Download Geometry Problem Solving Drill – 14: Special Quadrilaterals

Geometry Problem Solving Drill – 14: Special Quadrilaterals - Trapezoids and Kites
Question No. 1 of 10
Instruction: (1) Read the problem statement and answer choices carefully (2) Work the problems on paper as needed (3)
Pick the answer (4) Go back to review the core concept tutorial as needed.
Question 1. Mario’s pool is in the shape of a quadrilateral. It has two congruent sides and the other two
sides are parallel with each other. There are no right angles. Choose the most precise name for the
shape of Mario’s pool.
Question #01
(A)
(B)
(C)
(D)
trapezoid
isosceles triangle
isosceles trapezoid
kite
A. Incorrect!
Mario’s pool is a trapezoid but since it has two congruent sides, there is a more specific name for it.
B. Incorrect!
The word isosceles indicates that two sides are congruent, but his pool has four sides, not three.
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Choice
C. Correct!
Mario’s pool is an isosceles trapezoid.
D. Incorrect!
A kite does not have parallel sides.
The answer can easily be found by remembering the quadrilateral diagram. The term quadrilateral is
the most general way to describe a 4-sided figure. The diagram goes from general to specific as you
read it down the page. Quadrilaterals break down into figures that have two pairs of parallel sides,
parallelograms. Quadrilaterals that have one pair of parallel sides are called trapezoids. The isosceles
trapezoid is more specific than trapezoid. It has two congruent sides and its base angles are congruent.
Solution
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Question No. 2 of 10
Instruction: (1) Read the problem statement and answer choices carefully (2) Work the problems on paper as needed (3)
Pick the answer (4) Go back to review the core concept tutorial as needed.
Question 2. The figure is an isosceles trapezoid. Determine the value of the variable.
Question #02
(A)
(B)
(C)
(D)
x
x
x
x
=
=
=
=
3
2
18x + 48
51
A. Correct!
Since this figure is an isosceles trapezoid the base angles are congruent. You set up the equation and
solved.
B. Incorrect!
Check your arithmetic and try again.
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Choice
C. Incorrect!
Since the base angles are equal, 17x and x + 48 should be set equal to each other in an equation.
D. Incorrect!
The measure of each angle is 51, but the question was asking only for the value of x.
Since this figure is an isosceles trapezoid the base angles are congruent. 17x and X + 48 are equal to
each other. Set up and equation and solve.
17x = x + 48
-x
-x
(subtract x from both sides)
16x = 48
16
16
Solution
(Divide both sides by 16)
x=3
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Question No. 3 of 10
Instruction: (1) Read the problem statement and answer choices carefully (2) Work the problems on paper as needed (3)
Pick the answer (4) Go back to review the core concept tutorial as needed.
Question 3. Determine the value of x for which EFGH is a kite.
Question #03
(A)
(B)
(C)
(D)
3
4
5
10
A. Incorrect!
If you substitute 3 for the variable x, the adjacent sides will not be equal and EFGH will not be a kite.
B. Incorrect!
If you substitute 4 for the variable x, the adjacent sides will not be equal and EFGH will not be a kite.
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Choice
C. Correct!
If you substitute 5 for the variable x, the adjacent sides will be equal and EFGH will be a kite.
D. Incorrect!
Adjacent sides of a kite are congruent. Set up an equation and solve.
Remember the property: Adjacent sides of a kite are congruent.
Set up an equation and solve.
4x – 5 = 20 – x
+x
+x
(add x to both sides)
5x – 5 = 20
+5 +5
Solution
5x = 25
5
5
subtract 5 from both sides
(divide both sides by 5)
x=5
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Question No. 4 of 10
Instruction: (1) Read the problem statement and answer choices carefully (2) Work the problems on paper as needed (3)
Pick the answer (4) Go back to review the core concept tutorial as needed.
Question 4. Figure JKLM is a trapezoid. JM is parallel to KL . Find the measure of the median of the
trapezoid.
Question #04
(A)
(B)
(C)
(D)
180
180 + 5x
5x
90
A. Incorrect!
Remember, the median is between the two bases, 40 and 140.
B. Incorrect!
Remember, the only numbers you need to use are the measures of the bases, 40 and 140.
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Choice
C. Incorrect!
Remember, the median is half of the sum of the two bases, 40 and 140.
D. Correct!
The median is half of the sum of the two bases.
The median is half of the sum of the two bases.
Median = ½(40 + 140)
Median = ½(180)
Solution
Median = 90
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Question No. 5 of 10
Instruction: (1) Read the problem statement and answer choices carefully (2) Work the problems on paper as needed (3)
Pick the answer (4) Go back to review the core concept tutorial as needed.
Question 5. This figure is an isosceles trapezoid. Find the value of n.
Question #05
(A)
(B)
(C)
(D)
10.2
7
51 + 5n
30
A. Incorrect!
Check your arithmetic and try again.
B. Correct!
You set up and solved the equation correctly.
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Choice
C. Incorrect!
Legs of an isosceles trapezoid are congruent, which means that 43 and 5n + 8 are equal. Set up an
equation and solve.
D. Incorrect!
Legs of an isosceles trapezoid are congruent, which means that 43 and 5n + 8 are equal. Set up an
equation and solve.
Remember: Legs of an isosceles trapezoid are congruent. 43 is equal to 5n + 8. Set up an equation and
solve.
43 = 5n + 8
-8
-8
35 = 5n
5
5
Solution
7=n
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Question No. 6 of 10
Instruction: (1) Read the problem statement and answer choices carefully (2) Work the problems on paper as needed (3)
Pick the answer (4) Go back to review the core concept tutorial as needed.
Question 6. Figure RSTU is a kite. The diagonals intersect at V. Find m∠SVR and justify your answer.
Question #06
(A)
(B)
(C)
(D)
45° because all angles of a kite are acute
90° because diagonals of a kite are perpendicular
45° because diagonals of a kite bisect the angles
180° because consecutive angles are supplementary
A. Incorrect!
The diagonals of a kite are perpendicular.
B. Correct!
∠SVR is formed by the diagonals and is a right angle.
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C. Incorrect!
∠SVR is not one of the angles bisected by the diagonals.
D. Incorrect!
∠SVR is not a straight angle.
Step 1: Locate the angle and highlight it.
Step 2: Think about the properties of kites.
Remember that the diagonals of a kite form perpendicular lines. ∠SVR must be 90 degrees.
Solution
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Question No. 7 of 10
Instruction: (1) Read the problem statement and answer choices carefully (2) Work the problems on paper as needed (3)
Pick the answer (4) Go back to review the core concept tutorial as needed.
Question 7. Given WXYZ is a kite, m∠YWX = 2x + 30, and m∠WYX = 4x. Find m∠YWX.
Question #07
(A)
(B)
(C)
(D)
25
20
15
60
A. Incorrect!
The angles are congruent. Set the expressions equal to each other and solve.
B. Incorrect!
The angles are congruent. Set the expressions equal to each other and solve.
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Choice
C. Incorrect!
This is the value of x. There is one more step to find the measure of the angle.
D. Correct!
You found the value x and used that value to find the measure of the angle.
Step 1: Remember that the angles at the opposite endpoints of a diagonal are congruent.
Step 2: Set up an equation using the expressions given for each angle.
Step 3: 2x + 30 = 4x
Step 4: Solve the equation.
2x + 30 = 4x
-2x
-2x
Solution
30 = 2x
2
2
subtract 2x from both sides
divide each side by 2
x = 15
Step 5: Substitute 15 into either 2x + 30 or 4x and find that the angles measure 60 degrees.
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Question No. 8 of 10
Instruction: (1) Read the problem statement and answer choices carefully (2) Work the problems on paper as needed (3)
Pick the answer (4) Go back to review the core concept tutorial as needed.
Question 8. The figure is a trapezoid. The length of longer base is 20. The length of the median is 14.
What is the measure of the shorter base?
Question #08
(A)
(B)
(C)
(D)
8
17
28
34
A. Correct!
The sum of the bases divided by 2 gives the length of the median.
B. Incorrect!
Remember, the median length is known and one of the bases is unknown. You need to work backwards.
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Choice
C. Incorrect!
Remember, the median is half of the sum of the lengths of the two bases.
D. Incorrect!
Remember, the median length is known and one of the bases is unknown. You need to work backwards.
The median of a trapezoid is half of the sum of the lengths of the two bases. You can set up an equation
and solve.
(20 + n) ÷ 2 = 14
×2
×2
20 + n = 28
-20
-20
Solution
(multiply each side of the equation by 2)
(subtract 20 from each side)
n=8
The unknown base measures 8.
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Question No. 9 of 10
Instruction: (1) Read the problem statement and answer choices carefully (2) Work the problems on paper as needed (3)
Pick the answer (4) Go back to review the core concept tutorial as needed.
Question 9. Given ABCD is a trapezoid and JK is the median, find the value of x.
Question #09
(A)
(B)
(C)
(D)
3
5
7
9
A. Incorrect!
Don’t forget that the length of the median is half the sum of the lengths of the two bases of a trapezoid.
B. Incorrect!
Check your arithmetic and try again.
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Choice
C. Correct!
You setup an equation relating the median length to the lengths of the bases.
D. Incorrect!
Check your arithmetic and try again.
The median is half as long as the sum of the two bases. Set up an equation and double the median:
4x + 6x + 10 = 80
10x + 10 = 80
-10 -10
10x = 70
10
10
(combine like terms)
(subtract 10 from each side)
(divide each side by 10)
x=7
Solution
You could also write an equation that looks like this:
4x + 6x +10
2
= 40
The first step would be to multiply each side of the equation by 2. Then the rest of the solution would be
the same.
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Question No. 10 of 10
Instruction: (1) Read the problem statement and answer choices carefully (2) Work the problems on paper as needed (3)
Pick the answer (4) Go back to review the core concept tutorial as needed.
Question 10. HATS is a kite with diagonals that intersect at C. m∠TSC = 32o. Find m∠SHC.
Question #10
(A)
(B)
(C)
(D)
16°
29°
32°
58°
A. Incorrect!
Use the Angle Sum Theorem of triangles to find the unknown angle measure.
B. Incorrect!
Use the Angle Sum Theorem of triangles to find the unknown angle measure.
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Choice
C. Incorrect!
Use the Angle Sum Theorem of triangles to find the unknown angle measure.
D. Correct!
You used the Angle Sum Theorem of triangles to find the unknown angle measure.
Remember a property about triangles: The sum of the 3 angles of a triangle is 180.
Remember two properties about kites: the diagonals intersect to make 90o angles and the opposite
angles formed at the endpoints of the diagonal are congruent. Setup and solve the equation relating the
angles:
Solution
32 + 90 + n = 180
122 +n = 180
-122
-122
n = 58
∠STC measures 58 degrees.
The angle opposite it, ∠SHC, is also 58 degrees.
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