Chapter 7 Proofs and Conditional Probability Students will be able to prove properties of quadrilaterals and using these new properties prove what the shape is. Students will develope an understanding of conditional probability and a better definition of independent probability.

Section 7.1 Students will review proofs of triangles and apply these ideas to proving properties of quadrilaterals. Students will use a 2-column proof to determine what type of shape is given. Students will learn and apply information about midsegments in triangles and trapezoids.

Quadrilateral Summary List of definitions and properties of the 6 main quadrilaterals. We discussed these properties and ideas of the last 2 weeks.

Trapezoid Regular Def: 1 pair of parallel sides Property: Angles on the same leg are supplementary Right Trapezoid Def: 1 pair of parallel sides and 2 right angles on the same leg Isosceles Trapezoid: Def: 1 pair of parallel sides and legs are congruent Property: Angles on the same base are congruent diagonals are congruent Kite Def: 2 pairs of congruent consecutive sides Properties: Angles formed by non congruent sides are congruent Diagonal connecting non congruent angles bisects the other diagonal Diagonal connecting non congruent angles bisects both angles Diagonals are perpendicular

Parallelogram Def: 2 pairs of parallel sides Properties: opposite sides congruent Opposite angles are congruent Consecutive angles supplementary Diagonals bisect each other Rectangle Def: 2 pairs of parallel sides and 4 congruent angles or 4 right angles Properties: Same as a parallelogram Diagonals are congruent

Rhombus Def: 2 pairs of parallel sides and all sides congruent Properties: same as a parallelogram Diagonals are perpendicular and angle bisectors Square Def: 2 pairs of parallel sides, 4 congruent angles or right angles and 4 congruent sides Properties: Same as a parallelogram Diagonals perpendicular, angle bisectors, bisect each other and congruent

Midsegment What is it? Connects 2 midpoints of the sides (legs) How many does a triangle have? 3 How many does a trapezoid have? 1

Midsegment in a Triangle What is special about midsegments in Triangles? Midsegment bisects sides (midpoint) Midsegment parallel to side it doesn’t touch Corresponding angles Midsegment half the length of the side it doesn’t touch Can you prove any of these?

Example

Trapezoid Midsegment Trapezoid is part of a triangle Midsegment is parallel to both bases Midsegment bisects both legs Midsegment is average of both bases (b1+b2)/2

Example

Distance and Midpoint How to calculate Midpoint is halfway point between endpoints, add endpoints together and divide by 2 Distance you can draw in a triangle and use Pythagorean theorem Difference of x coordinates square + difference of the y coordinates square and square room Look at worksheet we used to review ideas on quads, problems on the bottom

Example Calculate where the midsegment should be and prove it is parallel and half the length

Example Plot the following points Using the definitions and properties determine what the best classification of the quadrilateral is

Other terms and ideas Perpendicular Bisector Perpendicular to a side and bisects the side Equidistant Same distance Any point that is on the perpendicular bisector is equidistant to the endpoints of the segment being bisected These form isosceles triangles Remember altitude and height

Isosceles Triangles Proving the perpendicular bisector Altitude or Height If coming from the vertex, non congruent angle it bisects base and vertex making 2 congruent triangles

Example Given segment PM is a perpendicular bisector of segment AB Prove P is equidistant to A and B Determine a point that would make an isosceles triangle

Examples Prove the following is a parallelogram

Homework Worksheet on properties of quadrilaterals Worksheet on midsegments Worksheet on proofs