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1 of 15 © Boardworks 2012 Information 2 of 15 © Boardworks 2012 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. 3 of 15 © Boardworks 2012 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°. 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 C vertices. Each of these triangles has an angle sum of 180 degrees. 4 of 15 B A D © Boardworks 2012 Quadrilateral angle sum Prove that the interior angle sum of quadrilateral ABCD is 360°. A 1 3 given: quadrilateral ABCD, C split by BC hypothesis: mA + mB + mC + mD = 360º triangle angle sum theorem: angle addition: 2 4 B D mA + m1 + m2 = 180° and mD + m3 + m4 = 180º m1 + m3 = mC and m2 + m4 = mB angle substitution: mA + mB + mC + mD = mA + (m2 + m4) + (m1 + m3) + mD group by triangles: (mA + m1 + m2) + (mD + m3 + m4) triangle angle sum theorem: 5 of 15 mA + mB + mC + mD = 180º + 180º = 360º © Boardworks 2012 Parallelograms In parallelogram ABCD we know that AD || BC and AB || CD. Diagonal BD divides the parallelogram into two triangles. A B D C alternate interior angles are congruent: reflexive property: ASA property: corresponding parts of congruent triangles are congruent (CPCTC): 6 of 15 Use parallelogram ABCD to prove that opposite sides and opposite angles in any parallelogram are congruent. ADB ≅ DBC and ABD ≅ BDC BD ≅ BD △ABD ≅ △BCD AD ≅ CB and AB ≅ CD Also BAD ≅ BCD © Boardworks 2012 Properties of parallelograms A bisector is 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 E D C congruence of alternate interior angles: congruence of opposite sides: ASA property: corresponding parts of congruent triangles are congruent (CPCTC): 7 of 15 ABD ≅ BDC and CAB ≅ ACD AB ≅ CD △ABE ≅ △CDE BE ≅ DE and AE ≅ CE © Boardworks 2012 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. 8 of 15 © Boardworks 2012 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 a B a b C D polygon angle sum theorem: A≅C and B≅D: group like terms: divide by 2: a and b are supplementary: converse of the alternate interior angle theorem: 9 of 15 mA + mB + mC + mD = 360° a + a + b + b = 360° 2a + 2b = 360° a + b = 180° A and D are supplementary D and C are supplementary AB || CD and AD || BC © Boardworks 2012 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 given: hypothesis: D C reflexive property: SSS congruence postulate: CPCTC: converse of the alternate interior angle theorem: given AB || CD and AD || BC: 10 of 15 quadrilateral ABCD with AB ≅ CD and AD ≅ BC ABCD is a parallelogram BD ≅ BD △ABD ≅ △CBD ABD ≅ BDC and CBD ≅ ADB Since ABD ≅ BDC, AB || CD Since CBD ≅ ADB, AD || BC ABCD is a parallelogram. © Boardworks 2012 Rectangles and parallelograms Prove that if the diagonals of a parallelogram are congruent, then the parallelogram must be a rectangle. A B given: hypothesis: D C congruence of opposite sides of a parallelogram: AD ≅ BC and AB ≅ CD SSS congruence postulate: △ADC ≅ △BCD CPCTC: since AD || BC and they are same side interior angles: ADC ≅ BCD ADC and BCD are congruent and supplementary: 11 of 15 parallelogram ABCD where AC ≅ BD ABCD is a rectangle ADC and BCD are supplementary mADC = mBCD = 90º © Boardworks 2012