Chapter 7
(369) parallelogram ABCD has length AB = x + 4 and CD = 12. solve for x
AB = CD > x + 4 = 12 = 8
(398) trapezoid
a quadrilateral with exactly one pair of parallel sides
(403) how are trapezoids and kites different
a trapezoid has one pair of parallel sides > a kite has two pairs of consecutive congruent sides
(361) equilateral polygon
all sides are congruent
(395) a square is __________ a rhombus
always
(372) why a parallelogram always a quadrilateral, but a quadrilateral is only sometimes a parallelogram.
anything with four sides is a quadrilateral, > for it to be a parallelogram, it must have two pairs of parallel sides
(360) in a polygon, two ____________ that are endpoints of the same side are called ________ vertices
endpoints / consecutive
(399) isosceles trapezoid base angles theorem
if a trapezoid is isosceles, then each pair of base angles is congruent
(376) PARALLELOGRAM OPPOSITE SIDES CONVERSE
if both pairs of opposite sides of a qaudrilateral ore congruent, then it is a parallelogram
(378) opposite sides parallel and congruent theorem
if one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parralelogram
(364) why do all vertices connected by a diagonal of a polygon have to be nonconsecutive.
if they are consecutive, they will be a side
(396) will a diagonal of a square ever divide the square into tow equilateral triangles.
no it always divides it into two right angles
(362) is home plate a regular polygon?
no. it is not equilateral or equiangular
(384) three interior angles of a quadrilateral are 67°, 67° and 113°. can you conclude that it is a parallelogram?
no. it might be a trapezoid
(388) rhombus
parallelogram with four congruent sides
(394) what quadrilaterals are always equiangular
rectangle and square
(410) if a parallelogram has 4 congruent sides and one interior angle of 56°, what is it
rhombus
(393) what is another name for an equilateral rectangle
square
(365) the four exterior angles of a polygon are x, 65°, 106° and 78°. what is x?
> x + 65 + 106 + 78 = 360 > x + 171 = 282 > x = 111
(389) a square is also a
1. parallelogram 2. rhombus 3. rectangle
(409) for quadrilateral DEFG, where > segment DE = 3x + 1 and > segment GF = 2x + 7 what value of x will make it a parallelogram
1. DE will have to be congruent to GF if a parallelogram 2. DE≅GF > 3x + 1 = 2x + 7 > x = 6
(408) what is the sumer of the measures of the interior angles of a hexagon.
1. a hexagon has 6 sides 2. interior angles theorem: > (n-2) * 180 = (6-2) * 180 = 4 * 180 = 720
(379) five ways to prove a quadrilateral is a parallelogram
1. both pairs of opposite sides are parallel 2. both paris of opposites sides are congruent 3. both pairs of opposite angles are congruent 4. one pair of opposite sides are congruent and parallel 5. diagonals bisect each other
(371) find the intersection of the diagonals of parallelogram LMNO with L (1,4) M (7,4) N (6,0) and O (0,0)
1. by the parallelogram diagonals theorem, the diagonals bisect each other 2. this you can find the midpoint of segment OM 3. x = (7+0) /2 = 3.5 > 7 = (4+0) / 2 = 2 4. intersection is (3.5,2)
(406) given EFGH is a kite and >segment EF≅segment FG and > segment EH≅segment GH prove ∠E≅∠G and > ∠F is not≅∠H
1. construct segment FH which is congruent to itself (reflextive property of congruence 2. so ∆FGH≅∆FEH (SSS congruence theorem) 3. ∠E≅∠G (corresponding parts of congruent triangles are congruent) 4. ∠F cannot be therefore ∠H under the definition of a kit
(357) solve 3(2+x) = -9 using (2+x) as single quantity.
1. divide both sides by 3 > [3(2+x)]/3 = -9 / 3 > 2+x = -3 2. solve for x >x = -5
(383) prove the parallelogram opposite angles converse given - parallelogram ABCD - m∠A≅m∠C and - m∠B≅m∠D
1. let m∠A = m∠C = x° and > m∠B = m∠D = y° (definition of congruent) 2. m∠A + m∠B + m∠C + m∠D > = x + y + x + y = 360° (corollary to poygon interior angles theorem) 3. 2x + 2y = 360° (simplify) 4. 2 (x + y ) = 360 (distributive property) 5. x + y = 180 (division property of equality) 6. m∠A + m∠B = 180 and > m∠C + m∠D = 180 (substitition) 7. ∠A and ∠B are supplementary and > ∠C and ∠D are supplementary (definition of supplementary angles) 8. segment BC ll segment AD and > segment AB ll segment DC (consecurtive interior angles converse) 9. ABCD is a parallelogram (definition of parallelogram)
(358) quadrilaterals can be classified into what three groups
1. parallelograms 2. trapezoids 3. kits
(405) for triangle JLN > given segment JL≅segment LN and > given segment KM is midsegement of ∆JLN Prove the JKMN is an isoceles trapezoid
1. segment KM ll segment JN (triangle midsegment theorem 2. JKMN is a trapezoid (definition of trapezoid) 3. ∠LJN ≅∠LNJ (base angles theorem 4. JKMN is an isosceles trapezoid (isoceles trapezoid base bangles converse
(361) the sum of the angles of a convex polygon is 900°?
1. use polygon interior angles theorem. > (n-2) * 180 = 900 > n-2 = 5 > n = 7
(374) in parallelogram STUV, m∠TSU = 32°, m ∠USV = x², m ∠TUV = 12x. what is m∠USV?
16°
(382) a diagonal of parallelogram ABCD is bisected so that the two segment of AC are 4x - 2 and 3x + 3. what is x
4x - 2 = 3x + 3 x = 5
(363) the exterior angles of a 4 side polygon are 67°, x°, 2x° & 89°. What is the value of x?
> 67 + x + 2x + 89 = 360 > 156 + 3x = 360 > 3x = 204 > x = 68°
(377) for parallelogram PQRS, PQ = x + 9, QR = x + 7, RS = 2x - 1 and SP = y. Solve for x & y.
PQ = RS > x + 9 = 2x - 1 > 10 = x > 10 = x QR = PS > x + 7 = y > 10 + 7 = y > 17 = y
(391) rectangle diagonals theorem
a parallelogram is a rectangle if and only if it's giagonals are congruent
(390) rhombus diagonals theorem
a parallelogram is a rhombus if and only if it's diagonals are perpendicular
(388) rectangle
a parallelogram with four right angles
(368) prallelogram
a quadrilateral > with both pairs of opposite sides parallel
(401) kite
a quadrilateral that has two pairs on sonsecutive congruent sides, > but opposite sides that are not congruent
(373) find the intersection of the diagonals of parallelogram WXYZ if W(-2,5) Y (2,5) X (4,0) Y (0,0)
since intersections bisect each other 1. find midpoint of WY 2. x = (2+0)/2 = 1 > y = (5+0)/2 = 2.5 3. (1,2.5)
(402) a ____________ that is a parallelogram is also a ___________ and rectangle
square / parallelogram
(400) mid segment of a trapezoid
the segment that connects the midpoints of its legs
(381) a quadrilateral has four congruent sides. is it a parallelogram?
yes. because both sets of opposite sides are congruent.
