SLT 42 Prove and apply the
diagonals of a parallelogram
bisect each other and its
converse.
MCPS
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Printed: October 25, 2014
AUTHOR
MCPS
www.ck12.org
Chapter 1. SLT 42 Prove and apply the diagonals of a parallelogram bisect each other and its converse.
C HAPTER
1
SLT 42 Prove and apply the
diagonals of a parallelogram bisect
each other and its converse.
What if a college wanted to build two walkways through a parallelogram-shaped courtyard between two buildings?
The walkways would be 50 feet and 68 feet long and would be built on the diagonals of the parallelogram with
a fountain where they intersect. Where would the fountain be? After completing this Concept, you’ll be able to
answer questions like this by applying your knowledge of parallelograms.
Guidance
A parallelogram is a quadrilateral with two pairs of parallel sides. Here are some examples:
Notice that each pair of sides is marked parallel. As is the case with the rectangle and square, recall that two lines
are parallel when they are perpendicular to the same line. Once we know that a quadrilateral is a parallelogram, we
can discover some additional properties.
1
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Parallelogram Diagonals Theorem: If a quadrilateral is a parallelogram, then the diagonals bisect each other.
If
then
Example A
Prove the Parallelogram Diagonals Theorem:
Given: ABCD is a parallelogram with diagonals BD and AC
Prove: AE ∼
= EC, DE ∼
= EB
TABLE 1.1:
Statement
1. ABCD is a parallelogram with diagonals BD and AC
2. ABkDC, ADkBC
3. 6 BAE ∼
= 6 DCE, 6 ABE ∼
= 6 CDE
4. AB ∼
DC
=
5. 4BAE ∼
= 4DCE
6. AE ∼
= EC, DE ∼
= EB
2
Reason
1. Given
2. Definition of a parallelogram
3. Alternate Interior Angles Theorem
4. Definition of a parallelogram
5. ASA
6. CPCTC
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Chapter 1. SLT 42 Prove and apply the diagonals of a parallelogram bisect each other and its converse.
Example B
Show that the diagonals of FGHJ bisect each other.
The easiest way to show this is to find the midpoint of each diagonal. If it is the same point, you know they intersect
at each other’s midpoint and, by definition, cuts a line in half.
−4 + 6 5 − 4
Midpoint of FH :
,
= (1, 0.5)
2
2
3−1 3−2
,
= (1, 0.5)
Midpoint of GJ :
2
2
Concept Problem Revisited
By the Parallelogram Diagonals Theorem, the fountain is going to be 34 feet from either endpoint on the 68 foot
diagonal and 25 feet from either endpoint on the 50 foot diagonal.
Vocabulary
A parallelogram is a quadrilateral with two pairs of parallel sides.
3
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Guided Practice
1. SAND is a parallelogram, SY = 4x − 11 and Y N = x + 10. Solve for x.
2. Show that the diagonals of FGHJ bisect each other.
Answers:
1. Because this is a parallelogram, the diagonals bisect each other and SY ∼
= Y N.
SY = Y N
4x − 11 = x + 10
3x = 21
x=7
2. Find the midpoint of each diagonal.
Midpoint of FH :
Midpoint of GJ :
−4 + 6 5 − 4
,
= (1, 0.5)
2
2
3−1 3−2
,
= (1, 0.5)
2
2
Because they are the same point, the diagonals intersect at each other’s midpoint. This means they bisect each other.
4
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Chapter 1. SLT 42 Prove and apply the diagonals of a parallelogram bisect each other and its converse.
Interactive Practice
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/113030
Practice
1. If m6 S = 143◦ in parallelogram PQRS, find the other three angles.
2. If AB ⊥ BC in parallelogram ABCD, find the measure of all four angles.
3. If m6 F = x◦ in parallelogram EFGH, find expressions for the other three angles in terms of x.
For questions 4-6, find the measures of the variable(s). All the figures below are parallelograms.
4.
5.
6.
5
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In the parallelogram SNOW, ST = 6, NW = 4, m6 OSW = 36◦ , m6 SNW = 58◦ and m6 NT S = 80◦ . ( diagram is not
drawn to scale)
7. SO
8. NT
9. m6 NW S
10. m6 SOW
Plot the points E(−1, 3), F(3, 4), G(5, −1), H(1, −2) and use parallelogram EFGH for problems 11 - 14.
11. Find the coordinates of the point at which the diagonals intersect. How did you do this?
12. Find the slopes of all four sides. What do you notice?
13. Use the distance formula to find the lengths of all four sides. What do you notice?
14. Make a conjecture about how you might determine whether a quadrilateral in the coordinate is a parallelogram.
Find the point of intersection of the diagonals to see if EFGH is a parallelogram.
15. E(−1, 3), F(3, 4), G(5, −1), H(1, −2)
16. E(3, −2), F(7, 0), G(9, −4), H(5, −4)
17. E(−6, 3), F(2, 5), G(6, −3), H(−4, −5)
18. E(−2, −2), F(−4, −6), G(−6, −4), H(−4, 0)
6