Geometry Questions

Examine angles whose Prove theorems involving angles Apply involving inscribed angles G lie on a circle Apply theorems involving angles formed by a and tangent

side of the angle that intercepts AC Prove The measure of ZABC is half the measure of AC ZADC and ZBDC are supplementary since they form a The measure of ZADC 180 180 2x which simplifies to or twice the measure of angle ABC Since ZADC is a central that of AC FLL LLL LLLL 180 2x D B thlebblebi bla bla kbEWERBLETT Since the measure of ZABC is x the measure of AC is 2x pair The sum of their measures is 180 xo its measure 2x is equal to that C

Review Key Concepts The measure of an inscribed angle is half the measure of its intercepted arc Two inscribed angles that interceptthe are congruent An angle inscribed in a semicircle is a angle The opposite sides of a quadrilateral inscribed in acirde are supplementary If a tangent and chord intersect on a cirde the measure of each angle they form is the measure of its intercepted arc

The Angle Formed by a Tangent and Chord Theorem Angle formed by a tangent and chord theorem If a tangent to a circle and a chord intersect at a point c on the circle then the measure of each angle they form is one half the measure of its intercepted arc MZ mAB This also means that ZEAB is going to be equal to the measure of its intercepted arc BCA E A B LL F

The trinket store is open for 14 hours per day They sell an average of 5 trinkets per hour The back storeroom currently has 560 trinkets in it 1 Write an equation that estimates the number of trinkets T that will be in the storeroom after H hours 2 How many days will it take to run out of trinkets

Find the volume of each object 1 the rectangular prism part of the milk container 2 the cylinder part of the measuring cup 6 3 cm 7 cm 10 cm 7 cm 7 cm

The Third Corollary to the Inscribed Angle Theorem Third corollary to the inscribed angle theorem The opposite angles of a quadrilateral inscribed in a circle are supplementary A B up to C Remember that the measures of supplementary anglesadd O D This theorem means the angles B and D are going to be and angles A and be supplementary are going to Remember thata circle has 360 and that the measure of angle is half the measure of the intercepted

Using the Third Corollary to the Inscribed Angle Theorem The measure of 20 is x 2 and the measure of 2S is 9x 7 What are the measures of angles Q and S Since angles Q and S are opposite in a they are supplementary x 2 9x 7 180 10x5 180 Our 185 10 10 x 18 5 Plug this value of x into the expressions m20 18 5 2 180 20 5 0 0 of a quadrilateral inscribed T S m s 9 18 5 7 You could have also used the fact that opposite angles are supplementary to find the measure of angle S Q Not drawn to scale R

Inscribed angle theorem The measure of an inscribed angle is measure of its intercepted arc The Case 1 is one side of the angle Case 2 The center of the circle lies the angle The Case 3 the the circle lies within the angle of

Second corollary to the inscribed angle theorem An angle inscribed in a semicircle is a right angle Since DF is a diameter the angle is inscribed in half the circle or a Coroll ry to the Inscribed Angle Theorem This means that the inscribed angle is a angle A semicircle measures 180 so the measure of the angle would be half of that or E C LL

The First Corollary to the Inscribed Angle Theorem First corollary to the inscribed angle theorem Two inscribed angles that intercept the same arc are congruent ZKJL and ZKML are their angles intercept the same arc KL same A JM because We can also say that JKM and congruent because their angles also intersect the are J K M

Central Angles The measure of a central angle is equal to the measure of its intercepted arc C 42 A B

An inscribed angle is an angle whose sides are chords is on a circle and whose An intercepted arc is the arc that lies in the interior of an and has endpoints on the angle C B angle

Find the area of polygon JKLMNO K 104 886 9 7 bl 2 10 9 8 7 6 5 4 3 2 11 123 2 456 M 1 2 3 4 5 6 7 8 9 10 N

Angle ABD measures 4x 10 Angle ACD measures 5x 2 A D E C B What is the measure of arc AD O 12 O 58 O 96 O 116

Angle BAC measures 56 D B O What is the measure of angle BDC O 28 O 34 O 56 O 112

In circle V angle WXZ measures 30 Line segments WV XV ZV and YV are radii of circle V W U X Y What is the measure of WUX in circle V O 60 O 90 O 120 O 150

104 D 79 G LL F What is the measure of arc ECF in circle G O 52 O 98 O 158 O 177

Line segment GJ is a diameter of circle L Angle K measures 4x 6 H K What is the value of x O 21 24 O 32 044

Line segment XY is tangent to circle Z at point U X T U W Y 12 If the measure of UV is 84 what is the measure of ZYUV A O O O O 42 84 96 168

In circle D angle ADC measures 7x 2 Arc AC measures 8x 8 A B C What is the measure of ZABE O 36 O 43 O 72 O 144

10 Scott set up a volleyball net in his backyard One of the poles which forms a right angle with the ground is 7 feet high To secure the pole he attached two ropes from the top of the pole to a stake 4 feet from the bottom of the pole Find the total length of the two ropes A 160 ft B 8 5 ft C 4 ft D 8 3 ft 10

K 70 116 1 What are the measures of JK and ZKIJ Measure of JK Measure of ZKIJ

88 D U 111 A What is the measure of A O 44 O 50 O 64 O 92

40 Q R PR and gs are diameters of circle T What is the measure of SR 50 80 100 120

Justifying the Second Corollary to the Inscribed Angle Theore Explain how you can use the inscribed angle theorem to justify its second corollary that an angle inscribed in a semicircle is a right angle

4 11 Given ZKJL and ZKML intercept arc KL Prove m2KJL mZKML 19 11 rst Co 11 orollary to the Inscribed Angle Theorem K M L Statements 1 ZKJL intercepts KL 2 ZKML intercepts KL 3 mZKJL mKL 4 mZKML mKL 5 mZKJL m KML 1 given 2 given 3 4 5 Reasons

A C B The measure of ZB is 3x 4 and the measure of ZD is 2x 6 What are the measures of angles B and D m2B mZD

Calculate the angle A in the following right angled triangle where a 53 2 m and b 5

B C E D F ZFAE measures 72 What other angle must measure 72 OZBFD OZDEA O ZEDF OZFBE

R T S The measure of arc QS is 4x 18 What is the value of x O 40 5 O 49 5 O 94 5 O 180

D B A P F O The measure of ADB is 162 What is the measure of ZEAB O

Add Work Question 16 Prove the following identity 1 cos 2y sin 2y degrees 1 cos 2y sin 2y tan y 1

Q 84 R S What is the measure of Qs

58 3 4 2 106 Is the measure of 21 equal to the measure of 22 Why O yes because they intercept the same arc O no because the sides of 22 are longer O no because they intercept the circle at different points

Ground anchor 2 is exactly 58 feet from the base of the tower dashed line in diagram The bottom ground support cable from level 6 to ground anchor 2 is attached to the tower at a height of 142 feet The top ground support cable from level 6 to ground anchor 2 is attached to the tower at a height of 150 feet How long is each support cable to level 6 The bottom cable is SCIEN feet long Round to the nearest hundredth Lesel Lami Support cables Grou aschin

J M P L K The measure of ZKJN is 92 What is the measure of JMK

e is a central angle that cuts off an arc of length s Find the radius of the circle if 8 155 s 5 km O 2 25 km O 1 95 km O 1 85 km 2 15 km O 2 05 km

X Y 86 N What is the measure of ZY O 43 O 68 86 172

10 9 8 7 16 1 5 A 3 2 11 10 9 8 7 6 5 4 3 2 11 1 2 3 4 5 6 7 8 9 IC 2 1 3 14 15 6 1 7 8 A X 9 Write the standard equation for the hyperbola graphed above

haruca a 270 Which of the following shows the correct rotation of parallelogram ABCD OA GOO OB GTS O C

Watch help video In circle F with m EFG 70 and EF Round to the nearest hundredth E F 19 units find the length of arc EG G 4

For triangle ABC we are given that mZC 89 1 AC 12 3 ft and BC 23 3 ft We can use the Law of Cosines Law of Sines and other properties of triangles to determine all of the missing information Note that the diagram may not be drawn to scale a AB B Determine all of the missing information about this triangle d m B 23 3 ft feet Preview b Draw the triangle and add the length of AB to your diagram c m A degrees Preview degrees Preview 89 10 A 12 3 ft

sin u v x 1 Points DETAILS Use the figure to find the exact value of the trigonometric function sin

K Use transformations of the graph of f x x to determine the graph of the given function h x x 3 What transformations are needed to graph the function h x x 3 2 Choose the correct answer below OA The graph of f x x should be horizontally shifted to the left by 3 units and reflected about the y axis OB The graph of f x x should be horizontally shifted to the left by 3 units and reflected about the x axis OC The graph of f x x should be horizontally shifted to the right by 3 units and reflected about the y axis OD The graph of f x x should be horizontally shifted to the right by 3 units and reflected about the x axis Use the graphing tool to graph the function Click to enlarge graph 10 2 Ay

Yvetia ordered two pizzas for her family Both pizzas are rectangular The larger pizza s width is equal to the length of the smaller pizza and the larger pizza s length is twice the width of the smaller pizza The sum of the perimeters of the two pizzas is 96 inches and the sum of the areas is 288 square inches Determine the dimensions of each pizza Answer The smaller pizza is Number inches by Number inches inches by Number inches and the larger pizza is Number

Evaluate each expression without using a calculator Hint Sketch a right triang tan arc a tan arccos cos arcsin 15 b

Complete the congruence statement M P N H O T U

a as x 2 c as x0 cotx tan x e as x cotx g as x0 csc x i as x cscx k as x 7 secx to complete each state b as x d as x 0 cotx f as x cotx tan x h as x 0 csc x j as x n csc x I as x I 2 7 secx

100 2 Move point C so the measure of arc AC is 50 What is the measure of ZABC 25 D 3 Make a conjecture Which measures will change if you move vertex B of the inscribed angle B A C mZABC 52 mAC 104