4.1 Quadrilaterals Quadrilateral Parallelogram Trapezoid Isosceles Trapezoid Rhombus Rectangle Square

4.1 Properties of a Parallelogram Definition: A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. A B D C

4.1 Properties of a Parallelogram Opposite angles are congruent Opposite sides are congruent Diagonals bisect each other Consecutive angles are supplementary

4.1 Properties of a Parallelogram In the following parallelogram: AB = 7, BC = 4, What is CD? What is AD? What is mABC? What is mDCB? A B D C

4.2 Proofs Proving a quadrilateral is a parallelogram: Show both pairs of opposite sides are parallel (definition) Show one pair of opposite sides are congruent and parallel Show both pairs of opposite sides are congruent Show the diagonals bisect each other

4.2 Kites Kite - a quadrilateral with two distinct pairs of congruent adjacent sides. Theorem: In a kite, one pair of opposite angles is congruent.

4.2 Midpoint Segments The segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to ½ the length of the third side. A M N C B

4.3 Rectangle, Square, and Rhombus Rectangle - a parallelogram that has 4 right angles. The diagonals of a rectangle are congruent. A square is a rectangle that has all sides congruent (regular quadrilateral).

4.3 Rectangle, Square, and Rhombus A rhombus is a parallelogram with all sides congruent. The diagonals of a rhombus are perpendicular.

4.3 Rectangles: Pythagorean Theorem Pythagorean Theorem: In a right triangle, with the hypotenuse of length c and legs of lengths a and b, it follows that c2 = a2 + b2 Note: You can use this to get the length of the diagonal of a rectangle. a c b

4.4 The Trapezoid Definition: A trapezoid is a quadrilateral with exactly 2 parallel sides. Base Leg Leg Base Base angles

4.4 The Trapezoid Isosceles trapezoid: 2 legs are congruent Base angles are congruent Diagonals are congruent

4.4 The Trapezoid A B Median of a trapezoid: connecting midpoints of both legs M N D C

4.4 Miscellaneous Theorems If 3 or more parallel lines intercept congruent segments on one transversal, then they intercept congruent segments on any transversal.

5.1 Ratios, Rates, and Proportions Ratio - sometimes written as a:b Note: a and b should have the same units of measure. Rate - like ratio except the units are different (example: 50 miles per hour) Extended Ratio: Compares more than 2 quantities example: sides of a triangle are in the ratio 2:3:4

5.1 Ratios, Rates, and Proportions two rates or ratios are equal (read “a is to b as c is to d”) Means-extremes property: product of the means = product of the extremes where a,d are the extremes and b,c are the means (a.k.a. “cross-multiplying”)

5.1 Ratios, Rates, and Proportions b is the geometric mean of a & c …..used with similar triangles

5.1 Ratios, Rates, and Proportions Ratios – property 2: (means and extremes may be switched) Ratios – property 3: Note: cross-multiplying will always work, these may lead to a solution faster sometimes

5.2 Similar Polygons Definition: Two Polygons are similar  two conditions are satisfied: All corresponding pairs of angles are congruent. All corresponding pairs of sides are proportional. Note: “~” is read “is similar to”

5.2 Similar Polygons Given ABC ~ DEF with the following measures, find the lengths DF and EF: E 10 5 B D A 6 4 C F

5.3 Proving Triangles Similar Postulate 15: If 3 angles of a triangle are congruent to 3 angles of another triangle, then the triangles are similar (AAA) Corollary: If 2 angles of a triangle are congruent to 2 angles of another triangle, then the triangle, then the triangles are similar. (AA)

5.3 Proving Triangles Similar AA - If 2 angles of a triangle are congruent to 2 angles of another triangle, then the triangle, then the triangles are similar. SAS~ - If a an angle of one triangle is congruent to an angle of a second triangle and the pairs of sides including the two angles are proportional, then the triangles are similar

5.3 Proving Triangles Similar SSS~ - If the 3 sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar CSSTP – Corresponding Sides of Similar Triangles are Proportional (analogous to CPCTC in triangle congruence proofs) CASTC – Corresponding angles of similar triangles are congruent.

5.3 Proving Triangles Similar (example proof using CSSTP) Statements Reasons 1. mA = m D Given 2. mB = m E 2. Given 3. ABC ~ DEF 3. AA 4. 4. CSSTP

5.4 Pythagorean Theorem Pythagorean Theorem: In a right triangle, with the hypotenuse of length c and legs of lengths a and b, it follows that c2 = a2 + b2 Converse of Pythagorean Theorem: If for a triangle, c2 = a2 + b2 then the  opposite side c is a right angle and the triangle is a right triangle. c a b

5.4 Pythagorean Theorem Pythagorean Triples: 3 integers that satisfy the Pythagorean theorem 3, 4, 5 (6, 8, 10; 9, 12, 15; etc.) 5, 12, 13 8, 15, 17 7, 24, 25

5.4 Classifying a Triangle by Angle If a, b, and c are lengths of sides of a triangle with c being the longest, c2 > a2 + b2  the triangle is obtuse c2 < a2 + b2  the triangle is acute c2 = a2 + b2  the triangle is right c a b

5.5 Special Right Triangles Leg opposite the 45 angle = a Leg opposite the 90 angle = 45 a 90 45 a

5.5 Special Right Triangles Leg opposite 30 angle = a Leg opposite 60 angle = Leg opposite 90 angle = 2a 60 2a a 30 90

5.6 Segments Divided Proportionally If a line is parallel to one side of a triangle and intersects the other two sides, then it divides these sides proportionally A D E B C

5.6 Segments Divided Proportionally When 3 or more parallel lines are cut by a pair of transversals, the transversals are divided proportionally by the parallel lines A D B E C F

5.6 Segments Divided Proportionally Angle Bisector Theorem: If a ray bisects one angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the length of the 2 sides which form that angle. C A B D