In the figure, square RSTV has its vertices on the sides of A, C, and F are three of the vertices of the cube shown in
2 a
A B C 60 ° 30 ° c a b It would be best to memorize the sketch in Figure 5.36. So that you will more eas- ily recall which expression is used for each side, remember that the lengths of the sides follow the same order as the angles opposite them. Thus, Opposite the 30° smallest angle is a length of shortest side. Opposite the 60° middle angle is length of middle side. Opposite the 90° largest angle is 2a length of longest side. ∠ a 1 3 ∠ ∠ EXAMPLE 4 Find the lengths of the missing sides of each triangle in Figure 5.37. Exs. 8–10 60 ° 30 ° a 2 a 3 a Figure 5.36 60 ° 30 ° a ? 5 S T R ? Figure 5.37 60 ° 30 ° b V W U 20 ? ? Z c Y X 60 ° 30 ° ? ? 9 The converse of Theorem 5.5.1 is true and is described in the following theorem. 5.5 Solution a b c XZ = 2 XY = 2 3 13 = 6 13 L 10.39 = 9 13 3 = 313 L 5.20 ZY = XY 13 : 9 = XY 13 : XY = 9 13 = 9 13 13 13 UV = VW 1 3 = 10 13 L 17.32 UW = 2 VW : 20 = 2 VW : VW = 10 ST = RS 1 3 = 5 13 L 8.66 RT = 2 RS = 2 5 = 10 쮿 쮿 EXAMPLE 5 Each side of an equilateral triangle measures 6 in. Find the length of an altitude of the triangle. 60 60 6 6 a Figure 5.383 a
a 602 a
b Solution The equilateral triangle shown in Figure 5.38a is separated into two 30°-60°-90° triangles by the altitude. In the 30°-60°-90° triangle in Figure 5.38b, the side of the equilateral triangle becomes the hypotenuse, so and . The altitude lies opposite the 60° angle of the 30°-60°-90° triangle, so its length is or 3 13 in. L 5.20 in. a 1 3 a = 3 2a = 6 Exs. 11–13 If the length of the hypotenuse of a right triangle equals the product of and the length of either leg, then the angles of the triangle measure 45°, 45°, and 90°. 12 THEOREM 5.5.3 GIVEN: The right triangle with lengths of sides a, a and See Figure 5.39. PROVE: The triangle is a 45°-45°-90° triangle Proof In Figure 5.39, the length of the hypotenuse is , where a is the length of either leg. In a right triangle, the angles that lie opposite the congruent legs are also congruent. In a right triangle, the acute angles are complementary, so each of the congruent acute angles measures 45°. a 1 2 a 1 22 a
45 45 a a Figure 5.39 쮿 The converse of Theorem 5.5.2 is also true and can be proved by the indirect method. Rather than construct the proof, we state and apply this theorem. See Figure 5.41. EXAMPLE 6 In right , . See Figure 5.40. What are the measures of the angles of the triangle? If , what is the length of or ? ST RS RT = 12 12 RS = ST 䉭RST R S T Figure 5.4030 a
2 a
Figure 5.41 30 c c 1 2 Figure 5.42 Solution The longest side is the hypotenuse , so the right angle is and . Because , the congruent acute angles are R and T and . Because , . RS = ST = 12 RT = 12 12 m ∠R = m∠T = 45° ∠s RS ⬵ ST m ∠S = 90° ∠S RT 쮿 If the length of the hypotenuse of a right triangle is twice the length of one leg of the tri- angle, then the angle of the triangle opposite that leg measures 30°. THEOREM 5.5.4 An equivalent form of this theorem is stated as follows: EXAMPLE 7 In right with right , and see Figure 5.43. What are the measures of the angles of the triangle? Also, what is the length of ? Solution Because is a right angle, and is the hypotenuse. Because , the angle opposite measures 30°. Thus, and . Because lies opposite the 60° angle, . AC = 12.3 13 L 21.3 AC m ∠B = 60° m ∠A = 30° BC BC = 1 2 AB AB m ∠C = 90° ∠C AC BC = 12.3 AB = 24.6 ∠C 䉭ABC If one leg of a right triangle has a length equal to one-half the length of the hypotenuse, then the angle of the triangle opposite that leg measures 30° see Figure 5.42. 12.3 24.6 B C A Figure 5.43 쮿1. For the 45°
- 45° - 90° triangle shown, suppose that . Find: a BC b AB2. For the 45°
- 45° - 90° triangle shown, suppose that Find: a AC b BC3. For the 30°
- 60° - 90° triangle shown, suppose that . Find: a YZ b XY4. For the 30°
- 60° - 90° triangle shown, suppose that . Find: a XZ b YZ In Exercises 5 to 22, find the missing lengths. Give your answers in both simplest radical form and as approximations to two decimal places.5. Given: Right with
and Find: YZ and XY XZ = 8 m ∠X = 45° 䉭XYZ XY = 2a XZ = a AB = a 1 2. AC = a A C B 45° 45° Exercises 1, 2 Y 60° 30° X Z Exercises 3, 4 X Y Z Exercises 5–8 Exercises 5.512. Given: Right with
and Find: DE and DF13. Given: Rectangle HJKL with diagonals
and Find: HL , HK, and MK m ∠HKL = 30° JL HK EF = 12 13 m ∠E = 2 m ∠F 䉭DEF6. Given: Right with
and Find: XZ and YZ7. Given: Right with
and Find: XZ and YZ8. Given: Right with
and Find: XZ and YZ9. Given: Right with
and Find: DF and FE DE = 5 m ∠E = 60° 䉭DEF XY = 12 12 m ∠X = 45° 䉭XYZ XY = 10 12 XZ ⬵ YZ 䉭XYZ XY = 10 XZ ⬵ YZ 䉭XYZ F D E Exercises 9–12 L H 6 3 K J M6 2
R T S V10. Given: Right with
and Find: DF and DE11. Given: Right with
and Find: DE and FE FD = 12 13 m ∠E = 60° 䉭DEF FE = 12 m ∠F = 30° 䉭DEF14. Given: Right with
and Find: RS and ST m ∠STV = 150° RT = 6 12 䉭RST In Exercises 15–19, create drawings as needed.15. Given: with and
Find: AC and AB16. Given: Right with
and Find: PM and PN17. Given: with , ,
and Find: RS and RT18. Given: with
with W on Find: ZW19. Given: Square ABCD with diagonals
and intersecting at E Find: DB20. Given: with angles
as shown in the drawing Find: NM , MP, MQ, PQ, and NQ21. Given: with angles
as shown in the drawing Find: XY HINT: Compare this drawing to the one for Exercise 20. 䉭XYZ MP ⬜ NQ 䉭NQM DC = 5 13 AC DB YZ = 6 XY ZW ⬜ XY XY ⬵ XZ ⬵ YZ 䉭XYZ ST = 12 m ∠S = 60° m ∠T = 30° 䉭RST MN = 10 12 MP = PN 䉭MNP BC = 6 m ∠A = m∠B = 45° 䉭ABC 60° 30° P 45° 45° 3 N Q M 60° 75° X 45° Y Z 1222. Given: Rhombus ABCD in which diagonals
and intersect at point E; Find: AC DB = AB = 8 DB AC 8 8 4 4 D 8 4 D 200 ? 400 W E T G S N 30°23. A carpenter is working with a board that
24. To unload groceries from a delivery truck at the Piggly
Wiggly Market, an 8-ft ramp that rises 4 ft to the door of the trailer is used. What is the measure of the indicated angle ? ∠D25. A jogger runs along two sides of an open rectangular lot.
If the first side of the lot is 200 ft long and the diagonal distance across the lot is 400 ft, what is the measure of the angle formed by the 200-ft and 400-ft dimensions? To the nearest foot, how much farther does the jogger run by traveling the two sides of the block rather than the diagonal distance across the lot?26. Mara’s boat leaves the dock at the same time that Meg’s
boat leaves the dock. Mara’s boat travels due east at 12 mph. Meg’s boat travels at 24 mph in the direction N 30° E. To the nearest tenth of a mile, how far apart will the boats be in half an hour? In Exercises 27 to 33, give both exact solutions and approximate solutions to two decimal places.27. Given: In ,
bisects and Find: DC and DB28. Given: In ,
bisects and Find: DC and DB29. Given: is equiangular and
bisects bisects Find: NQ ∠MQN QR ∠MNQ NR NR = 6 䉭MNQ AC = 10 AB = 20 ∠BAC AD 䉭ABC AB = 12 m ∠B = 30° ∠BAC AD 䉭ABC C A B D Exercises 27, 28 R M 6 N Q30. Given: is an isosceles right triangle
M and N are midpoints of and Find: MN SV ST 䉭STV N 20 V T M S B C D A Exercises 31, 3231. Given: Right with
and ; point D on ; bisects and Find: BD AB = 12 ∠BAC AD BC m ∠BAC = 60° m ∠C = 90° 䉭ABC32. Given: Right with
and ; point D on ; bisects and Find: BD AC = 2 13 ∠BAC AD BC m ∠BAC = 60° m ∠C = 90° 䉭ABC is wide. After marking off a point down the side of length , the carpenter makes a cut along with a saw. What is the measure of the angle that is formed? ∠ACB BC 33 4
in. 33 4
in. C A B 33 4
33 4
33. Given: with , ,
and Find: AB HINT: Use altitude from C to as an auxiliary line. AB CD BC = 12 m ∠B = 30° m ∠A = 45° 䉭ABC A B C 30° 45° M N T Q P 12 120°34. Given:
Isosceles trapezoid MNPQ with and ; the bisectors of MQP and NPQ meet at point T on Find: The perimeter of MNPQ MN ∠s m ∠M = 120° QP = 1235. In regular hexagon ABCDEF,
. Find the exact length of: a Diagonal b Diagonal36. In regular hexagon ABCDEF, the
length of AB is x centimeters. In terms of x, find the length of: a Diagonal b Diagonal CF BF CF BF AB = 6 inches F E B A D C Exercises 35, 36 Y V Z X W D E C A B37. In right triangle XYZ, and . Where
V is the midpoint of and , find VW. HINT: Draw XV. m ∠VWZ = 90° YZ YZ = 4 XY = 338. Diagonal separates pentagon
ABCDE into square ABCE and isosceles triangle DEC. If and , find the length of diagonal . HINT: Draw DF ⬜ AB. DB DC = 5 AB = 8 EC Segments Divided Proportionally The Angle-Bisector Theorem Ceva’s Theorem Segments Divided Proportionally 5.6 KEY CONCEPTS In this section, we begin with an informal description of the phrase divided proportion- ally . Suppose that three children have been provided with a joint savings account by their parents. Equal monthly deposits have been made to the account for each child since birth. If the ages of the children are 2, 4, and 6 assume exactness of ages for sim- plicity and the total in the account is 7200, then the amount that each child should re- ceive can be found by solving the equation Solving this equation leads to the solution 1200 for the 2-year-old, 2400 for the 4-year- old, and 3600 for the 6-year-old. We say that the amount has been divided proportion- ally. Expressed as a proportion, this is 1200 2 = 2400 4 = 3600 6 2x + 4x + 6x = 7200Parts
» Elementary Geometry for College Students
» TEST 174 Elementary Geometry for College Students
» 4 + 3 ⫽ 7 and an angle has two sides. 2. 4 + 3 ⫽ 7 or an angle has two sides.
» If P, then Qf premises P C. ‹ Q
» If it is raining, then Tim will stay in the house. 2. It is raining.
» If a man lives in London, then he lives in England. 2. William lives in England.
» If P, then Q 2. P If P, then Q If P, then Q P Q
» If an angle is a right angle, then it measures 90°. 2. Angle A is a right angle.
» a Chicago is located in the state of Illinois.
» a Christopher Columbus crossed the Atlantic Ocean.
» If Alice plays, the volleyball team will win. 6. Alice played and the team won.
» The first-place trophy is beautiful. 8. An integer is odd or it is even.
» Matthew is playing shortstop. 10. You will be in trouble if you don’t change your ways.
» If the diagonals of a parallelogram are perpendicular, then
» Corresponding angles are congruent if two parallel lines
» Vertical angles are congruent when two lines intersect. 17. All squares are rectangles.
» Base angles of an isosceles triangle are congruent.
» If a number is divisible by 6, then it is divisible by 3. 20. Rain is wet and snow is cold.
» Rain is wet or snow is cold. 22. If Jim lives in Idaho, then he lives in Boise.
» Triangles are round or circles are square. 24. Triangles are square or circles are round.
» While participating in an Easter egg hunt, Sarah notices
» You walk into your geometry class, look at the teacher,
» Albert knows the rule “If a number is added to each side
» You believe that “Anyone who plays major league
» As a handcuffed man is brought into the police station,
» While judging a science fair project, Mr. Cange finds that
» You know the rule “If a person lives in the Santa Rosa
» As Mrs. Gibson enters the doctor’s waiting room, she
» In the figure, point M is called the midpoint of line
» The two triangles shown are similar to each other.
» Observe but do not measure the following angles.
» Several movies directed by Lawrence Garrison have won
» On Monday, Matt says to you, “Andy hit his little sister at
» While searching for a classroom, Tom stopped at an
» At a friend’s house, you see several food items, including
» If the sum of the measures of two angles is 90°, then these
» If a person attends college, then he or she will be a
» All mathematics teachers have a strange sense of humor.
» If Stewart Powers is elected president, then every family
» If Tabby is meowing, then she is hungry. Tabby is hungry.
» If a person is involved in politics, then that person will be
» If a student is enrolled in a literature course, then he or
» If a person is rich and famous, then he or she is happy.
» If you study hard and hire a tutor, then you will make an
» 1 If an animal is a cat, then it makes a “meow” sound.
» 1 All Boy Scouts serve the United States of America.
» Using the same outer scale, read the angle size by reading the degree
» If line segment AB and line segment CD are drawn to
» How many endpoints does a line segment have? How
» How many lines can be drawn that contain both points A
» Consider noncollinear points A, B, and C. If each line
» Name all the angles in the figure.
» Which of the following measures can an angle have?
» Must two different points be collinear? Must three or
» Which symbols correctly expresses the order in which the
» Which symbols correctly name the angle shown?
» A triangle is named 䉭ABC. Can it also be named 䉭ACB?
» Consider rectangle MNPQ. Can it also be named rectangle
» Suppose ⬔ABC and ⬔DEF have the same measure.
» When two lines cross intersect, they have exactly one
» Judging from the ruler shown not to scale, estimate the
» Judging from the ruler, estimate the measure of each line
» A trapezoid is a four-sided figure that contains one pair of
» An angle is bisected if its two parts have the same Find AC if AB
» Find m ⬔1 if m⬔ABC Find x if m ⬔1
» Find an expression for m ⬔ABC if m⬔1 A compass was used to mark off three congruent
» Use your compass and straightedge to bisect . In the figure, m ⬔1
» Find the bearing of airplane B relative to the control
» Find the bearing of airplane C relative to the control
» Axioms or postulates 4. Theorems
» It is reversible. The definition distinguishes the line segment as a specific part of a line.
» The definition is reversible.
» There exists a number measure for each line segment. 2. Only one measure is permissible.
» Convert 6.25 feet to a measure in inches. 4. Convert 52 inches to a measure in feet and inches.
» Convert meter to feet. 6. Convert 16.4 feet to meters. AB
» In the figure, the 15-mile road
» A cross-country runner jogs at a rate of 15 meters per
» Name three points that appear to be
» How many lines can be drawn through
» Explain the difference, if any, between
» Name two lines that appear to be
» Classify as true or false: Given:
» Can a segment bisect a line? a segment? Can a line bisect
» Suppose that a point C lies in plane X and b point D
» Suppose that a planes M and N intersect, b point A lies
» Suppose that a points A, B, and C are collinear and
» Suppose that points A, R, and V are collinear. If AR
» Using the number line provided, name the point that
» What type of angle is each of the following?
» What relationship, if any, exists between two angles:
» ⬔7 and ⬔8 Suppose that TEST 534
» Must two rays with a common endpoint be coplanar?
» Without using a protractor, name the type of angle
» What, if anything, is wrong with the claim
» ⬔FAC and ⬔CAD are adjacent and
» Using variables x and y, write an equation that expresses
» Using variables x and y, write an equation that expresses Given: m Given: m
» Draw a triangle with three acute angles. Construct angle
» What seems to be true of two of the sides in the triangle
» Refer to the circle with center O. If m If m
» Refer to the circle with center P.
» On the hanging sign, the three angles
» Provide reasons for this proof. “If a Write a proof for: “If a
» Given: ⬔1 ⬵ ⬔3 Given: intersects at
» Given: m Given: Given: Given: Given:
» Given: Triangle ABC TEST 534
» Draw a conclusion based on the results of Exercise 9.
» Given: ⬔s 1 and 3 are complementary
» The Segment-Addition Postulate can be generalized as
» In the proof to the right, provide the missing reasons.
» Construct the Proof. This formal proof is
» If m ⬔1 If two angles are complementary to the same angle, then
» If two lines intersect, the vertical angles formed are
» Any two right angles are congruent. 29. If the exterior sides of two adjacent acute angles form
» If two angles are supplementary to the same angle, then
» If two line segments are congruent, then their midpoints
» If two angles are congruent, then their bisectors separate
» The bisectors of two adjacent supplementary angles form
» The supplement of an acute angle is an obtuse angle.
» Name the four components of a mathematical system. 2. Name three types of reasoning.
» Name the four characteristics of a good definition.
» While watching the pitcher warm up, Phillip thinks, “I’ll
» Laura is away at camp. On the first day, her mother brings
» Sarah knows the rule “A number not 0 divided by itself
» If the diagonals of a trapezoid are equal in length, then the
» The diagonals of a parallelogram are congruent if the
» 1. If a person has a good job, then that person has a
» 1. If the measure of an angle is 90°, then that angle is a
» A, B, and C are three points on a line. AC
» Use three letters to name the angle shown. Also use one
» Figure MNPQ is a rhombus. Draw diagonals and
» Points A, B, C, and D are coplanar. A, B, and C are the
» Line intersects plane X at point P.
» On the basis of appearance, what type of angle is shown? Given: Given:
» Suppose that r is parallel to s
» Does the relation “is parallel to” have a
» In the three-dimensional TEST 534
» If two parallel lines are cut by a transversal, then the
» If a transversal is perpendicular to one of two parallel
» Suppose that two lines are cut by a transversal in such a
» If Matt cleans his room, then he will go to the movie. 2. Matt does not get to go to the movie.
» For each statement in Exercise 9 that can be proved by the
» If lines 艎 and m are not perpendicular, then the angles
» If all sides of a triangle are not congruent, then the
» If no two sides of a quadrilateral figure with four sides
» 1. If the areas of two triangles are not equal, then the two
» 1. If two triangles do not have the same shape, then the
» A periscope uses an indirect method of observation. This
» Some stores use an indirect method of observation. The
» If two angles are not congruent, then these angles are not If , then
» If alternate interior angles are not congruent when two
» If a and b are positive numbers, then .
» In a plane, if two lines are parallel to a third line, then the
» In a plane, if two lines are intersected by a transversal so
» If two lines are parallel to the same line, then these lines
» Explain why the statement in Exercise 33 remains true
» Given that point P does not lie on line 艎, construct the line
» Given that point Q does not lie on , construct the line
» A carpenter drops a plumb line from point A to .
» If two lines are cut by a transversal so that the alternate
» If two lines are cut by a transversal so that the exterior
» Describe the auxiliary line segment as determined,
» a All sides of are of the same length.
» Use an indirect proof to establish the following theorem: Given:
» Given: bisects Given: In rt. ,
» What is the measure of each interior angle
» Consider a polygon of n sides For the concave quadrilateral ABCD,
» Which words have a vertical line of symmetry? Which words have a vertical line of symmetry?
» Complete each figure so that it has symmetry with respect
» Complete each figure so that it reflects across line 艎.
» Complete each figure so that it reflects across line m.
» Suppose that slides to the right to the position of
» Suppose that square RSTV slides point for point to form
» Given that the vertical line is a line of symmetry,
» Given that the horizontal line is a line of symmetry,
» Given that each letter has symmetry with respect to the
» What word is produced by a 180° rotation about the
» What word is produced by a 360° rotation about the
» In which direction clockwise or counterclockwise will
» Considering that the consecutive dials on the electric
» Considering that the consecutive dials on the natural gas
» A regular hexagon is rotated about a centrally located
» A regular octagon is rotated about a centrally located
» Describe the types of symmetry displayed by each of
» Given a figure, which of the following pairs of
» Reflexive Property of Congruence If , then If and
» In the figure for Exercise 2, write a statement that the
» 8. Suppose that you wish to prove that SAS
» Given: and Given that , does it follow that
» Establishing a further relationship, like bisects
» Mark the figures systematically, using:
» Given: and are Given: P Given: Given: and are Given: and Given: and are
» lies in the structural support system of the Ferris Given: In the figure,
» 90° and then 45° 16. 60° and then 30°
» 30° and then 15° 18. 45° and then 105°
» Describe how you would construct an angle
» Construct the complement of the acute
» Construct the right triangle with hypotenuse of length
» Construct a line segment of length 2b. 2. Construct a line segment of length
» Construct a line segment of length . 4. Construct a line segment of length
» Construct an angle that is congruent to acute .
» Construct an angle that is congruent to obtuse .
» Construct an angle that has one-half the measure of .
» Construct an angle that has a measure equal to
» Construct an angle that has twice the measure of .
» Construct an angle whose measure averages the measures
» Construct the triangle that has sides of lengths r and t with
» Construct the triangle that has a side of length t included
» Construct an isosceles triangle with base of length c and
» Construct an isosceles triangle with a vertex angle of 30°
» Construct a right triangle with base angles of 45° and
» Construct the right triangle in which acute angle R has a
» Complete the justification of the construction of the To construct a regular hexagon, what
» Construct an equilateral triangle and its three altitudes. A carpenter has placed a
» One of the angles of an isosceles triangle measures 96°.
» NASA in Huntsville, Alabama at point H, has called a
» Given: Equilateral and TEST 534
» A tornado has just struck a small Kansas community at
» Given: Quadrilateral RSTU with diagonal
» Given: Quadrilateral ABCD with
» In not shown, point Q lies on
» The sides of a triangle have lengths of 4, 6, and x. Write
» The sides of a triangle have lengths of 7, 13, and x. As in
» If the lengths of two sides of a triangle are represented by
» Prove by the indirect method: “The length of a diagonal of
» Prove by the indirect method:
» The length of the median from the vertex of an isosceles
» The length of an altitude of an acute triangle is less than
» ABCD is a parallelogram. Given that ,
» Assuming that in Suppose that diagonals and
» a Which line segment is the a Which line segment is
» In quadrilateral RSTV, the midpoints of consecutive sides ABCD is a parallelogram.
» MNPQ is a parallelogram. Suppose that , MNPQ is a parallelogram. Suppose that ,
» Given that and Given that and
» Given: Parallelogram ABCD with and
» The bisectors of two consecutive angles of are
» When the bisectors of two consecutive angles of a
» Draw parallelogram RSTV with and
» Draw parallelogram RSTV so that the diagonals have
» The following problem is based on the Parallelogram
» In the drawing for Exercise 35, the bearing direction in
» Two streets meet to form an obtuse angle at point B.
» To test the accuracy of the foundation’s measurements,
» For quadrilateral ABCD, the measures of its angles are
» Prove: In a parallelogram, the sum of squares of the
» a As shown, must RSTV be a parallelogram?
» In kite WXYZ, the measures of selected angles are shown.
» In the drawing, suppose that and
» In the drawing, suppose that is the perpendicular
» A carpenter lays out boards of lengths 8 ft, 8 ft, 4 ft, and
» A carpenter joins four boards of lengths 6 ft, 6 ft, 4 ft, and
» In parallelogram ABCD not shown, , In , In ,
» If the perimeter sum of the lengths of all three sides of
» Given: Kite HJKL with diagonal Given: with diagonals
» Prove that when the midpoints of consecutive sides of a
» If diagonal is congruent to each side of rhombus
» A line segment joins the midpoints of two opposite sides
» If the diagonals of a parallelogram are perpendicular, what
» If the diagonals of a parallelogram are congruent, what
» If the diagonals of a parallelogram are perpendicular and
» If the diagonals of a quadrilateral are perpendicular
» If the diagonals of a rhombus are congruent, what can you
» and R Given: Quadrilateral PQST with midpoints A, B, C, and
» Find the measures of the remaining angles of trapezoid
» What type of quadrilateral is formed when the midpoints
» In trapezoid ABCD, is the median. Without writing a
» Would RSTV have symmetry with respect to
» Given: Isosceles with TEST 534
» In trapezoid RSTV, , TEST 534
» Each vertical section of a suspension bridge is in the
» In trapezoid WXYZ with bases and
» In isosceles trapezoid MNPQ with , diagonal
» In the figure, a b c and B is
» both sides inverted TEST 534
» Assume that AD is the geometric mean of BD and DC in
» All pairs of corresponding sides are proportional. DEFINITION
» a What is true of any pair of corresponding angles of two
» a Are any two quadrilaterals similar?
» a Are any two regular pentagons similar?
» a Are any two equilateral hexagons similar?
» Given , a second triangle Given
» has an inscribed rhombus ARST. If A square with sides of length 2 in. rests as shown on a
» and DF = 3 and and Given: ; ;
» Prove that the line segment joining the midpoints of two with with
» In quadrilateral RSTU, and . Given: is not a right
» In right triangle XYZ, and . Where Diagonal separates pentagon
» Given: , In preparing a certain recipe, a chef uses 5 oz of ingredient
» Given that , are the following proportions true?
» Use Theorem 5.6.1 and the drawing to complete the proof
» In shown in Exercise 27, suppose that
» Given point D in the interior of , suppose that
» Complete the proof of this property:
» Given point D in the interior of
» In right not shown with right
» Use Exercise 33 and the following drawing to complete
» In , the altitudes of the triangle intersect at a point
» In the figure, the angle bisectors of intersect at a
» Given: not shown is isosceles with
» An amusement park ride the “Octopus” has eight Given: Diameters and in
» Given: Circle O with diameter Given: Find if .
» Find if . Is it possible for
» Given: and are Given: Tangents and
» Given: Given: Given: Given: Given: Given: a How are and
» a How are and A quadrilateral RSTV is circumscribed about a circle so
» Given: and are Given: Given: Tangent to at
» Given: with Given: and Given: in Given:
» Sketch two circles that have:
» Two congruent intersecting circles B and D not shown
» For the two circles in Figures a, b, and c, find the
» If a tangent segment and a secant segment are drawn to a
» Construct a circle O and choose some point D on the
» Construct a circle P and choose three points R, S, and T on
» X, Y, and Z are on circle O such
» Construct the two tangent segments to circle P not
» Point V is in the exterior of circle Q not shown such that
» Given circle P and points R-P-T such that R and T are in
» Given parallel chords TEST 534
» In not shown, the length of radius Provide the missing statements and reasons in the
» A circle is inscribed in an isosceles triangle with legs of
» The inscribed circle’s radius is any line segment from the center drawn
» What condition must be satisfied for it to be possible to
» In a regular polygon with each side of length 6.5 cm, the
» If the perimeter of a regular dodecagon 12 sides is
» If the apothem of a square measures 5 cm, find the
» Find the lengths of the apothem and the radius of a square
» Find the lengths of the apothem and the radius of a regular
» Find the lengths of the side and the radius of an
» Find the lengths of the side and the radius of a regular
» Find the measure of the central angle of a regular polygon of
» Find the number of sides of a regular polygon that has a
» Inscribe a regular octagon within a circle. 6. Inscribe an equilateral triangle within a circle.
» Circumscribe a square about a circle. 8. Circumscribe an equilateral triangle about a circle.
» Find the perimeter of a regular octagon if the length of
» Find the measure of each interior angle of a regular
» Find the measure of each exterior angle of a regular
» Find the number of sides for a regular polygon in which
» Is there a regular polygon for which each central angle
» Given regular hexagon ABCDEF with each side of
» Given regular octagon RSTUVWXY with each side of
» Given that RSTVQ is a regular pentagon and is
» Given: Regular pentagon RSTVQ with equilateral
» Given: Regular pentagon JKLMN not shown with
» Prove: If a circle is divided into n congruent arcs
» 14. Consider the information in Exercise 2, but suppose you
» If MNPQ is a rhombus, which formula from this section 17.
» 16. Are and congruent? TEST 534
» is an obtuse triangle with obtuse angle A.
» A square yard is a square with sides 1 yard in length.
» The following problem is based on this theorem: “A
» Gary and Carolyn plan to build the deck shown.
» The roof of the house shown needs to be shingled.
» The exterior wall the gabled
» Carpeting is to be purchased
» A triangular corner of a store has been roped off to be
» Given region R ´ S, explain The algebra method of FOIL multiplication is illustrated Given region
» In the right triangle, find the length of the altitude drawn
» For cyclic quadrilateral ABCD, For cyclic quadrilateral ABCD, find
» Find the ratio of the areas of two similar rectangles if: Given: Equilateral
» Given: Isosceles with Given: In ,
» In a triangle of perimeter 76 in., the length of the first side In a triangle whose area is 72 in
» A trapezoid has an area of 96 cm 14.
» Given: Hexagon RSTVWX with Given: Pentagon ABCDE
» Find the area of a square with
» Find the area of an equilateral triangle with
» Find the area of an equiangular triangle with
» In a regular polygon, each central angle measures 30°. If
» In a regular polygon, each interior angle measures 135°. If
» For a regular hexagon, the length of the apothem is 10 cm.
» For a regular hexagon, the length of the radius is 12 in.
» In a particular type of regular polygon, the length of the
» In one type of regular polygon, the measure of each
» If the area and the perimeter of a regular
» Find the area of a square with apothem and
» Find the area of an equilateral triangle with apothem
» Find the area of an equiangular triangle with apothem
» Find the area of a regular pentagon with an apothem of
» Find the area of a regular octagon with an apothem of
» Find the area of a regular hexagon whose sides have
» Find the approximate area of a regular pentagon whose
» In a regular octagon, the approximate ratio of the length
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