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4.5-4.8 without 4.7

4.5-4.8 without 4.7. Proving quadrilateral properties Conditions for special quadrilaterals Constructing transformations. By: Tyler Register and Tre Burse. geometry. The Vocabulary and Theorems. A diagonal of a parallelogram divides the parallelogram into two equal triangles

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4.5-4.8 without 4.7

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  1. 4.5-4.8 without 4.7 Proving quadrilateral properties Conditions for special quadrilaterals Constructing transformations By: Tyler Register and Tre Burse geometry

  2. The Vocabulary and Theorems • A diagonal of a parallelogram divides the parallelogram into two equal triangles • Opposite sides of a parallelogram are congruent • Opposite angles of a parallelogram are congruent • Diagonals of a parallelogram bisect each other

  3. Theorems cont. • A rhombus is a parallelogram • A rectangle is a parallelogram • The diagonals and sides of a rhombus form 4 congruent triangles • The diagonals of a rhombus are perpendicular • The diagonals of a rectangle are congruent • A square is a rhombus

  4. Theorems cont. • The diagonals of a square are perpendicular and are bisectors of the angles • If two pairs of opposite sides of a quadrilateral are congruent then the quadrilateral is a parallelogram • If the diagonals of a quadrilateral bisect each other then the quadrilateral is a parallelogram

  5. Theorems • If one angle of a parallelogram is a right angle then the parallelogram is a rectangle • If the diagonals of a parallelogram are congruent then the parallelogram is a rectangle • If one pair of adjacent sides of a quadrilateral are congruent then the quadrilateral is a rhombus

  6. More Theorems • If the diagonals of a parallelogram bisect the angles of the parallelogram then it is a rhombus • If the diagonals of a parallelogram are perpendicular than it is a rhombus • Triangle mid-segment theorem- A mid-segment of a triangle is parallel to a side of the triangle and its length is equal to half the length of than side

  7. The Last Theorem Slide • Betweenness postulate- given the three points: P, Q, and R PQ+QR=PR then Q is between P and R on a line. • The Triangle inequality theorem- The sum of any two sides of a triangle are greater than the other side.

  8. 4-5 Statements Reasons • PLGM is a parallelogram and LM is a diagonal • Given • Def of parallelogram • Objective- Prove quadrilateral conjectures by using triangle congruence postulates and theorems. • PL II GM Given: parallelogram PLGM with diagonal LM • < 3 = < 2 • Alt. Int. angles • PM II GL • Def of parallelogram Prove: triangle LGM= triangle MPL • <1 = <4 • Alt. Int. angles • LM=LM • reflexive LGM= MPL • ASA P L M G

  9. 4-6 • There are many theorems in this section that state special cases in quadrilaterals • The most notable of these theorems is the House Builder Theorem • There is also the Triangle Mid- segment Theorem House Builder Theorem: If the diagonals of a parallelogram are congruent then the parallelogram is a rectangle Conditions of special quadrilaterals Triangle Mid-segment Theorem: A mid-segment of a triangle is parallel to a side of the triangle and its length is equal to half the length of than side The list of the theorems in 4-6 are on page 5 and 6

  10. 4-8Constructing transformations • This section has one theorem and one postulate • The Betweenness postulate (converse of the segment addition postulate) and the Triangle Inequality Theorem The Betweenness postulate: given the three points: P, Q, and R PQ+QR=PR then Q is between P and R on a line. Triangle Inequality Theorem: The sum of any two sides of a triangle are greater than the other side. 5+7>X X+5>7 X+7>5 2<X<13 5 7 X

  11. Quiz • Which of the following are possible lengths of a triangle? A. 14,8,25 B.16,7,23 C.18,8,24 • If one angle of a quadrilateral is a right angle than the quadrilateral is a ___________. • Find the measure of the following angles: <Q= <RPQ= <PRQ= Rectangle 60 P Q S R 40 80 60 40

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