pd 3notes 5.4 November 09, 2016 Based on work from pages , complete In an isosceles triangle, the &

Transcription

1 chapter 5 Based on work from pages , complete In an isosceles triangle, the & & & drawn from the vertex angle of an isosceles triangle are the! 5.1 Indirect proof. G: DB AC F is the midpt. of AC P: AD == CD D A B F C

2 G: BD bisects <ABC, <ADB is acute P: AB = BC

3 G: ABC P: BCD > B draw median from A, through seg. BC, at M, such that AM = MP What is true about ^ABM and ^PCM? what is true about <1, <3? explain how the Prove statement may be conclude.

4 5.2 Proving that lines are parallel The measure of an exterior angle of a triangle is greater than either of the two remote interior angles. Theorems If two lines are cut by a transversal such that two alternate interior angles are congruent OR alternate exterior angles are congruent OR corresponding angles are congruent OR same-side interior angles are supplementary OR same-side exterior angles are supplementary THEN the lines are parallel If two coplanar lines are parallel to a third line then the lines

5 E G: <1 comp. to <2 <3 comp. to <2 C 1 2 A P: CA // DB D 3 B

6 G: <1 supp. to <2 <3 supp. to <2 P: FLOR is a parallelogram

7

8

9 5.3 Congruent angles associated with parallel lines Through point P, how many lines are parallel to line k? a // b, Find <1: 4x + 36 x + 2x Look at the theorems numbered

10 G: FH // JM, <1 = <2 K JM = FH P: GJ = HK F 2 G H J M 1

11 G: CY AY, YZ // CA P: YZ bis. <AYB C Y A Z B

12 THE famous crook problem 50 deg x deg 132 deg.

13 5.4 Four sided polygons BE able to define the basic quadrilaterals as described on page 236. What does convex mean? Can you draw a convex polygon? What does concave mean? Can you draw a concave polygon? examine carefully, what are some properties? examine carefully, what are some properties?

14 examine, list properties examine, list properties examine, list properties examine, list properties

15 examine, list properties 13 4 find the area of the trapezoid 21 5

16 A S N 1) a square is a rhombus 2) a rectangle is a square 3) a parallelogram has at least two sides parallel 4)the diagonals of a square are congruent 5)a trapezoid has at most two sides parallel 6)a kite is a trapezoid 7)the diagonals of a trapezoid are congruent

17

18

19

20

21

22 5.5 Properties of quadrilaterals Prove that (1) the opposite sides of a parallelogram are congruent (2)the opposite angles of a parallelogram are congruent (3) the diagonals of a parallelogram bisect each other

23 Prove that the diagonals of a kite are perpendicular

24 kite rectangle square parallelogram rhombus quadrilateral isosc. trapezoid trapezoid

25 What am I?

26 5.6 Proving that a quadrilateral is a parallelogram given BCDF is a kite with BC=3x+4y, B CD=20, BF=12 and FD=x+2y, find the perimeter. C F D Prove that if both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram.

27 Prove that if the diagonals of a quadrilateral bisect each other then it is a parallelogram (x^5)(x^2) (x^2-25) (x-5)(x+5) x^7 Show that the figure above is a parallelogram

28 5.7 Proving that figures are special quadrilaterals How do you prove that a figure is >>Rectangle parallelogram with at least one right angle parallelogram with congruent diagonals quadrilateral with 4 right angles >>Kite 2 disjoint pairs of consecutive sides of quadrilateral are congruent 1 diagonal is the perpendicular bisector of the other diagonal >>Rhombus parallelogram contains a pair of consecutive sides congruent either diagonal of a parallelogram bisects two angles the diagonals of a quadrilateral are perpendicular bisectors of each other >>Square quadrilateral is both a rhombus and a rectangle >>Isosceles Trapezoid non-parallel sides of a trapezoid are congruent lower or upper pair of base angles of a trapezoid are congruent diagonals of a trapezoid are congruent

29 G: AB // CD, <ABC <ADC AB AD P: ABCD is a rhombus B C A D

30 G: FR bisects ED, FE RE P: FRED is a kite F E R D

31 Prove that the segments joining the midpoints of the sides of a rectangle form a rhombus. Use coordinate geometry. The distance formula is d= (x2-x1)^2 + (y2-y1)^2

32