chapter 5 Based on work from pages 178-179, 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
G: BD bisects <ABC, <ADB is acute P: AB = BC
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.
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 31-36 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
E G: <1 comp. to <2 <3 comp. to <2 C 1 2 A P: CA // DB D 3 B
G: <1 supp. to <2 <3 supp. to <2 P: FLOR is a parallelogram
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 37-44...
G: FH // JM, <1 = <2 K JM = FH P: GJ = HK F 2 G H J M 1
G: CY AY, YZ // CA P: YZ bis. <AYB C Y A Z B
THE famous crook problem 50 deg x deg 132 deg.
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?
examine, list properties examine, list properties examine, list properties examine, list properties
examine, list properties 13 4 find the area of the trapezoid 21 5
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
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
Prove that the diagonals of a kite are perpendicular
kite rectangle square parallelogram rhombus quadrilateral isosc. trapezoid trapezoid
What am I?
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.
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
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
G: AB // CD, <ABC <ADC AB AD P: ABCD is a rhombus B C A D
G: FR bisects ED, FE RE P: FRED is a kite F E R D
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