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Geometry – Complete Solutions Manual
90° - answers The sum of an angle and its complement is _____? Parallel - answers A line which joins the midpoints of two sides of a triangle is _____ to the third side. Trapezoid - answers A quadrilateral that has only two sides parallel is a(n) _____. Perpendicular - answers A line parallel to one of two perpendicular lines is _____ to the other line. Radii (or diameters) - answers Two circles are congruent if they have equal _____. Right - answers An angle inscribed in a semicircle is a(n) _____ angle. Parallel - answers Tangents to a circle at opposite ends of a diameter are _____ to each other. Secant - answers A line that intersects a circle in two points is a(n) _____. Bisect - answers A diameter that is perpendicular to a chord will _____ the chord. 540° (or 3 straight angles) (n-2) 180° = (5-2) 180° = 3 × 180° = 540° - answers The sum of the interior angles of a pentagon is _____. Show your work. 2 (x + 2) = 2 - answers x = - 3 (5x - 2) + 2 = 26 - answers x = 2 5 (n + 2) = 3n - answers x = - 5 (5 - x) = -5 - answers x = 6 1+ 4 (2x + 1) = 5 - answers x = 0 x + 3 = 3 (11 - 3x) - answers x = 3 2x - 4 = -4 (x - 5) - answers x = 4
- 3x = 2 (7 + 5x) - answers x = -
- 2 (3x + 1) = 7x - 5 - answers x =
- 2 (x + 1) = 6x - 46 - answers x =
- 7 (2 - x) - 5= 30 - answers x = -
- 5 (x - 6) = 3 + 8 (x + 9) - answers x = -
- 245 = -5 (4x - 17) - answers x = -
- 17 (3x + 2) = -272 - answers x = -
- 5 (7x - 13) = -30 - answers x =
- 12 (-2x - 9) = -12 - answers x = -
- -10 (-4x + 7) = 130 - answers x =
- -210 = -7 (9x - 60) - answers x =
- 12 (-x + 5) = 144 - answers x = -
- 9 (-4x - 29) = 99 - answers x = -
- 32 - answers x-7 =
- -5 - answers 3x + 12 = -
- 4 - answers 5x - 7 =
- 10 - answers 3x -30 =
- 5 - answers 5 - 3x = -
- -13 - answers 2x + 9 = x -
- 1 - answers 3x - 7 = 5x -
- -2 - answers 4x + 5 = -2x -
- 2 - answers 12x - 9 = 2x +
- 3 - answers 7(x - 4) = -
Distance formula on a coordinate plane - answers d = √[( x₂ - x₁)² + (y₂ - y₁)²] Horizontal distance - answers d = | x2 -x1| Vertical Distance - answers d = |y2 -y1| Pythagorean Theorem - answers a²+b²=c² (another way to find distance on triangles) Midpoint formula on a coordinate plane - answers M = ((x₁+x₂)/2, (y₁+y₂)/2) Partition formula - answers (Px,Py) = ((M(x2)+N(x1))/(M+N) ,(M(y2)+N(y1))/(M+N)) or 1.Find fraction 2. Distance between x's and y's then multiply by fraction. 3. Take INITIAL point and add the 'fraction of distance'. ORDER MATTERS Sum of the Exterior Angles of a Polygon - answers Sum = 360 degrees sum of interior angles of a polygon - answers Sum = 180(n-2) One interior angle of a regular polygon - answers 1 angle = (n-2)180/n One exterior angle of a regular polygon - answers 1 angle = 360/n Translation Rule - answers (x,y)->(x+a, y+b) Rotate 90° - answers (x,y)->(-y,x) Switch and change the FIRST Remember - Implies CCW rotate 180° - answers (x,y)->(-x,-y) Negate Both Rotate 270° - answers (x,y)->(y,-x) Switch and change the SECOND Remember implies CCW Reflect across y = x - answers (x,y)->(y,x) JUST SWITCH Reflect across y = -x - answers (x,y)->(-y,-x) Switch and Negate BOTH Total degrees in a Triangle (triangle sum) - answers Sum of THREE angles = 180°
Exterior Angle of a Triangle Theorem - answers The measure of an exterior angle of a triangle equals the sum of the measures of the two REMOTE INTERIOR angles Equilateral Triangle Corollary - answers If a triangle is equilateral, then it is equiangular Equilateral Triangle Theorem - answers If a triangle is equilateral, then it is also equiangular. Each angle is 60° and all sides congruent. Isosceles Triangle Theorem - answers If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Isosceles Triangle Conjecture - answers If a triangle is isosceles, then its base angles are congruent. Isosceles Triangle - answers a triangle with two congruent sides and two congruent base angles Have vertex angle? 180-VERTEX / 2 = base angle The Triangle Congruence Postulates are : _____, _____, _____, _____, _____ - answers SSS ,SAS, ASA, AAS, HL Similarity Postulates - answers ~SSS, ~AA, ~SAS When SIMILAR or COMPARING LIKE PIECES set up.... - answers CORRESPONDING PROPORTIONS! (good for MANY applications - Set them equal and CROSS X MULTIPLY) A/B = C/D midpoint / bisector - answers A point that divides a segment into two congruent segments. A point, line, or ray (anything) that Cuts 🔪 side or angle into.... 2 = PARTS! Mid-segment of triangle - answers 2(small MIDSEGMENTs) = BIG BASE (look for tick marks or 2:1 ratios) Midsegment of a Trapezoid Theorem - answers The Midsegment of trapezoid is parallel to each base and it's length = 1/2 (b1 + b2) Slope formula needed if given 2 ordered pairs - answers m = (y2 - y1) / (x2 - x1) REMEMBER you can you slope in combination with PYTHAGOREAN'S THEOREM to find LENGTH!! Perpendicular - answers IMPLIES Right Angles Perpendicular Slopes: FLIP/NEGATE SLOPE {Neg. reciprocal a/b to -b/a}
X on the top? MULTIPLE! X on the BOTTOM? Take what's on the top and divide by what's on the left! INVERSE TRIG RATIOS (Also known as ARC ratios) - answers SIN(O/H)¯ ¹ = β° COS(A/H)¯ ¹ = β° TAN(O/a)¯ ¹ = β° ArcSIN, ArcCOS ArcTan Need an angle? Go BACK and get it! Put (ratio) in calculator hit 'second' trig function!!! EASY Special Right Triangles - answers 30°60°90° S, M=S√3 , Hyp = 2S 45°45°90° Hyp =Leg√ Area Square - answers A = s² A = bh A = lw A = ½ d1d Area of a rhombus - answers A=1/2d1d2, A = bh Area of a kite - answers A=1/2d1d Area of a rectangle - answers A = bh A = lw Area of a parallelogram - answers A=bh (height is PERPENDICULAR) Area of a trapezoid - answers A=1/2(b1+b2)h Area of a regular polygon - answers A=1/2aP Circumference of a circle - answers C = 2πr or C = πd Arc Length Formula - answers AL/2πr = (Degrees)°/360° AL/2πr = (Radians)/2π (AL is Arc length) Area of a Circle - answers A = πr² area of a sector formula - answers SA/πr^2 = (Degrees)°/360° SA/πr^2 = (Radians)/2π (SA is Sector Area) Area of a triangle - answers A=1/2bh Equation of a circle in standard form - answers (x-h)^2 + (y-k)^2 = r^ Remember RADIUS HAS BEEN SQUARED!!
central angle - answers angle with vertex is at center of the circle and angle = arc measure inscribed angle - answers angle whose vertex is on the circle 2(angle) = Arc measure middle angle in a circle - answers angle 'somewhere' in the middle of circle 1/2(arc1 + arc2) = angle measure Angle outside the circle - answers 1/2(big arc-small arc) = measure of exterior angle Density - answers d =mass/volume d = STUFF/SPACE Read the UNITS, they will tell you what to divide: Bacteria in a petri dish, means: BACTERIA/in^ (STUFF / SPACE) population density - answers number of people / Area of land Surface Area of a Prism/Cylinder - answers SA = Ph+2B PERIMETER of the BASE x HEIGHT + 2 AREA of the BASES (Remember formulas change depending on the shape of the BASE!!!!) Surface area of a sphere - answers SA = 4πr² surface area of a pyramid/cone - answers 1/2 Pl+B l = SLANT HEIGHT (careful here) 1/2 PERIMETER of the BASE x SLANTE HEIGHT + 1 AREA of the BASE (Remember formulas change depending on the shape of the BASE!!!!) Volume of a cylinder/prism - answers V=Bh AREA of the BASE x perpendicular height (Remember formulas change depending on the shape of the BASE!!!!) Volume of a cone/pyramid - answers V=1/3Bh 1/3 AREA of the BASE x perpendicular height (Remember formulas change depending on the shape of the BASE!!!! Volume of a Sphere Formula - answers V=4πr³/ Scale factor of a dilation - answers The ratio of a side length of the IMAGE to the corresponding side length of the PREIMAGE or New (x,y)/Old (x,y) on the coordinate plane. K = (New/Old)
If Consecutive __________ ___________ are suppl., then 2 lines are parallel - answers interior angles Rhombus - answers //gram with 4 congruent sides Reactangle - answers //gram with 4 right angles Square - answers //gram with 4 right angles and 4 congruent sides The diagonals of a rectangle are - answers Congruent The diagonals of a Rhombus are - answers Perpendicular Each diagonal of a rhombus _______ 2 _________ or the Rhombus - answers Bisects, angles Trapezoid - answers Quad. with exactly 1 pair of // sides Isosceles Trapezoid - answers Trapezoid with congruent legs mid segment of a trapezoid - answers a segment that connects the midpoints of the legs of a trapezoid The mid segment of a trapezoid is ________ to the bases - answers parallel Mid Segment length formula - answers 1/2(Length of base 1 + Length or base 2) Kite - answers Quad. wit exactly 2 pairs of consecutive congruent sides If a quad. is a kite, then the ______ are _________ - answers diagonals, perpendicular If a quad. is a kite, then exactly ____ pair of opposite angles are ______ - answers 1, congruent Similarity transformation - answers Figure and its image have the same shape but not necessarily size Dilation - answers (Scaling) similarity Transformation that enlarges or reduces a figure proportionally Center of Dilation - answers The fixed point in a dilation Scale Factor - answers The ratio of side lengths of image the to the corresponding length of pre-imagon
Coordinate Notion for a Dilation with respect to the origin. - answers (x,y) -> (kx, ky) Reduction - answers 0 < k < Enlargement - answers k> Ratio - answers a comparison of 2 quantities Proportion - answers a statement that 2 ratios are equal Polygons are similar if... - answers corresponding angles are congruent ad corresponding side are proportional Angle-Angle similarity (AA~) - answers If 2 angles of 1 triangle are congruent to 2 angles of another triangle then the 2 triangles are ~ Side-Side-Side Similarity (SSS~) - answers All sides of 1 triangle are proportional to all of the sides of another triangle then the triangles are ~ Side-Angle-Side Similarity (SAS~) - answers If 2 sides of 1 triangle are congruent to 2 sides of another triangle and the included angle of the sides are congruent then the triangles are ~ Triangle Proportionality Theorem - answers If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. Converse of the Triangle Proportionality Theorem - answers If a line divides two sides of a triangle proportionally, then it is parallel to the third side. Triangle Midsegment - answers Segment with endpoints tat are the midpoints of 2 sides of a triangle Triangle Midsegment Theorem - answers 1. The midsegment is parallel to 1 side of a triangle
- The midsegment is 1/2 the length of the parallel side. Proportional Parts of Parallel Lines Corollary - answers If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. Congruent Part of Parallel Sides Corollary - answers If three or more parallel lines intersect two transversals, then they cut off the transversals congruentlly. Parts of Similar Triangles - answers If 2 triangles are similar the lengths of corresponding altitudes, angle bisectors, medians are proportional to the lengths of corresponding sides.
What does CAH stand for? - answers Cosine Adjacent Hypotenuse What does TOA stand for? - answers Tangent Opposite Adjacent What is the Input of an inverse trig function? - answers ratio of sides What is the output of an inverse trig function? - answers angle measure Angle of Elevation - answers The angle that you line of sight makes with a line that is parallel to the ground Angle of Depression - answers congruent to the angle of elevation Circle - answers set of all parts in a plane equidistant from the center Radius - answers Segment whose endpoints are the center and any point on the circel Chord - answers Segment whose endpoints are on the circle Diameter - answers Chord that contains the center of the circle Central Angle - answers Angle whose vertex is the center of the circle and sides are chords Circles are congruent i.f.f. - answers they have congruent radii Circles are similar.... - answers All circles are similar Circle are concentric if - answers they are co-planer and share the same center Circumfrence - answers distance around a circle A polygon is ___________ in a circle if all its vertices lie on the circle - answers inscribed A circle is __________ about a polygon if it contains all the vertices of the polygon - answers circumscribed Arc - answers a section of the circle Semicircle - answers an arc whose measure is 180 arc length formula - answers (measure of the arc)/360 x πd circumfrence formula - answers C=2πr
Diameter formula - answers d=πr² Chord Thm - answers In the same or congruent circle, two minor arcs are congruent i.f.f. their chords are congruent. If one chord is a perpendicular bisector of another chord, then... - answers the 1st chord is a diameter If a diameter of a circle is perpendicular to a chord, then - answers the diameter bisects the chord and its arc In the same or congruent circle, two chords are congruent i.f.f. - answers they are equidistant from the center Inscribed angle - answers an angle whose vertex is on a circle and whose sides contain chords of the circle Intercepted arc - answers an arc that consists of endpoints that lie on the sides of an inscribed angle and all the points of the circle between the endpoints If you have an inscribed quadrilateral in a circle, - answers then its opposite angles are supplements Tangent - answers a line that intersects the circle in exactly one point point of tangency - answers the point where a circle and a tangent intersect Common tangents - answers A line, ray, or segment that is tangent to two co-planar circles Tangent thm - answers In a plane, a line is tangent to a circle i.f.f. it is perpendicular to the radius drawn to the point of tangency Party Hat Thm - answers If two segments from the same exterior point are tangent to a circle, then they are congruent Secant - answers A line that intersects a circle in exactly 2 points Intersection on a Circle Thm - answers m<1 = 1/2 (m arc AB) Angles Inside the Circle Thm - answers If 2 chords intersect inside a circle, then the measure of each angle is 1/2 the sum of the measures of the arcs intercepted by the angle and its vertical angle
Probability of Compliments - answers consists of all the outcomes in the sample space that are not included as an outcome of Event A Permuntation - answers arrangement of objects in which order is important Combination - answers An arrangement of objects in which order is not important Equation for a Permutation - answers nPr = n!/(n-r)! equation for a Combination - answers nCr = n!/(n-r)! r! Compound Event - answers 2 or more simle events Independent Event - answers compound event where the outcome of event A doesn't affect the outcome of event B Dependent Event - answers Compoud event where the outcome of event A affects the outcoem of event B Independent Event Equation - answers P(A n B) = P(A)P(B) Dependent Event Equation - answers P(A n B) = P(A)P(B|A) Conditional Probability - answers The probability tat event B occurs given that event A had already occurred Notation of conditional probability - answers P(B|A) Overlapping Events - answers One or more outcome in common Mutually Exclusive Events - answers No outcomes in common Apothom - answers perpendicular distance rfom the center to a side Apothom equation - answers A=1/2pa Composite Figure - answers Figures that can be separated into regions that are basic figures Polyhedron - answers 3 dimensional figure whose surfaces are polygons Net - answers two dimensional pattern that can be folded up to a three dimensional figure Prism - answers three dimensional figure with excactly two congruent, parallel faces,
Lateral Area - answers sum of the are of the latteral faces Lateral Area formula (Prism) - answers LA=Ph Surface Area - answers Sum of each face plus the area of the two bases Surface Are formula (Prism) - answers SA= LA+2B Lateral Area formula (Cylinder) - answers LA=2πrh Surface Area formula (Cylinder) - answers SA=2πrh+2πr² Lateral Area formula (Pyramid) - answers LA= 1/2pl Surface Area formula (Pyramid) - answers SA=LA+B Lateral faces - answers All congruent isosceles triangle Slant height - answers the height of eah lateral face Altitude - answers perpendicular segment from vertex to center of base Lateral Area formula (Cone) - answers LA=πrl Surface Area formula (Cone) - answers SA=πrl+πr² Formula: Area of a Parallelogram - answers A=bh Formula: Area of a Triangle - answers A=1/2bh Formula: Area of a Trapezoid - answers A=1/2h(b1+b2) Formula: Area of a Rhombus and Kite - answers A=1/2d1d Base of a Parallelogram or Triangle - answers any of its sides Altitude (Triangle) - answers segment perpendicular to the line containing he base frawn from the side to the opposite base Height (Triangle) - answers Length of altitude Perimeters and Areas of Similar Figures Thm. - answers Similarity ratio of 2 similar figures is a/b or a²/b²
- wz≅ wz : reflexive property of congruence
- wxz ≅ yxz : hl congruence theorem
- ∠yzx≅∠wzx : corresponding parts of congruent angles are congruent use the information and diagram to complete the problem. given: ab ∥ed bc∥df bc ≅df prove: △abc ≅ △edf match each statement in the proof with the correct reason. - answers 0.67 of 1
- ab ∥ed bc∥df bc ≅df : given
- ∠bca ≅∠dfe, ∠bac ≅∠def : alternative interior angles theorem
- △abc ≅ △edf : hl congruence theorem use the information and diagram to complete the problem. given: j is the midpoint of hl △hij and △lkj are right triangles. ij ≅kj prove: △hij ≅△lkj match each statement in the proof with the correct reason. - answers 0.5 of 1
- j is the midpoint of hl △hij and △lkj are right triangles. ij ≅kj : given
- j bisects hl : definition of bisect
- hj ≅ lj : definition of midpoint
- △hij ≅△lkj : hl congruence theorem use the triangles to answer the question. if △abc ≅ △def, what is the length of df? - answers 13 use the triangles to answer the question. if △abc ≅ △fde, what is the measure of ∠acb? - answers 34.
lesson 17 - answers properties of parallelograms examine parallelogram abcd. sides cd and ab have lengths of y + 18 and 4y, respectively. determine the value of y and answer the following question. what is the length of ab? - answers 24 use the figure and the information to complete the proof. given: abcd is a parallelogram. prove: the diagonals ac and bd bisect each other. match each statement in the proof with the correct reason. - answers 0.67 of 1
- abcd is a parallelogram : given
- ab ≅ cd, ad ≅ bc : corresponding parts of congruent triangles are congruent
- 9 ≅ 11 : vertical angles theorem
- ab cd : definition of parallelogram
- 2 ≅ 5 : alternative interior angles
- aeb ≅ ced : asa congruence theorem
- ae ≅ ce and be ≅ de : opposite sides of a parallelogram are congruent
- e is the midpoint of ac, e is the midpoint of bd : definition of midpoint
- ac and bd bisect each other : definition of bisect examine parallelogram abcd. angles a and d have measures of x and (x+30), respectively. what is the value of x? - answers 75 examine parallelogram abcd and carefully read the description of paola's proof. given: ab∥cd and ad∥bc and ac is a diagonal of both sets of parallel lines.
