How to find if right triangles are similar - PSAT Math
Card 0 of 35
In the figure above, line segments DC and AB are parallel. What is the perimeter of quadrilateral ABCD?
In the figure above, line segments DC and AB are parallel. What is the perimeter of quadrilateral ABCD?
Tap to see back →
Because DC and AB are parallel, this means that angles CDB and ABD are equal. When two parallel lines are cut by a transversal line, alternate interior angles (such as CDB and ABD) are congruent.
Now, we can show that triangles ABD and BDC are similar. Both ABD and BDC are right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that angles CDB and ABD are congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus triangles ABD and BDC are similar triangles.
We can use the similarity between triangles ABD and BDC to find the lengths of BC and CD. The length of BC is proportional to the length of AD, and the length of CD is proportional to the length of DB, because these sides correspond.
We don’t know the length of DB, but we can find it using the Pythagorean Theorem. Let a, b, and c represent the lengths of AD, AB, and BD respectively. According to the Pythagorean Theorem:
_a_2 + _b_2 = _c_2
152 + 202 = _c_2
625 = _c_2
c = 25
The length of BD is 25.
We now have what we need to find the perimeter of the quadrilateral.
Perimeter = sum of the lengths of AB, BC, CD, and DA.
Perimeter = 20 + 18.75 + 31.25 + 15 = 85
The answer is 85.
Because DC and AB are parallel, this means that angles CDB and ABD are equal. When two parallel lines are cut by a transversal line, alternate interior angles (such as CDB and ABD) are congruent.
Now, we can show that triangles ABD and BDC are similar. Both ABD and BDC are right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that angles CDB and ABD are congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus triangles ABD and BDC are similar triangles.
We can use the similarity between triangles ABD and BDC to find the lengths of BC and CD. The length of BC is proportional to the length of AD, and the length of CD is proportional to the length of DB, because these sides correspond.
We don’t know the length of DB, but we can find it using the Pythagorean Theorem. Let a, b, and c represent the lengths of AD, AB, and BD respectively. According to the Pythagorean Theorem:
_a_2 + _b_2 = _c_2
152 + 202 = _c_2
625 = _c_2
c = 25
The length of BD is 25.
We now have what we need to find the perimeter of the quadrilateral.
Perimeter = sum of the lengths of AB, BC, CD, and DA.
Perimeter = 20 + 18.75 + 31.25 + 15 = 85
The answer is 85.
and 
is a right angle.
Which angle or angles must be complementary to 
?
I) 
II) 
III) 
IV) 
V) 
Which angle or angles must be complementary to
I)
II)
III)
IV)
V)
Tap to see back →
is a right angle, and, since corresponding angles of similar triangles are congruent, so is 
. A right angle cannot be part of a complementary pair so both can be eliminated.
can be eliminated, since it is congruent to 
; congruent angles are not necessarily complementary.
Since 
is right angle, 
is a right triangle, and 
and 
are its acute angles. That makes 
complementary to 
. Since 
is congruent to 
, it is also complementary to 
.
The correct response is II and V only.
Since
The correct response is II and V only.
Note: Figures NOT drawn to scale.
Refer to the above figure. Given that 
, evaluate 
.
Note: Figures NOT drawn to scale.
Refer to the above figure. Given that
Tap to see back →
By the Pythagorean Theorem, since 
is the hypotenuse of a right triangle with legs 6 and 8, its measure is
.
The similarity ratio of 
to 
is
.
Likewise,
By the Pythagorean Theorem, since
The similarity ratio of
Likewise,
Note: Figure NOT drawn to scale.
Refer to the above figure. Given that 
, give the area of 
.
Note: Figure NOT drawn to scale.
Refer to the above figure. Given that
Tap to see back →
By the Pythagorean Theorem,
The similarity ratio of 
to 
is
,
This can be used to find 
:
The area of 
is therefore
By the Pythagorean Theorem,
The similarity ratio of
This can be used to find
The area of
Refer to the above figure. Given that , give the perimeter of .
Refer to the above figure. Given that
Tap to see back →
By the Pythagorean Theorem,
The similarity ratio of 
to 
is
,
which is subsequently the ratio of the perimeter of 
to that of 
.
The perimeter of 
is
,
so the perimeter of 
can be found using this ratio:
By the Pythagorean Theorem,
The similarity ratio of
which is subsequently the ratio of the perimeter of
The perimeter of
so the perimeter of
In the figure above, line segments DC and AB are parallel. What is the perimeter of quadrilateral ABCD?
In the figure above, line segments DC and AB are parallel. What is the perimeter of quadrilateral ABCD?
Tap to see back →
Because DC and AB are parallel, this means that angles CDB and ABD are equal. When two parallel lines are cut by a transversal line, alternate interior angles (such as CDB and ABD) are congruent.
Now, we can show that triangles ABD and BDC are similar. Both ABD and BDC are right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that angles CDB and ABD are congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus triangles ABD and BDC are similar triangles.
We can use the similarity between triangles ABD and BDC to find the lengths of BC and CD. The length of BC is proportional to the length of AD, and the length of CD is proportional to the length of DB, because these sides correspond.
We don’t know the length of DB, but we can find it using the Pythagorean Theorem. Let a, b, and c represent the lengths of AD, AB, and BD respectively. According to the Pythagorean Theorem:
_a_2 + _b_2 = _c_2
152 + 202 = _c_2
625 = _c_2
c = 25
The length of BD is 25.
We now have what we need to find the perimeter of the quadrilateral.
Perimeter = sum of the lengths of AB, BC, CD, and DA.
Perimeter = 20 + 18.75 + 31.25 + 15 = 85
The answer is 85.
Because DC and AB are parallel, this means that angles CDB and ABD are equal. When two parallel lines are cut by a transversal line, alternate interior angles (such as CDB and ABD) are congruent.
Now, we can show that triangles ABD and BDC are similar. Both ABD and BDC are right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that angles CDB and ABD are congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus triangles ABD and BDC are similar triangles.
We can use the similarity between triangles ABD and BDC to find the lengths of BC and CD. The length of BC is proportional to the length of AD, and the length of CD is proportional to the length of DB, because these sides correspond.
We don’t know the length of DB, but we can find it using the Pythagorean Theorem. Let a, b, and c represent the lengths of AD, AB, and BD respectively. According to the Pythagorean Theorem:
_a_2 + _b_2 = _c_2
152 + 202 = _c_2
625 = _c_2
c = 25
The length of BD is 25.
We now have what we need to find the perimeter of the quadrilateral.
Perimeter = sum of the lengths of AB, BC, CD, and DA.
Perimeter = 20 + 18.75 + 31.25 + 15 = 85
The answer is 85.
and is a right angle.
Which angle or angles must be complementary to ?
I)
II)
III)
IV)
V)
Which angle or angles must be complementary to
I)
II)
III)
IV)
V)
Tap to see back →
is a right angle, and, since corresponding angles of similar triangles are congruent, so is . A right angle cannot be part of a complementary pair so both can be eliminated.
can be eliminated, since it is congruent to ; congruent angles are not necessarily complementary.
Since is right angle, is a right triangle, and and are its acute angles. That makes complementary to . Since is congruent to , it is also complementary to .
The correct response is II and V only.
Since
The correct response is II and V only.
Note: Figures NOT drawn to scale.
Refer to the above figure. Given that , evaluate .
Note: Figures NOT drawn to scale.
Refer to the above figure. Given that
Tap to see back →
By the Pythagorean Theorem, since is the hypotenuse of a right triangle with legs 6 and 8, its measure is
.
The similarity ratio of to is
.
Likewise,
By the Pythagorean Theorem, since
The similarity ratio of
Likewise,
Note: Figure NOT drawn to scale.
Refer to the above figure. Given that , give the area of .
Note: Figure NOT drawn to scale.
Refer to the above figure. Given that
Tap to see back →
By the Pythagorean Theorem,
The similarity ratio of to is
,
This can be used to find :
The area of is therefore
By the Pythagorean Theorem,
The similarity ratio of
This can be used to find
The area of
Refer to the above figure. Given that , give the perimeter of .
Refer to the above figure. Given that
Tap to see back →
By the Pythagorean Theorem,
The similarity ratio of to is
,
which is subsequently the ratio of the perimeter of to that of .
The perimeter of is
,
so the perimeter of can be found using this ratio:
By the Pythagorean Theorem,
The similarity ratio of
which is subsequently the ratio of the perimeter of
The perimeter of
so the perimeter of
In the figure above, line segments DC and AB are parallel. What is the perimeter of quadrilateral ABCD?
In the figure above, line segments DC and AB are parallel. What is the perimeter of quadrilateral ABCD?
Tap to see back →
Because DC and AB are parallel, this means that angles CDB and ABD are equal. When two parallel lines are cut by a transversal line, alternate interior angles (such as CDB and ABD) are congruent.
Now, we can show that triangles ABD and BDC are similar. Both ABD and BDC are right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that angles CDB and ABD are congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus triangles ABD and BDC are similar triangles.
We can use the similarity between triangles ABD and BDC to find the lengths of BC and CD. The length of BC is proportional to the length of AD, and the length of CD is proportional to the length of DB, because these sides correspond.
We don’t know the length of DB, but we can find it using the Pythagorean Theorem. Let a, b, and c represent the lengths of AD, AB, and BD respectively. According to the Pythagorean Theorem:
_a_2 + _b_2 = _c_2
152 + 202 = _c_2
625 = _c_2
c = 25
The length of BD is 25.
We now have what we need to find the perimeter of the quadrilateral.
Perimeter = sum of the lengths of AB, BC, CD, and DA.
Perimeter = 20 + 18.75 + 31.25 + 15 = 85
The answer is 85.
Because DC and AB are parallel, this means that angles CDB and ABD are equal. When two parallel lines are cut by a transversal line, alternate interior angles (such as CDB and ABD) are congruent.
Now, we can show that triangles ABD and BDC are similar. Both ABD and BDC are right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that angles CDB and ABD are congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus triangles ABD and BDC are similar triangles.
We can use the similarity between triangles ABD and BDC to find the lengths of BC and CD. The length of BC is proportional to the length of AD, and the length of CD is proportional to the length of DB, because these sides correspond.
We don’t know the length of DB, but we can find it using the Pythagorean Theorem. Let a, b, and c represent the lengths of AD, AB, and BD respectively. According to the Pythagorean Theorem:
_a_2 + _b_2 = _c_2
152 + 202 = _c_2
625 = _c_2
c = 25
The length of BD is 25.
We now have what we need to find the perimeter of the quadrilateral.
Perimeter = sum of the lengths of AB, BC, CD, and DA.
Perimeter = 20 + 18.75 + 31.25 + 15 = 85
The answer is 85.
and is a right angle.
Which angle or angles must be complementary to ?
I)
II)
III)
IV)
V)
Which angle or angles must be complementary to
I)
II)
III)
IV)
V)
Tap to see back →
is a right angle, and, since corresponding angles of similar triangles are congruent, so is . A right angle cannot be part of a complementary pair so both can be eliminated.
can be eliminated, since it is congruent to ; congruent angles are not necessarily complementary.
Since is right angle, is a right triangle, and and are its acute angles. That makes complementary to . Since is congruent to , it is also complementary to .
The correct response is II and V only.
Since
The correct response is II and V only.
Note: Figures NOT drawn to scale.
Refer to the above figure. Given that , evaluate .
Note: Figures NOT drawn to scale.
Refer to the above figure. Given that
Tap to see back →
By the Pythagorean Theorem, since is the hypotenuse of a right triangle with legs 6 and 8, its measure is
.
The similarity ratio of to is
.
Likewise,
By the Pythagorean Theorem, since
The similarity ratio of
Likewise,
Note: Figure NOT drawn to scale.
Refer to the above figure. Given that , give the area of .
Note: Figure NOT drawn to scale.
Refer to the above figure. Given that
Tap to see back →
By the Pythagorean Theorem,
The similarity ratio of to is
,
This can be used to find :
The area of is therefore
By the Pythagorean Theorem,
The similarity ratio of
This can be used to find
The area of
Refer to the above figure. Given that , give the perimeter of .
Refer to the above figure. Given that
Tap to see back →
By the Pythagorean Theorem,
The similarity ratio of to is
,
which is subsequently the ratio of the perimeter of to that of .
The perimeter of is
,
so the perimeter of can be found using this ratio:
By the Pythagorean Theorem,
The similarity ratio of
which is subsequently the ratio of the perimeter of
The perimeter of
so the perimeter of
In the figure above, line segments DC and AB are parallel. What is the perimeter of quadrilateral ABCD?
In the figure above, line segments DC and AB are parallel. What is the perimeter of quadrilateral ABCD?
Tap to see back →
Because DC and AB are parallel, this means that angles CDB and ABD are equal. When two parallel lines are cut by a transversal line, alternate interior angles (such as CDB and ABD) are congruent.
Now, we can show that triangles ABD and BDC are similar. Both ABD and BDC are right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that angles CDB and ABD are congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus triangles ABD and BDC are similar triangles.
We can use the similarity between triangles ABD and BDC to find the lengths of BC and CD. The length of BC is proportional to the length of AD, and the length of CD is proportional to the length of DB, because these sides correspond.
We don’t know the length of DB, but we can find it using the Pythagorean Theorem. Let a, b, and c represent the lengths of AD, AB, and BD respectively. According to the Pythagorean Theorem:
_a_2 + _b_2 = _c_2
152 + 202 = _c_2
625 = _c_2
c = 25
The length of BD is 25.
We now have what we need to find the perimeter of the quadrilateral.
Perimeter = sum of the lengths of AB, BC, CD, and DA.
Perimeter = 20 + 18.75 + 31.25 + 15 = 85
The answer is 85.
Because DC and AB are parallel, this means that angles CDB and ABD are equal. When two parallel lines are cut by a transversal line, alternate interior angles (such as CDB and ABD) are congruent.
Now, we can show that triangles ABD and BDC are similar. Both ABD and BDC are right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that angles CDB and ABD are congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus triangles ABD and BDC are similar triangles.
We can use the similarity between triangles ABD and BDC to find the lengths of BC and CD. The length of BC is proportional to the length of AD, and the length of CD is proportional to the length of DB, because these sides correspond.
We don’t know the length of DB, but we can find it using the Pythagorean Theorem. Let a, b, and c represent the lengths of AD, AB, and BD respectively. According to the Pythagorean Theorem:
_a_2 + _b_2 = _c_2
152 + 202 = _c_2
625 = _c_2
c = 25
The length of BD is 25.
We now have what we need to find the perimeter of the quadrilateral.
Perimeter = sum of the lengths of AB, BC, CD, and DA.
Perimeter = 20 + 18.75 + 31.25 + 15 = 85
The answer is 85.
and is a right angle.
Which angle or angles must be complementary to ?
I)
II)
III)
IV)
V)
Which angle or angles must be complementary to
I)
II)
III)
IV)
V)
Tap to see back →
is a right angle, and, since corresponding angles of similar triangles are congruent, so is . A right angle cannot be part of a complementary pair so both can be eliminated.
can be eliminated, since it is congruent to ; congruent angles are not necessarily complementary.
Since is right angle, is a right triangle, and and are its acute angles. That makes complementary to . Since is congruent to , it is also complementary to .
The correct response is II and V only.
Since
The correct response is II and V only.
Note: Figures NOT drawn to scale.
Refer to the above figure. Given that , evaluate .
Note: Figures NOT drawn to scale.
Refer to the above figure. Given that
Tap to see back →
By the Pythagorean Theorem, since is the hypotenuse of a right triangle with legs 6 and 8, its measure is
.
The similarity ratio of to is
.
Likewise,
By the Pythagorean Theorem, since
The similarity ratio of
Likewise,
Note: Figure NOT drawn to scale.
Refer to the above figure. Given that , give the area of .
Note: Figure NOT drawn to scale.
Refer to the above figure. Given that
Tap to see back →
By the Pythagorean Theorem,
The similarity ratio of to is
,
This can be used to find :
The area of is therefore
By the Pythagorean Theorem,
The similarity ratio of
This can be used to find
The area of
Refer to the above figure. Given that , give the perimeter of .
Refer to the above figure. Given that
Tap to see back →
By the Pythagorean Theorem,
The similarity ratio of to is
,
which is subsequently the ratio of the perimeter of to that of .
The perimeter of is
,
so the perimeter of can be found using this ratio:
By the Pythagorean Theorem,
The similarity ratio of
which is subsequently the ratio of the perimeter of
The perimeter of
so the perimeter of