorem, angles opposite congruent sides of a triangle are congruent. Therefore, a scalene triangle, having three noncongrent sides, must have three noncongruent angles.
For the other three choices, we need to find the measure of the third angle using the fact that the degree measures of the angles of a triangle must total
The third angle measure is
This triangle has two
The third angle measure is
This triangle has two
The third angle measure is
This triangle has three angles of different measure, making it scalene. This is the correct choice.
For a triangle, suppose a given interior angle is 
degrees and the other two angles are 
. What is the value of 
?
For a triangle, suppose a given interior angle is
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The sum of the three interior angles of a triangle is 180 degrees.
Set up an equation such that all three angles are equal to 180 degrees.
Combine like-terms.
Subtract 34 on both sides.
Divide by 6 on both sides.
The answer is: 
The sum of the three interior angles of a triangle is 180 degrees.
Set up an equation such that all three angles are equal to 180 degrees.
Combine like-terms.
Subtract 34 on both sides.
Divide by 6 on both sides.
The answer is:
If a triangle is isosceles, and the vertex angle is 100 degrees, what must be another angle?
If a triangle is isosceles, and the vertex angle is 100 degrees, what must be another angle?
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The isosceles triangle has two congruent sides and two congruent angles.
The interior angles will sum to 180 degrees.
Let the unknown angles be 
. These angles are congruent to each other because the triangle is isosceles.
Divide by 2 on both sides.
The answer is: 
The isosceles triangle has two congruent sides and two congruent angles.
The interior angles will sum to 180 degrees.
Let the unknown angles be
Divide by 2 on both sides.
The answer is:
Figure NOT drawn to scale.
Refer to the above diagram. Evaluate 
Figure NOT drawn to scale.
Refer to the above diagram. Evaluate
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is indicated to be isosceles, with 
, so, by the isosceles triangle theorem,
.
Since the measures of the interior angles of a triangle total 
,
Substitute 
for 
and 
for 
, then solve for 
:
Subtract 
from both sides:
Multiply both sides by 
:
is an exterior angle of 
; its measure is the sum of those of its remote interior angle, 
and 
, so
Substitute and add:
.
Since the measures of the interior angles of a triangle total
Substitute
Subtract
Multiply both sides by
Substitute and add: