Knowing Essential Properties of Triangles - SAT Math
Card 0 of 40
If 
and 
are the lengths of two sides of a triangle, which of the following can be the length of the third side?
I. 
II. 
III. 
If 
I.
II.
III.
The third side of a triangle is always greater than the difference of the other two sides and less than the sum of the other two sides. This applies to every side of a triangle. In other words, you can arbitrarily pick any one side to be the “third side,” and then that side must be greater than the difference of the other two and less than the sum of those two.
Here that means that the third side must greater than the difference of 
and 
. Since 
, that means that 
is not an option. It also means that the third side must be less than the sum of 
and 
. Since 
, that rules out 
as an option. You know that the third side must be greater than 
and less than 
: only 
, option II, fits.
The third side of a triangle is always greater than the difference of the other two sides and less than the sum of the other two sides. This applies to every side of a triangle. In other words, you can arbitrarily pick any one side to be the “third side,” and then that side must be greater than the difference of the other two and less than the sum of those two.
Here that means that the third side must greater than the difference of 
Compare your answer with the correct one above
Triangles ABC and BCD are each isosceles. If 
, what is the value of 
?
Triangles ABC and BCD are each isosceles. If 
One extremely helpful tool that you’ll often find in your geometry toolkit is the presence of isosceles triangles. Here, once you’ve filled in 
for angle 
, you should notice that even though there are two angles remaining to solve for within triangle ABC, those two angles each equal the same thing. So since the sum of all three has to be 
, and angle a already accounts for 
, 
has to equal the remaining 
. 
means that 
. And then you get to use the same logic all over again. Within triangle BCD, you know that 
and that the sum of the three angles must be 
. That means that 
, so 
.
One extremely helpful tool that you’ll often find in your geometry toolkit is the presence of isosceles triangles. Here, once you’ve filled in 
Compare your answer with the correct one above
What is the value of 
?
What is the value of 
An important, fundamental rule of triangles is that the sum of the interior angles equals 
degrees. For triangle ABC above, those three angles are expresses as 
, 
, and 
, meaning that the sum of the interior angles is 
. If 
, then you can divide both sides by 
to recognize that 
.
An important, fundamental rule of triangles is that the sum of the interior angles equals 
Compare your answer with the correct one above
What is the area of Triangle XYZ?
What is the area of Triangle XYZ?
Whether you use the Pythagorean Theorem or you quickly recognize that this is a 
triangle, you can solve for the length of side 
.
You can prove that this is a 
triangle because the hypotenuse is twice as long as one of the legs of the triangle. This then fits the side ratio 
, so you can fill in the middle side as 
.
Or you can use Pythagorean Theorem. Since you know side YZ is the hypotenuse, you can set it up as 
. This means that 
, so 
and 
.
Once you've determined that 
, you can calculate the area. The area is half the base times the height, where the base and height form a 
-degree angle. This means that you can use 
and 
. 
.
Whether you use the Pythagorean Theorem or you quickly recognize that this is a 
You can prove that this is a 
Or you can use Pythagorean Theorem. Since you know side YZ is the hypotenuse, you can set it up as 
Once you've determined that 
Compare your answer with the correct one above
In the isosceles triangle EFG above, angle FEG measures 
degrees and side FG measures 
centimeters. What is the length in centimeters, of side EG?
In the isosceles triangle EFG above, angle FEG measures 
There is a rule about isosceles triangles that isn't obvious the first time you see it, but that the SAT likes to test: if an isosceles triangle includes a 
-degree angle, then it must be an equilateral triangle.
You can prove this by testing the cases: if you know that you have an isosceles triangle with a measure of 
for one angle, then you can call your angles 
, 
, and 
. You know that 
must sum to 
(a rule of triangles), and that one of the following cases must be true:
matches 
. if that's true, then your three angles are 
, 
, and 
. Since 
, that means that 
. Here, all three angles are 
.
matches 
. The same as the above.
matches 
. This then means that 
, so 
. If 
, then 
so x = 60, which means that 
. Here, again, all angles are 
.
Since you know that this is an equilateral triangle, all sides have the same measure. That means that all sides are 
.
You can prove this by testing the cases: if you know that you have an isosceles triangle with a measure of 
Since you know that this is an equilateral triangle, all sides have the same measure. That means that all sides are 
Compare your answer with the correct one above
In isosceles triangle XYZ above, angle XYZ measures 
degrees. What is the measure of angle XZY?
In isosceles triangle XYZ above, angle XYZ measures 
This problem blends two important rules related to triangles:
-
The sum of the interior angles of a triangle is 
.
-
In an isosceles triangle, two angles have the same measure.
Here, although you're not explicitly told which angles have the same measure, you can deduce that it must be angles XZY and YXZ - the two angles that do not measure 
degrees. Note that if 
were to be the "match," then you would already have 
degrees assigned to those two angles, but that would violate the 
rule.
Therefore, you know that your three angles are 
, 
, and 
(where 
represents the unknown, matching angle). You can then say that 
, so 
and 
.
This problem blends two important rules related to triangles:
-
The sum of the interior angles of a triangle is
-
In an isosceles triangle, two angles have the same measure.
.Here, although you're not explicitly told which angles have the same measure, you can deduce that it must be angles XZY and YXZ - the two angles that do not measure 
Therefore, you know that your three angles are 
Compare your answer with the correct one above
Isosceles right triangle ABC has an area of 
. What is its perimeter?
Isosceles right triangle ABC has an area of 
One of the most convenient things about isosceles right triangles is that you can use the two shorter sides as the base and the height to find the area, since they're connected by a right angle:
So if you know that 
is the area in an isosceles right triangle, you can use 
to solve for 
as the length of each of the shorter sides. This means that 
, which you can simplify to:
And then solve for 
.
Because this is an isosceles right triangle, the sides will form the ratio 
, meaning that the hypotenuse will measure 
. If you sum the two shorter sides of 
with the hypotenuse of 
, you reach 
.
One of the most convenient things about isosceles right triangles is that you can use the two shorter sides as the base and the height to find the area, since they're connected by a right angle:
So if you know that 
And then solve for 
Because this is an isosceles right triangle, the sides will form the ratio 
Compare your answer with the correct one above
Starting from her home, Mirinda rides her bike in a straight line due south for 
kilometers, then turns and rides in a straight line due west for 
kilometers, at which point she stops. Assuming that she can ride in any direction with no barriers, what is the distance of her shortest route back home?
Starting from her home, Mirinda rides her bike in a straight line due south for 
This problem forces you to visualize a right triangle from the information provided in the story. When Mirinda makes a turn from heading directly south to heading directly west, that is a 
-degree right angle. And the shortest distance to get from her endpoint (
km south and 
km west) back home is a diagonal line that connects the triangle:
So that shortest distance home is the hypotenuse of a right triangle with shorter sides 
and 
.
To solve, you can use the Pythagorean Theorem: 
, where 
and 
are lengths of the shorter sides and 
is the length of the hypotenuse. This then means that:
This simplifies to:
So 
and 
.
Note that 
is a relatively-common Pythagorean triplet and is one you should consider memorizing. You should absolutely memorize 
and 
as side ratios, with 
and 
as much lower priority (as they are tested much less frequently, and you can always use Pythagorean Theorem if you do not know them offhand).
This problem forces you to visualize a right triangle from the information provided in the story. When Mirinda makes a turn from heading directly south to heading directly west, that is a 
So that shortest distance home is the hypotenuse of a right triangle with shorter sides 
To solve, you can use the Pythagorean Theorem: 
This simplifies to:
So 
Note that 
Compare your answer with the correct one above
What is the sum 
?
What is the sum 
The key to this problem is recognizing the two relationships here that must sum to 
degrees: the three angles in a triangle must sum to 
, and supplementary angles (those that are formed by splitting a straight line) must also sum to 
. Using those relationships, you should see that:
In the big triangle (JXZ), two angles are already given as 
and 
, meaning that angle KZY must equal 
, based on the sum of angles in that triangle needing to be 
.
Since that angle KZY is 
, that means that 
, as it has to sum with its supplementary angle to 
.
And using the small triangle to the right, KYZ, if you know that the two bottom angles (KYZ and KZY) are 
and 
, then the third angle (YKZ at the top of that triangle) must account for the remaining 
.
And then since a is supplementary to that 
-degree angle, it has to be 
.
So 
and 
, meaning that the sum is 
.
The key to this problem is recognizing the two relationships here that must sum to 
In the big triangle (JXZ), two angles are already given as 
Since that angle KZY is 
And using the small triangle to the right, KYZ, if you know that the two bottom angles (KYZ and KZY) are 
And then since a is supplementary to that 
So 
Compare your answer with the correct one above
In triangle LMN above what is 
in terms of 
?
In triangle LMN above what is 
One triangle rule you must know is that the sum of a triangle's three angles must equal 
. Here that is important because you're given one angle 
and part of another 
, and then asked to relate two variables together. You therefore know that 
must equal 
. Using that equation:
Your job is to solve for 
in terms of 
. So first combine like terms:
Then subtract 
and 
from both sides to isolate 
:
And you'll see that that is your answer.
One triangle rule you must know is that the sum of a triangle's three angles must equal 
Your job is to solve for 
Then subtract 
And you'll see that that is your answer.
Compare your answer with the correct one above
If and are the lengths of two sides of a triangle, which of the following can be the length of the third side?
I.
II.
III.
If
I.
II.
III.
The third side of a triangle is always greater than the difference of the other two sides and less than the sum of the other two sides. This applies to every side of a triangle. In other words, you can arbitrarily pick any one side to be the “third side,” and then that side must be greater than the difference of the other two and less than the sum of those two.
Here that means that the third side must greater than the difference of and . Since , that means that is not an option. It also means that the third side must be less than the sum of and . Since , that rules out as an option. You know that the third side must be greater than and less than : only , option II, fits.
The third side of a triangle is always greater than the difference of the other two sides and less than the sum of the other two sides. This applies to every side of a triangle. In other words, you can arbitrarily pick any one side to be the “third side,” and then that side must be greater than the difference of the other two and less than the sum of those two.
Here that means that the third side must greater than the difference of
Compare your answer with the correct one above
Triangles ABC and BCD are each isosceles. If , what is the value of ?
Triangles ABC and BCD are each isosceles. If
One extremely helpful tool that you’ll often find in your geometry toolkit is the presence of isosceles triangles. Here, once you’ve filled in for angle , you should notice that even though there are two angles remaining to solve for within triangle ABC, those two angles each equal the same thing. So since the sum of all three has to be , and angle a already accounts for , has to equal the remaining . means that . And then you get to use the same logic all over again. Within triangle BCD, you know that and that the sum of the three angles must be . That means that , so .
One extremely helpful tool that you’ll often find in your geometry toolkit is the presence of isosceles triangles. Here, once you’ve filled in
Compare your answer with the correct one above
What is the value of ?
What is the value of
An important, fundamental rule of triangles is that the sum of the interior angles equals degrees. For triangle ABC above, those three angles are expresses as , , and , meaning that the sum of the interior angles is . If , then you can divide both sides by to recognize that .
An important, fundamental rule of triangles is that the sum of the interior angles equals
Compare your answer with the correct one above
What is the area of Triangle XYZ?
What is the area of Triangle XYZ?
Whether you use the Pythagorean Theorem or you quickly recognize that this is a triangle, you can solve for the length of side .
You can prove that this is a triangle because the hypotenuse is twice as long as one of the legs of the triangle. This then fits the side ratio , so you can fill in the middle side as .
Or you can use Pythagorean Theorem. Since you know side YZ is the hypotenuse, you can set it up as . This means that , so and .
Once you've determined that , you can calculate the area. The area is half the base times the height, where the base and height form a -degree angle. This means that you can use and . .
Whether you use the Pythagorean Theorem or you quickly recognize that this is a
You can prove that this is a
Or you can use Pythagorean Theorem. Since you know side YZ is the hypotenuse, you can set it up as
Once you've determined that
Compare your answer with the correct one above
In the isosceles triangle EFG above, angle FEG measures degrees and side FG measures centimeters. What is the length in centimeters, of side EG?
In the isosceles triangle EFG above, angle FEG measures
There is a rule about isosceles triangles that isn't obvious the first time you see it, but that the SAT likes to test: if an isosceles triangle includes a -degree angle, then it must be an equilateral triangle.
You can prove this by testing the cases: if you know that you have an isosceles triangle with a measure of for one angle, then you can call your angles , , and . You know that must sum to (a rule of triangles), and that one of the following cases must be true:
matches . if that's true, then your three angles are , , and . Since , that means that . Here, all three angles are .
matches . The same as the above.
matches . This then means that , so . If , then so x = 60, which means that . Here, again, all angles are .
Since you know that this is an equilateral triangle, all sides have the same measure. That means that all sides are .
You can prove this by testing the cases: if you know that you have an isosceles triangle with a measure of
Since you know that this is an equilateral triangle, all sides have the same measure. That means that all sides are
Compare your answer with the correct one above
In isosceles triangle XYZ above, angle XYZ measures degrees. What is the measure of angle XZY?
In isosceles triangle XYZ above, angle XYZ measures
This problem blends two important rules related to triangles:
-
The sum of the interior angles of a triangle is .
-
In an isosceles triangle, two angles have the same measure.
Here, although you're not explicitly told which angles have the same measure, you can deduce that it must be angles XZY and YXZ - the two angles that do not measure degrees. Note that if were to be the "match," then you would already have degrees assigned to those two angles, but that would violate the rule.
Therefore, you know that your three angles are , , and (where represents the unknown, matching angle). You can then say that , so and .
This problem blends two important rules related to triangles:
-
The sum of the interior angles of a triangle is
-
In an isosceles triangle, two angles have the same measure.
Here, although you're not explicitly told which angles have the same measure, you can deduce that it must be angles XZY and YXZ - the two angles that do not measure
Therefore, you know that your three angles are
Compare your answer with the correct one above
Isosceles right triangle ABC has an area of . What is its perimeter?
Isosceles right triangle ABC has an area of
One of the most convenient things about isosceles right triangles is that you can use the two shorter sides as the base and the height to find the area, since they're connected by a right angle:
So if you know that is the area in an isosceles right triangle, you can use to solve for as the length of each of the shorter sides. This means that , which you can simplify to:
And then solve for .
Because this is an isosceles right triangle, the sides will form the ratio , meaning that the hypotenuse will measure . If you sum the two shorter sides of with the hypotenuse of , you reach .
One of the most convenient things about isosceles right triangles is that you can use the two shorter sides as the base and the height to find the area, since they're connected by a right angle:
So if you know that
And then solve for
Because this is an isosceles right triangle, the sides will form the ratio
Compare your answer with the correct one above
Starting from her home, Mirinda rides her bike in a straight line due south for kilometers, then turns and rides in a straight line due west for kilometers, at which point she stops. Assuming that she can ride in any direction with no barriers, what is the distance of her shortest route back home?
Starting from her home, Mirinda rides her bike in a straight line due south for
This problem forces you to visualize a right triangle from the information provided in the story. When Mirinda makes a turn from heading directly south to heading directly west, that is a -degree right angle. And the shortest distance to get from her endpoint ( km south and km west) back home is a diagonal line that connects the triangle:
So that shortest distance home is the hypotenuse of a right triangle with shorter sides and .
To solve, you can use the Pythagorean Theorem: , where and are lengths of the shorter sides and is the length of the hypotenuse. This then means that:
This simplifies to:
So and .
Note that is a relatively-common Pythagorean triplet and is one you should consider memorizing. You should absolutely memorize and as side ratios, with and as much lower priority (as they are tested much less frequently, and you can always use Pythagorean Theorem if you do not know them offhand).
This problem forces you to visualize a right triangle from the information provided in the story. When Mirinda makes a turn from heading directly south to heading directly west, that is a
So that shortest distance home is the hypotenuse of a right triangle with shorter sides
To solve, you can use the Pythagorean Theorem:
This simplifies to:
So
Note that
Compare your answer with the correct one above
What is the sum ?
What is the sum
The key to this problem is recognizing the two relationships here that must sum to degrees: the three angles in a triangle must sum to , and supplementary angles (those that are formed by splitting a straight line) must also sum to . Using those relationships, you should see that:
In the big triangle (JXZ), two angles are already given as and , meaning that angle KZY must equal , based on the sum of angles in that triangle needing to be .
Since that angle KZY is , that means that , as it has to sum with its supplementary angle to .
And using the small triangle to the right, KYZ, if you know that the two bottom angles (KYZ and KZY) are and , then the third angle (YKZ at the top of that triangle) must account for the remaining .
And then since a is supplementary to that -degree angle, it has to be .
So and , meaning that the sum is .
The key to this problem is recognizing the two relationships here that must sum to
In the big triangle (JXZ), two angles are already given as
Since that angle KZY is
And using the small triangle to the right, KYZ, if you know that the two bottom angles (KYZ and KZY) are
And then since a is supplementary to that
So
Compare your answer with the correct one above
In triangle LMN above what is in terms of ?
In triangle LMN above what is
One triangle rule you must know is that the sum of a triangle's three angles must equal . Here that is important because you're given one angle and part of another , and then asked to relate two variables together. You therefore know that must equal . Using that equation:
Your job is to solve for in terms of . So first combine like terms:
Then subtract and from both sides to isolate :
And you'll see that that is your answer.
One triangle rule you must know is that the sum of a triangle's three angles must equal
Your job is to solve for
Then subtract
And you'll see that that is your answer.
Compare your answer with the correct one above