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Quadrilaterals

Quadrilaterals. Information. What is a quadrilateral?. A quadrilateral is a four-sided polygon. A quadrilateral is the polygon with the fewest number of sides that allows for it to be a concave polygon (where one of the interior angles is greater than 180 degrees).

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Quadrilaterals

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  1. Quadrilaterals

  2. Information

  3. What is a quadrilateral? A quadrilateral is a four-sided polygon. A quadrilateral is the polygon with the fewest number of sides that allows for it to be a concave polygon (where one of the interior angles is greater than 180 degrees). Why is it impossible to have a concave triangle? Since the interior angles of a triangle have to sum to 180°, there can not be an interior angle greater than 180 degrees.

  4. Quadrilateral angle sum Polygon angle sum theorem: The sum of the measures of the interior angles of a convex n-sided polygon is (n – 2) 180° Using this theorem, we can see that the sum of the measures of a quadrilateral is: 2 × 180° = 360°. B The angle sum theorem of a quadrilateral can also be explained in a diagram. A quadrilateral can be divided into two triangles by drawing a line between two opposite vertices. Each of these triangles has an angle sum of 180 degrees. A D C

  5. Quadrilateral angle sum B A Prove that the interior angle sum of quadrilateral ABCD is 360°. 2 4 1 3 quadrilateral ABCD, split by BC given: C D mA + mB + mC + mD = 360º hypothesis: mA + m1 + m2 = 180° and mD + m3 + m4 = 180º triangle angle sum theorem: m1 + m3 = mC and m2 + m4 = mB angle addition: mA + mB + mC + mD= mA + (m2 + m4) + (m1 + m3) + mD angle substitution: (mA + m1 + m2) + (mD + m3 + m4) group by triangles: triangle angle sum theorem:  mA + mB + mC + mD = 180º + 180º = 360º

  6. Types of quadrilaterals

  7. Types of parallelograms

  8. Parallelograms In parallelogram ABCD we know that AD || BC and AB || CD. Diagonal BD divides the parallelogram into two triangles. A B Use parallelogram ABCD to prove that opposite sides and opposite angles in any parallelogram are congruent. D C alternate interior angles are congruent: ADB≅DBC and ABD≅BDC reflexive property: BD≅BD ASA property: △ABD≅△BCD corresponding parts of congruent triangles are congruent (CPCTC):  AD≅CB and AB≅CD  Also BAD≅BCD

  9. Properties of parallelograms A bisectoris a line that goes through the midpoint of a segment and divides the line into two congruent parts. Given parallelogram ABCD, with diagonals AC and BD intersecting at E, prove that AE≅ CE and BE ≅ DE (that the diagonals of the parallelogram bisect each other) A B congruence of alternate interior angles: ABD ≅ BDCand CAB ≅ ACD E AB ≅ CD congruence of opposite sides: D C △ABE≅△CDE ASA property: BE ≅ DEand AE ≅ CE corresponding parts of congruent triangles are congruent (CPCTC): 

  10. Parallelogram properties

  11. Proving that ABCD is a parallelogram If ABCD is a quadrilateral, then how can we prove that it is also a parallelogram? We must prove that both pairs of opposite sides are parallel. How can we prove that lines are parallel in a quadrilateral? To prove that lines are parallel, we must prove one of the following: 1) Alternate interior angles are congruent. 2) Corresponding angles are congruent. 3) Same side interior angles are supplementary.

  12. Proving that ABCD is a parallelogram If the opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Given quadrilateral ABCD with A ≅ C and B ≅ D, prove that ABCD is a parallelogram. A B polygon angle sum theorem: mA + mB + mC + mD = 360° a b a + a + b + b = 360° A≅C and B≅D: a b 2a + 2b = 360° group like terms: D C a + b = 180° divide by 2: A and D are supplementary D and Care supplementary a and b are supplementary: converse of the alternate interior angle theorem:  AB || CD and AD || BC

  13. Proof of a parallelogram With a partner, use congruent triangles to prove that if the opposite sides of a quadrilateral are congruent, then it is a parallelogram. A B quadrilateral ABCD with AB ≅ CDand AD ≅ BC given: ABCD is a parallelogram hypothesis: BD ≅ BD reflexive property: D C △ABD≅△CBD SSS congruence postulate: ABD ≅ BDCand CBD ≅ ADB CPCTC: Since ABD ≅ BDC, AB || CDSince CBD ≅ ADB, AD || BC converse of the alternate interior angle theorem:  ABCD is a parallelogram. given AB || CD and AD || BC:

  14. Problems

  15. Rectangles and parallelograms Prove that if the diagonals of a parallelogram are congruent, then the parallelogram must be a rectangle. A B parallelogram ABCD where AC ≅ BD given: ABCD is a rectangle hypothesis: D C congruence ofopposite sides of a parallelogram: AD ≅ BC and AB ≅ CD SSS congruence postulate: △ADC≅△BCD CPCTC: ADC ≅ BCD since AD || BC and they are same side interior angles: ADC and BCD are supplementary ADC and BCD are congruent and supplementary:  mADC = mBCD = 90º

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