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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.
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Challenge Practice
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
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.