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Parallelogram Properties and Midpoint Theorem, Lecture notes of Geometry

The answer key for proving that the midpoints of adjacent sides of a quadrilateral form a parallelogram through the use of the given theorem. It includes statements, reasons, and sample answers. The document also covers the properties of parallelograms such as alternate interior angles, corresponding angles, and angle-angle similarity.

What you will learn

  • What properties of parallelograms are used to prove that the midpoints of adjacent sides form a parallelogram?

Typology: Lecture notes

2021/2022

Uploaded on 09/27/2022

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Answer Key
Lesson 8.2
Challenge Practice
1. (21, 5), (7, 5)
2. (2, 1), (2, 3)
3. (a 1 2, b 1 3), (a 1 6, b 1 3)
4.
(
a2, b2
)
,
(
a 1 a2, b2
)
5. 2
6. All four vertices of a parallelogram can only have at most two equal x-values and two equal y-values.
So the third and fourth vertices of the parallelogram cannot have the same x-values or y-values as the other
vertices if there are already two equal x-values or two equal y-values.
7. Sample answer: Draw the diagonal of one vertex to the opposite vertex to create two congruent trian-
gles. This can be done in two ways.
8. Sample answer: Draw a diagonal connecting two opposite vertices and then draw another diagonal
connecting the other two opposite vertices.
9. Sample answer: First Way: Connect one vertex to the midpoint of the opposite side. Repeat for the op-
posite vertex. Next draw the diagonal connecting the remaining opposite vertices. Second Way: Repeat the
first method but connect the other opposite vertices with a diagonal.
10. No.
11. Sample answer:
Statements Reasons
1. RSTU and WXYZ are 1. Given
parallelograms.
2.
}
RS i
}
UT ,
}
RU i
}
ST , 2. Definition of a
}
WX i
}
ZY , and parallelogram
}
WZ i
}
XY
3. RWX > YXW 3. Alternate Interior Angles Theorem
4. YXW > WZY 4. Opposite angles of a parallelogram are congruent.
5. WZY > TYZ 5. Alternate Interior Angles Theorem
6. RWX > TYZ 6. Substitution Prop. of Equality
7. XRW > UZR 7. Alternate Interior Angles Theorem
8. SXT > YTZ 8. Alternate Interior Angles Theorem
9. XRW > SXT 9. Corresponding Angles Postulate
10. XRW > YTZ 10. Substitution Prop. of Equality
11. n RWX , n TYZ 11. Angle-Angle Similarity Post.
12.
Given: WXYZ is a parallelogram and
}
WY and
}
XZ are diagonals of WXYZ.
Prove: D is the midpoint of the segment with endpoints on opposite sides passing through the point of inter-
section D.
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Lesson 8.

Challenge Practice

  1. ( 2 1, 5), (7, 5)
  2. (2, 1), (2, 3)
  3. ( a 1 2, b 1 3), ( a 1 6, b 1 3)
  4. ( a^2 , b^2 ), ( a 1 a^2 , b^2 )^ 5. 2
  5. All four vertices of a parallelogram can only have at most two equal x -values and two equal y -values. So the third and fourth vertices of the parallelogram cannot have the same x -values or y -values as the other vertices if there are already two equal x -values or two equal y -values.
  6. Sample answer: Draw the diagonal of one vertex to the opposite vertex to create two congruent trian- gles. This can be done in two ways.
  7. Sample answer: Draw a diagonal connecting two opposite vertices and then draw another diagonal connecting the other two opposite vertices.
  8. Sample answer: First Way: Connect one vertex to the midpoint of the opposite side. Repeat for the op- posite vertex. Next draw the diagonal connecting the remaining opposite vertices. Second Way: Repeat the first method but connect the other opposite vertices with a diagonal.
  9. No.
  10. Sample answer:

Statements Reasons

1. RSTU and WXYZ are 1. Given parallelograms. 2.

} RS i^ } UT , } RU i^

} ST , 2. Definition of a } WX i } ZY , and parallelogram } WZ i } XY

3.RWX > ∠ YXW 3. Alternate Interior Angles Theorem 4.YXW > ∠ WZY 4. Opposite angles of a parallelogram are congruent. 5.WZY > ∠ TYZ 5. Alternate Interior Angles Theorem 6.RWX > ∠ TYZ 6. Substitution Prop. of Equality 7.XRW > ∠ UZR 7. Alternate Interior Angles Theorem 8.SXT > ∠ YTZ 8. Alternate Interior Angles Theorem 9.XRW > ∠ SXT 9. Corresponding Angles Postulate 10.XRW > ∠ YTZ 10. Substitution Prop. of Equality 11. n RWX , n TYZ 11. Angle-Angle Similarity Post.

Given: WXYZ is a parallelogram and } WY and } XZ are diagonals of WXYZ.

Prove: D is the midpoint of the segment with endpoints on opposite sides passing through the point of inter- section D.

Statements Reasons

1. WXYZ is a 1. Given parallelogram, } WX i } YZ , and } WZ i } XY. 2. DW } > } DY and 2. Theorem 8. DX^ } > DZ } 3. WX } > } YZ , and 3. Theorem 8. } WZ > } YX 4. n WDX > n YDZ , 4. SSS Cong. Post. n WDZ > n YDZ 5. Altitude of n WDX 5 5. If two triangles alt. of n YDZ , and alt. are congruent, of n WDZ 5 alt. of their altitudes n YDZ. are equal. 6. Let AB } pass through 6. Assume D and have its endpoints on WX^ } and } YZ. 7. The angle formed by 7. Vertical Angles AD^ } and the altitude Cong. Theorem of n WDX > the angle formed by } BD and the alt. of n YDZ. 8. ∠ 1 and ∠ 2 are 8. Definition of an right angles. altitude 9. ∠ 1 > ∠ 2 9. Right Angle Congruence Thm. 10. The triangles formed 10. ASA Congruence by the altitudes and Postulate AD^ } and } BD are cong. 11. AD } > } BD 11. Corresponding parts of cong. triangles are congruent. 12. D is the midpoint 12. Def. of a of AB }. midpoint

  1. Sample answer:

Given: ABCD is a quadrilateral and E , F , G , and H are midpoints of their respective segments.Prove: When the midpoints of adjacent sides are connected by segments, a parallelogram is formed.Step 1: Place ABCD and assign coordinates. Let E , F , G , and H be midpoints of their respective segments. Find the coordinates of the midpoints.