Given: bisects Given: In rt. ,
3 4
5 7 8 1 6 9 A Figure 2.33 The total number of diagonals D in a polygon of n sides is given by the formula . D = n n - 3 2 THEOREM 2.5.1 Theorem 2.5.1 reaffirms the fact that a triangle has no diagonals; when , D = 33 - 3 2 = 0. n = 3 쮿 REGULAR POLYGONS Figure 2.34 shows polygons that are, respectively, a equilateral, b equiangular, and c regular both sides and angles are congruent. Note the dashes that indicate congru- ent sides and the arcs that indicate congruent angles. EXAMPLE 2 Find the sum of the measures of the interior angles of a hexagon. Then find the measure of each interior angle of an equiangular hexagon. Solution For the hexagon, , so the sum of the measures of the interior angles is or 4180° or 720°. In an equiangular hexagon, each of the six interior angles measures , or 120°. 쮿 720 ° 6 S = 6 - 2 180° n = 6 EXAMPLE 3 Find the number of sides in a polygon whose sum of interior angles is 2160°. Solution Here in the formula of Theorem 2.5.2. Because , we have . Then The polygon has 14 sides. 쮿 n = 14 180n = 2520 180n - 360 = 2160 n - 2 180 = 2160 S = 2160 Exs. 6–9 a Figure 2.34 b c A regular polygon is a polygon that is both equilateral and equiangular. DEFINITION The polygon in Figure 2.34c is a regular pentagon. Other examples of regular polygons include the equilateral triangle and the square. Based upon the formula from Theorem 2.5.2, there is also a formula for the measure of each interior angle of a regular polygon having n sides. It applies to equiangular polygons as well. S = n - 2 180° The measure I of each interior angle of a regular polygon or equiangular polygon of n sides is . I = n - 2 180° n COROLLARY 2.5.3 A second corollary to Theorem 2.5.2 concerns the sum of the interior angles of any quadrilateral. For the proof, we simply let in the formula . Then . Also, see the Discover at the left. S = 4 - 2 180° = 2 180° = 360° S = n - 2 180° n = 4 EXAMPLE 4 Find the measure of each interior angle of a ceramic floor tile in the shape of an equiangular octagon Figure 2.35. Solution For an octagon, . Then Each interior angle of the tile measures 135°. NOTE: For the octagonal tiles of Example 4, small squares are used as “fillers” to cover the floor. The pattern, known as a tessellation, is found in Section 8.3. 쮿 = 1080 8 , so I = 135° = 6 180 8 I = 8 - 2 180 8 n = 8 EXAMPLE 5 Each interior angle of a certain regular polygon has a measure of 144°. Find its number of sides, and identify the type of polygon it is. Solution Let n be the number of sides the polygon has. All n of the interior angles are equal in measure. The measure of each interior angle is given by Then multiplying by n With 10 sides, the polygon is a regular decagon. 쮿 n = 10 36n = 360 180n - 360 = 144n n - 2 180 = 144n n - 2 180 n = 144 I = n - 2 180 n where I = 144 Discover From a paper quadrilateral, cut the angles from the “corners.” Now place the angles so that they have the same vertex and do not overlap. What is the sum of measures of the four angles? ANSWER 360° Exs. 10–12 Figure 2.35 On the basis of Corollary 2.5.4, it is clearly the case that each interior angle of a square or rectangle measures 90°. The following interesting corollary to Theorem 2.5.2 can be established through algebra. The sum of the four interior angles of a quadrilateral is 360°. COROLLARY 2.5.4 We now consider an algebraic proof for Corollary 2.5.5. Proof A polygon of n sides has n interior angles and n exterior angles, if one is considered at each vertex. As shown in Figure 2.36, these interior and exterior angles may be grouped into pairs of supplementary angles. Because there are n pairs of angles, the sum of the measures of all pairs is degrees. Of course, the sum of the measures of the interior angles is . In words, we have Let S represent the sum of the measures of the exterior angles. The next corollary follows from Corollary 2.5.5. The claim made in Corollary 2.5.6 is applied in Example 6. ‹ S = 360 - 360 + S = 0 180n - 360 + S = 180n n - 2 180 + S = 180n Sum of Measures Sum of Measures Sum of Measures of All of Interior Angles of Exterior Angles Supplementary Pairs n - 2 180° 180 n The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°. COROLLARY 2.5.5 EXAMPLE 6 Use Corollary 2.5.6 to find the number of sides of a regular polygon if each interior angle measures 144°. Note that we are repeating Example 5. Solution If each interior angle measures 144°, then each exterior angle measures 36° they are supplementary, because exterior sides of these adjacent angles form a straight line. Now each of the n exterior angles has the measure In this case, , and it follows that , so . The polygon a decagon has 10 sides. 쮿 n = 10 36n = 360 360 n = 36 360° n The measure E of each exterior angle of a regular polygon or equiangular polygon of n sides is . E = 360° n COROLLARY 2.5.6 POLYGRAMS A polygram is the star-shaped figure that results when the sides of convex polygons with five or more sides are extended. When the polygon is regular, the resulting poly- gram is also regular—that is, the interior acute angles are congruent, the interior reflex 1 n 1 2 2 33 4
4 n Figure 2.36 Exs. 13, 14 ⫹ ⫽ 쮿 angles are congruent, and all sides are congruent. The names of polygrams come from the names of the polygons whose sides were extended. Figure 2.37 shows a pentagram, a hexagram, and an octagram. With congruent angles and sides indicated, these figures are regular polygrams. Exs. 15, 16 Pentagram Figure 2.37 Hexagram Octagram Geometry in Nature The starfish has the shape of a pentagram. Exercises 2.51. As the number of sides of a regular polygon increases,
does each interior angle increase or decrease in measure?2. As the number of sides of a regular polygon increases,
does each exterior angle increase or decrease in measure?3. Given:
, , , with angle measures as indicated Find: x , y, and z4. In pentagon ABCDE with
, find the measure of interior angle D.5. Find the total number of diagonals for a polygon of n
sides if: a b6. Find the total number of diagonals for a polygon of n
sides if: a b7. Find the sum of the measures of the interior angles of a
polygon of n sides if: a b8. Find the sum of the measures of the interior angles of a
polygon of n sides if: a b9. Find the measure of each interior angle of a regular
polygon of n sides if: a b n = 12 n = 4 n = 8 n = 6 n = 10 n = 5 n = 8 n = 6 n = 10 n = 5 ∠B ⬵ ∠D ⬵ ∠E AE 7 FC AD 7 BC AB 7 DC10. Find the measure of each interior angle of a regular
polygon of n sides if: a b11. Find the measure of each exterior angle of a regular
polygon of n sides if: a b12. Find the measure of each exterior angle of a regular
polygon of n sides if: a b13. Find the number of sides that a polygon has if the sum of
the measures of its interior angles is: a 900° b 1260°14. Find the number of sides that a polygon has if the sum of
the measures of its interior angles is: a 1980° b 2340°15. Find the number of sides that a regular polygon has if the
measure of each interior angle is: a 108° b 144°16. Find the number of sides that a regular polygon has if the
measure of each interior angle is: a 150° b 168°17. Find the number of sides in a regular polygon whose
exterior angles each measure: a 24° b 18°18. Find the number of sides in a regular
polygon whose exterior angles each measure: a 45° b 9°19. What is the measure of each interior angle
of a stop sign? n = 10 n = 6 n = 12 n = 4 n = 10 n = 6 z 36 ° x A F B D E C 77 ° y A C D E B 93 9320. Lug bolts are equally
spaced about the wheel to form the equal angles shown in the figure. What is the measure of each of the equal acute angles? In Exercises 21 to 26, with P ⫽ {all polygons} as the universe, draw a Venn Diagram to represent the relationship between these sets. Describe a subset relationship, if one exists. Are the sets described disjoint or equivalent? Do the sets intersect?21. T
⫽ {triangles}; I ⫽ {isosceles triangles}22. R
⫽ {right triangles}; S ⫽ {scalene triangles}23. A
⫽ {acute triangles}; S ⫽ {scalene triangles}24. Q
⫽ {quadrilaterals}; L ⫽ {equilateral polygons}25. H
⫽ {hexagons}; O ⫽ {octagons}26. T
⫽ {triangles}; Q ⫽ {quadrilaterals}27. Given: Quadrilateral RSTQ with exterior
at R and T Prove: m ∠1 + m∠2 = m∠3 + m∠4 ∠s31. A father wishes to make a home plate for his son to use in
practicing baseball. Find the size of each of the equal angles if the home plate is modeled on the one in a and if it is modeled on the one in b. 1 3 2 4 S R T Q a b ? ? ?28. Given: Regular hexagon
ABCDEF with diagonal and exterior Prove:29. Given: Quadrilateral RSTV with diagonals
and intersecting at W Prove: m ∠1 + m∠2 = m∠3 + m∠4 SV RT m ∠2 + m∠3 = m∠1 ∠1 AC A F C D E B 2 3 1 1 23 4
W S R V T30. Given:
Quadrilateral ABCD with and Prove: B and D are supplementary ∠s BC ⬜ DC BA ⬜ AD A B C D32. The adjacent interior and exterior angles of a certain
polygon are supplementary, as indicated in the drawing. Assume that you know that the measure of each interior angle of a regular polygon is a Express the measure of each exterior angle as the supplement of the interior angle. b Simplify the expression in part a to show that each exterior angle has a measure of . 360 n n - 2180 n . 1 233. Find the measure of each acute interior angle of a regular
pentagram.34. Find the measure of each acute interior angle of a regular
octagram.35. Consider any regular polygon; find and join in order the
midpoints of the sides. What does intuition tell you about the resulting polygon?36. Consider a regular hexagon RSTUVW. What does intuition
tell you about 䉭 , the result of drawing diagonals , , and ?37. The face of a clock has the shape of a
regular polygon with 12 sides. What is the measure of the angle formed by two consecutive sides?38. The top surface of a picnic table is in
the shape of a regular hexagon. What is the measure of the angle formed by two consecutive sides?39. Consider a polygon of n sides
determined by the n noncollinear vertics A, B, C, D, and so on. a Choose any vertex of the polygon. To how many of the remaining vertices of the polygon can the selected vertex be joined to form a diagonal? b Considering that each of the n vertices in a can be joined to any one of the remaining vertices to form diagonals, the product appears to represent the total number of diagonals possible. However, this number n n - 3 n - 3 VR TV RT RTV ? A B C D E ? LINE SYMMETRY In the figure below, rectangle ABCD is said to have symmetry with respect to line 艎 be- cause each point to the left of the line of symmetry or axis of symmetry has a correspon- ding point to the right; for instance, X and Y are corresponding points. 2.6 includes duplications, such as and . What expression actually represents D, the total number of diagonals in a polygon of n sides?40. For the concave quadrilateral ABCD,
explain why the sum of the interior angles is 360°. HINT: Draw .41. If , , and
, find the measure of the reflex angle at vertex D. HINT: See Exercise 40. m ∠C = 31° m ∠B = 88° m ∠A = 20° BD CA AC42. Is it possible for a polygon to have the following sum of
measures for its interior angles? a 600° b 720°43. Is it possible for a regular polygon to have the following
measures for each interior angle? a 96° b 140° A B C D Exercises 40, 41 Symmetry Line of Symmetry Axis of Symmetry Point Symmetry Transformations Slides Translations Reflections Rotations Symmetry and Transformations 2.6 KEY CONCEPTS A X D C B Y Figure 2.38 A D C B X Y A figure has symmetry with respect to a line 艎 if for every point A on the figure, there is a second point B on the figure for which 艎 is the perpendicular bisector of AB. DEFINITION In particular, ABCD of Figure 2.38 has horizontal symmetry with respect to line 艎. That is, a vertical axis of symmetry leads to a pairing of corresponding points on a horizon- tal line. In Example 1 on page 108, we see that a horizontal axis leads to vertical symmetry for points.Parts
» 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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