Given parallelogram ABCD and diagonal t Prove that triangles ABC and CDA are congruent (Theorem 5.14) and opposite angles and sides are congruent

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1 t m l parallelogram and diagonal t Prove that triangles and are congruent (Theorem 5.14) and opposite angles and sides are congruent (Theorem 5.15) n p l m ef of a Parallelogram n p ef of a Parallelogram , m 1 m 4 m 2 m 3 m m 1 m 4 m m 2 m 3 reflexive S P P PE P P Substitution

2 l If the transversals are perpendicular and l and m are parallel re lines and parallel? m re segments and congruent? Parallel lines are everywhere equidistant How would we prove Theorem 5.17 If both pairs of opposite sides of a quadrilateral are congruent then it is a parallelogram.

3 Theorem 5.18 If both pairs of opposite angles of a quadrilateral are congruent, then it is a parallelogram. b c If you were trying to prove this you could start with the sum of all of the angles. a b c d 360 a b a b 360 2a 2b 360 a b 180 c b 180 o o o o o a d Theorem 5.19 If a quadrilateral has two sides that are parallel and congruent, then it is a parallelogram. Try to prove this using SS

4 E Theorem 5.20 The diagonals of a parallelogram bisect each other. First provethat E E E Theorem 5.21 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. First provethat E E and use Theorem 5.19.

5 Some pplications Linkages Forces and Resultant Vectors -the combination of two forces acting on an object.

6 Theorem 5.17 If both pairs of opposite sides of a quadrilateral are congruent then it is a parallelogram. : and Prove: Quadrilateral is a parallelogram. Δ Δ Reflexive SSS P 5.1 is a Parallelogram ef of P 5.1 a Parallelog ram

7 E Theorem 5.20 The diagonals of a parallelogram bisect each other. : Quadrilateral is a parallelogram. Prove: and bisect eachother is a parallelogram ef of a parallelog ram E E 5.5 E E Vertical 's Theorem ΔE ΔE S E E and E E and bisect eachother P efinition of a Segment isector

8 Theorem 5.22 Every Rhombus is a parallelogram. (oth pairs of opposite sides are congruent) Since the rhombus is a parallelogram, what can we say about the diagonals? The four triangles in the diagram are congruent by SSS So you can say the opposite angles are bisected by P The diagonals are perpendicular by E The ongruent Supplements Theorem Or Theorem 4.9

9 E The diagonals are bisected (5.20) and the trianlges are congruent by SS, so all the sides are congruent by P. If the diagonals of a parallelogram are perpendicular, why can we say it s a rhombus? E If the opposite angles of a parallelogram are bisected, why can we say it s a rhombus? Opposite angles are congruent (5.15) and bisected (given). The diagonals are bisected (5.20), so the triangles are congruent by SS and the sides are congruant by P

10 Number 46 from Section 5.3 asks what will happen to a cue ball, shot at a 45 degree angle on a pool table with dimensions of 9 by 4.5 feet.

11 Now, what if we changed the dimensions of the pool table to 9 by 6 feet with no side pockets.

12 t home, use the Paper Pool pplet in the Unit 5 notes to experiment with a variety of table dimensions and begin recording your results in a table. Will the ball always land in a pocket? re there any dimensions for which the ball will not land in a pocket? an you generalize a rule for determining how many bounces it will take for any size pool table? Extensions: What impact does changing the starting point or angle of departure have on your conclusions.

13 E Prove Theorems 5.22 and 5.23 : Quadrilateral is a rhombus. Prove: is a parallelogram and. efinition of a Rhombus is a Parallelogram 5.17 and bisect eachother 5.20 E E efinition of a Segment isector E E Reflexive ΔE ΔE SSS E E P E is a straight angle E and E are supplementary E and E are right angles efinition of a straight angle ongruent Supplements Theorem efinition of

14 E Prove Theorems 5.22 and 5.23 : Quadrilateral is a rhombus. Prove: is a parallelogram and. efinition of a Rhombus is a Parallelogram 5.17 and bisect eachother 5.20 E E efinition of a Segment isector E E Reflexive ΔE ΔE SSS E E P E is an angle bisector ef of an angle bisector is isosceles efinition of an isosceles triangle E is the bisector of 4.9 E efinition of bisector

15 What is the definition of a rectangle? Single house architecture from Tamizo rchitects Group. Theorem 5.26 Every Rectangle is a parallelogram (y Theorem 5.18 the opposite angles are congruent) Theorem 5.27 parallelogram with one right angle is a rectangle? How would you prove this Theorem? (Use Theorem 5.6) Theorem 5.28 The diagonals of a rectangle are congruent. Prove this by getting

16 Theorem 5.30 The base angles of an isosceles trapezoid are congruent. E oorways of Machu Picchu Introduce a segment through parallel to segment to form a parallelogram and classify triangle E. Is the converse of this theorem true? If the base angles of a trapezoid are congruent, are the sides opposite them congruent? E Introduce a segment E such that angle E is congruent to angle E.

17 Theorem 5.27 parallelogram with one right angle is a rectangle : Prove: Parallelogram and is a is a rectangle. right is a parallelogram and is a right angle is a right angle is a right angle is a right angle is a rectangle efinition of a Parallelogram efinition of 5.6 efinition of 5.6 efinition of 5.6 efinition of efinition of a rectangle

18 Theorem 5.28 The diagonals of a rectangle are congruent. : is a rectangle Prove: is a rectangle is a parallelogram Reflexive and are right angles efinition of a rectangle Right ngle Theorem SS P

19 Theorem 5.29 If a parallelogram has congruent diagonals, then it is a rectangle. is a parallelogram Δ Δ and are supplementary and are right angles is a rectangle given given Theorem 5.15 Reflexive SSS P Same-side Interior ngles ongruent Supplements Theorem Theorem 5.27

20 1 : lmand transversal t Prove: 1 and 2 are supplementary 3 2 m 1 = m 3 2 and 3 form a straight angle m 2 + m 3 =180 o m 2 + m 1 =180 o 1 and 2 are supplementary 5.7 ef. of a Straight ngle Substitution ef. of Supplementary

21 3 1 : lmand transversal t Prove: 1 and 2 are supplementary and 4 are supplementary m 3 + m 4 =180 o m 1 + m 2 =180 o 1 and 2 are supplementary 5.8 ef. of Supplementary Vertical ngles Thm. Vertical ngles Thm. Substitution ef. of Supplementary

22 : Prove : E E and form a straight m +m 180 o and E form a straight m +m E 180 o m +m E m m m E m E ase ngles Thm. ef. of a Straight ef. of a Straight Substitution PE ef. of ngles

23 : Prove : E E and form a straight and are supplementary and E form a straight and E are supplementary E ase ngles Thm. ef. of a Straight ef. of a Straight Supplements Theorem