Geometry Questions

angle I can prove properties of angles for a quadrilateral ascribed in a circle Lesson 8 2 Checkpoint Once you have completed the above problems and checked your solutions complete the below Complete the Lesson Reflection above by circling your current understanding of the Le 1 Find the measure of BC C 6y 16 2 Find AD B 15x 22 A B 4y 6 D C D 3x 74 Starting Gettin A

HW OA OB C OD ework Complete problems 1 3 below for independent practice 1 Which statement correctly describes the perpendicular bisector construction below RO Construction of TS to bisect QR Midpoint between point S and T Construction of QR to bisect ST Midpoint between point Q and R 10 When you are finished check the solutions with your teacher 3 DX EX FX are perpendicular bisectors of AABC Select two options that are true B 12 D F 8 C 2 Which statement correctly describes the perpendicular bisector construction below G A Construction of GH to bisect JK B Midpoint between point J and K OC Midpoint between point G and H OD Construction of JK to bisect GH XC 10 BX 12 CAC 20 BA 24 A OB H OD Optional Complete problems 5 7 15 from your textbook on separate lined paper When you are finished check the odd problems for solutions at the back of the textbook

B HW R When you are finished check the solutions with your teacher 1 In the figure below PU 12 VT 5 and VQ 6 Which of the following is FALSE Complete problems 1 2 below for independent practice T H H C 2 In the figure below FE 8 EH 8 and HD 18 Which of the following is FALSE F S H Q E ARU 12 OBUV 3 CPR 17 D PV 10 D A FD BE B BH FD CEH ED OD BE 24 Optional Complete problems 5 7 11 from your textbook on separate lined paper When you are finished check the odd problems for solutions at the back of the forth

160 gle order its side measures from least to greatest 2 C 15 B A 60 30

A des are congruent Lesson 7 2 Checkpoint Once you have completed the above problems and checked your solutions complete the Lesson Checkpoint Complete the Lesson Reflection above by circling your current understanding of the Learning Goal below 4x 2 1 Look at the isosceles triangle below Determine which of the following angle measures are correct Select two that apply B 7x 34 74 T C Starting D S DA B C D 2 In the figure below LRQT and LRST are bisected forming isosceles triangle QRS and QTS What is the value of LRSQ R m BCA 58 mzBCD 58 m2DCB 122 mZBAC 48

Turn For each triangle order its angle measures from least to greatest

struct the inscribed and circumscribed circles of a angle I can prove properties of angles for a quadrilateral mithid in a circle B Lesson 8 1 Checkpoint Once you have completed the above problems and checked your solutions complete the Lesso below AD Complete the Lesson Reflection above by circling your current understanding of the Learning ZD ZE and ZF are perpendicular bisectors of AABC If FZ 36 ZA 85 and AB 150 fine of AD and ZB ZB E N A Lesson Reflection Cim F Starting C Getting thera

A manufacturer sells backpacks and tents The profit function for the manufacturer is a function z f x y where x is the quantity in thousands of backpacks that is sold y is the quantity in thousands of tents that is sold and z is in thousands of dollars Suppose that f 10 14 23 and fx 10 14 4 and fy 10 14 3 Estimate the profit obtained if the manufacturer sells 12 000 backpacks and 14 000 tents O a 3 000 b 4 000 O c 26 000 O d 27 000 O e 29 000 O f 31 000 O g 49 000 O h 50 000

Lesson 7 3 Checkpoint Once you have completed the above problems and checked your solutions complete below Complete the Lesson Reflection above by circling your current understanding of the I 1 Write the side lengths in order from shortest to greatest A B 45 18 2 Write the angles in order from least to greatest S T 25 15 65 U 70 C

Determine which translations would map Figure W onto Figure X Press try to test a translation or sequence of translations A translation 12 11 10 8765 units and Figure X W Figure W 3 2 2 3 4 5 6 7 8 9 10 11 12 4 5 6 10 11 12 units

Question 13 of 13 The table below gives the number of calories burned per minute B f w s for someone roller blading as a function of the person s weight w and speed s The number of calories w s The number of calories 120 lbs 140 lbs 160 lbs 180 lbs 200 lbs 9 mph 11 mph 13 mph 15 mph by about i 7 1 by about i 8 3 9 5 10 7 11 9 9 3 10 9 12 4 14 0 15 6 11 7 13 7 15 6 17 6 19 5 13 3 15 6 a Is fe positive or negative Is f positive or negative What do your answers tell us about the effect of weight and speed on calories burned per minute fuis positive C f is positive A heavier person will burn more calories per minute while roller blading than a lighter person as the speed goes up The number of calories burned per minute while roller blading goes up b Estimate f 160 11 and f 160 11 Interpret your answers fe 160 11 160 11 17 8 20 0 22 2 if the weight is increased by 1 pound if the speed is increased by 1 mph SUP

Question The right triangle on the right is a scaled copy of the right triangle on the left Identify the scale factor Expre fraction in simplest form 2 2 8 8

Which transformation would take Figure A to Figure B Answer 10 47 5 4 341 A clockwise rotation of 180 about the origin A reflection over the line y x O A clockwise rotation of 270 about the origin A reflection over the line y r 241 301 14 H 7 Teh El 10 111 2 4 5 6 B H 30

Cool Down All Together Now Here is the image of triangle ABC and its dilation using center D and scale factor 2 DC CB A C A B C B H D 1 Why is segment B C on the same line as its corresponding side segment BC 2 Explain how you know that segment A C is parallel to its corresponding side segment AC 3 What relationship does angle ACB have with its corresponding angle A C B Explain or show your reasoning

AR TOONOS H HO SA U VAN SHOE OML 25 sine 11x 47 T 7 What is the value of x in the figure below Q SA Suur W cosine 54 9 Find the value of x Using that value what is the measurement of LUTQ V 3x 14 to 105 adi wy V J YOU 8 What is the value of x in the figure below 3x 2 2x R 10 Find the value of x Using that value what is the measurement of 2BCD D Chilman 16x 2 80 B 152 6x 2 C EVALUATO

Wy 3 Dilate line with a scale factor of 2 The image is line g Which labeled point could be the center of this dilation A A B B C C D D A B D 8 4 Quadrilateral A B C E is the image of quadrilateral ABCE after a dilation centered at F What is the scale factor of this dilation 3 A B 1 5 A B 6 3 Mathematics F E C C 2 5 5 E

Lesson 4 Practice Problems 1 Angle ABC is taken by a dilation with center P and scale factor 3 to angle A B C The measure of angle ABC is 21 What is the measure of angle A B C 2 Select all lines that could be the image of line m by a dilation A C B m C n D 0 E p 0 l m n P K

Four Thern 1 Use the Paragraph Proof of the Isosceles Triangle Theorem to complete a Theorem Hint Refer to the reasen bank below The first one is done for you Two Column Proof of the Isasceles Triangle Theorem AB a AC Prove S 5 Statements Draw AD the angle bisector of CAB LBAD LCAD AB AC 4 AD AD 5 BAD CAD 6 LB 2 C 1 2 3 Given Statements 1 Draw EH the angle bisector of ZFEG 2 LFEHLGEH 3 LF 2G 2 Complete a two column proof of the Converse of the Isosceles Triangle Theorem Hint Refer to the reason bank below Two Column Proof of the Converse of the Isosceles Triangle Theorem Prove EF EG LF LG A FEH BAGEH 6 EF EG Reason Bank CPCTC Corresponding Parts of Congruent Triangles are Congruent 1 Construction of angle bosector Given Reasons AAS Triangle Congruence en prost of the SAS Triangle Congruence B Reflexive Property of Congruence E Definition of angle bisecto Construction of angle bisector

Solve the following trigonometric equations for all solutions Enter exact answers whenever possible Choose th smallest positive angle for your base angle a sin 0 3 Solution with smaller positive base angle 0 n for all integers n Solution with larger positive base angle 0 n for all integers n b cos 0 4 Solution with smaller positive base angle 0 1 n for all integers 7 Solution with larger positive base angle 0 nto for all integers n

6 Given the polar equation r 2 8 sin 0 The eccentricity is e The equation of the directrix is The shape of the curve is Select an answer

2 Find the measure of ZPRS 1260 R M 63 3x 8 5x 50 P x 2

TOS 5 ddo 87 Wyp O ASA Your Turn 1 OASA O ASA hyr tan O AAS O AAS Write whether the AAS Congruence Theorem the ASA Congruence Theorem or neither can be used to prove whether each pair of triangles are congruent O AAS a quis O Neither ONeither O Neither 6 0 6 2 4 O ASA O ASA O ASA U O AAS O AAS O AAS O Neither O Neither W X ONeither

Part 4 6 Let x 2 x 1 2 a Find the vertex b Find x intercepts zeros c Graph f x by plotting the points from parts a and b

Part 5 7 Let g x x 2x 3 a Find the vertex b Find the x intercepts zeros c Graph f x by plotting the points from parts a and b

Cool Down Missing Length What must be true about segment MN 2 1 N 1 8 M 3 6

Step 1 Given Information A dot plot for so simulated player took 30 shots is Step 2 a games To given in which a out a simulation to estimate the carry probability that a 50 shooter who took 30 shots in a game would have a streak of 10 or more made shots first need to the assign numbers to the outcomes Let 1 represent the made shot and 2 represent the missed shot Now generate 30 random digits from 1 to 2 Record whether there is a sequence of at least 10 made shots or not Repeat the

Solution a n 100 SRS size The number of pupils who claimed they recycle is 55 We must decide whether there is persuasive evidence that more than half of the pupils at the school would indicate they recycle regularly using the Fathom dot plot which displays the results of taking 200 SRSs of 100 students from a population in which the genuine proportion who recycle is 0 50 A simulation is a model of random behaviour that is usually carried out with random numbers It is insufficient proof that more than half of the students at the school recycle because Yes was replied by 55 out of 100 pupils 55 100 0 55 We can see that the proportion of 0 55 has a lot of dots above it in the dot plot As a result a proportion of 0 55 is quite likely to be obtained when the genuine proportion is 0 5 and there is insufficient evidence to support the assertion that more than half of the school s students recycle b Because 63 out of 100 students answered Yes this statistic strongly suggests that the majority of the school s students recycle 0 63 63 100 We can see that the proportion of 0 63 has only one dot above it and no dots to the right of it in the supplied dot plot Thus while the genuine proportion is 0 5 a proportion of 0 63 is quite likely to be obtained and there is adequate evidence to support the assertion that more than half of the school s students recycle As a result the outcome

S of interior angles of a triangle sum to 180 50 Lesson 7 1 Checkpoint Once you have completed the above problems and checked your solutions complete the Lesson Checkpoint below Complete the Lesson Reflection above by circling your current understanding of the Learning Goal 1 Determine which of the following angle measures are correct Select two that apply A B 4x 11 5 10x 5x 5 R 60 11x 6 2 Determine which of the following angle measures are correct Select two that apply C A A OB C OD J easures Starting Getting there A OB C OD mLAQR 95 mzSQR 100 mZAQR 55 mzSRQ 45 Got it mZCAB 75 m2JAB 115 mzCBA 65 mzCBA 55

2 Your Turn 1 Remote interior angles For the figure shown find the unknown angle measure x For the figure shown find the unknown angle measure x Exterior angle OLUTASH EXPLAIN 3 Using Exterior Angles Read Explain 3 and complete Your Turn 1 2 adapted from Lesson 7 1 Show all your work 0 In the figure to the right we have the interior angles 21 22 and 23 of a triangle We also have the exterior angle 24 of the triangle 140 150 21 22 23 23 24 130 When we extend the side of a polygon the angle formed by the side of the polygon and the extended side is called an exterior angle Exterior angles form linear pairs with the interior angles The remote interior angles of a triangle are the angles that are not adjacent to the exterior angle m21 m2 m 4 Because they form a linear pair 23 and 24 are supplementary So 23 24 180 Also the sum of the measures of any triangle is 180 So 21 22 23 180 After substituting for 180 and simplifying we have 21 22 24 105 Thus the measure of the triangle s exterior angle 24 is equal to the sum remote interior angles 21 and 22 This conjecture can be formally stated as a theorem Exterior Angle Theorem

HW 1 The interior angles of a triangle have measures of 2 What is the value of x in the figure below 55 25 and x Find the value of x D Complete problems 1 10 below for independent practice 3 Find the value of the missing angle using the Exterior Angle Theorem U 94 GT When you are finished check the solutions with your teacher 42 5 Find the value of the missing angle using the Exterior Angle Theorem E 105 155 V C Y 3x D 127 119 4 Find the value of the missing angle using the Exterior Angle Theorem S 53 W B 54 100 63 6 Find the value of the missing angle using the Exterior Angle Theorem T C V 80 U8 Camaldos que andag 0

87 opposite bypolerice hgr 8 6 5 B 26 A median of a triangle is a line that connects a vertex with the midpoint of the opposite leg For AABC below AF is one of the medians of the triangle Draw the medians for the other two vertices Then locate the location where the lines intersect D F E Havia C For each of the figures you drew the point where the lines intersect have special names In this credit you will be learning what those names are and the relationship between that point and the segments in the triangles

Lesson 7 1 Interior and Exterior Angles Learning Goal I can prove the following theorems about triangles The measures of interior angles of a triangle sum to 180 Standard G CO C 10 EXPLORE Exploring Interior Angles in Polygons Read and complete the following adapted from Lesson 7 1 Show all your work interior angles The Triangle Angle Sum Theorem An interior angle is an angle formed by two sides of a polygon with a common vertex When we rearrange the interior angles of a triangle we find that they form a straight angle measuring 180 The sum of the angle measures of a triangle is 180 To determine the sum of the interior angles for any polygon you can consider how many triangles there are in other polygons If a convex po is broken up into triangles then the sum of the interior angles is the number of triangles times 180 For example by drawing the diagonal from a vertex of a quadrilateral you can form two triangles Since each triangle has an angle sum of 180 the quadrilateral must have an angle sum of 2 180 360 In each of the figures below how many triangles do you see Triangle Quadrilateral Pentagon Polygon Triangle Quadrilateral Pentagon Number of Sides 3 4 LAL Math Resources http bit ly L4LMath Hexagon Triangle s Triangle s Triangle s Triangle s Triangle s Triangle s For each polygon identify the number of sides the number of triangles and determine the angle sums of those triangles Remember that the sum of the angle measures of a triangle is 180 Number of Triangles 1 2 Heptagon Octagon Sum of Interior Angle Measures x 180 x 180 x 180 x 180

EVALUATE Lesson 6 3 Homework HW Independent Practice Complete problems 1 8 below for independent practice 11 When you are finished check the solutions with your teacher 1 Are the triangles below congruent by the Hypotenuse Leg Theorem 11 3 Are the triangles below congruent by the Hypotenuse Leg Theorem 2 Are the triangles below congruent by the Hypotenuse Leg Theorem 4 Are the triangles below congruent by the Hypotenuse Leg Theorem

whether 2 Reason Bank Your Turn are congruent 1 Given Prove Given Prove HL Triangle Congruence Theorem Reflexive Property of Congruence me wo Isosceles triangle A PQS Mogy with PQPS and PR 1 QS A PQR A PSR Statements 1 PQ PS and PR I QS 1 LPRQ and ZPRS are right angles 3 A PQR and A PSR are right triangles 4 PR PR 5 A PQR A PSR Use the HL Congruence Theorem to prove that the triangles Statements ZCAB and ZDBA are right angles and AD BC A ABC A BAD insbusqe Perpendicular lines form alwright angles 1 ZCAB and ZDBA are right angles 2 AD BC 3 AB AB 4 A ABC A BAD Chidman Reasons triangle Definition of right Q Given Reasons P A DE RS A D

Learning Goal I can prove theorems about geometric figures using triangle congruence and similarity 19 07 Starting Lesson 6 3 Checkpoint Once you have completed the above problems and checked your solutions complete the Lesson Checkpoint below Complete the Lesson Reflection above by circling your current understanding of the Learning Goal G Getting there 1 Determine whether enough information is given to prove that the triangles are congruent by the HL Congruence Theorem Explain your answer AGKJ and AJHG K Got H

550 du site y potenuce Your Turn 1 For each pair of triangles use the HL Theorem Congruence Criteria to determine whethe the triangles are congruent by the HL Theorem HL Theorem Congruence Criteria The triangles are right angles A pair of legs are congruent 2 HL Theorem Congruence Criteria The triangles are right angles A pair of legs are congruent Reason Bank

Your Turn 2 Use the figure to find the mzM Lesson 7 2 Ch 3 Use the figure to find the mzP N 7y 4y 15 M 3x 3 R 5x 2

5 What additional information would be enough to prove that AHIJ AKLM by HL I A IH LK B ZILL C LJ LM or I L D ZJ LM H J 7 What additional information would be enough to prove that AKLM AWVM by HL L domany M K March A LK LW B LK VW C ML MV or LK VW D ZKML LWMV K WAS APP pogrens 6 What additional information would be enough to prove that AXYZ AXYJ by HL Z bezmsidots 12 X A LZXY LJXY or LXYZ LXYJ B YZ YJ C XY XY D YZ YJ or ZX JX K 8 What additional information would be enough to prove that AWUVAKUL by HL W V A UV UL or WU KU B LWUV LKUL or V L C LWUV KUL or W 2K D LV ZL Optional Complete problems 2 3 5 10 13 16 17 19 20 from your textbook on separate lined paper When you are finished check the odd problems for solutions at the back of the textbook

Lesson 6 2 Homework 1 Write whether the triangles below are congruent by ASA AAS or neither W Complete problems 1 6 below for independent practice 3 What additional information would be enough to 4 What additional information would be enough to prove that the two triangles are congruent by prove that AWVU NMLA by AAS AAS 0000 When you are finished check the solutions with your teacher U A B C D N M A ZWLN B VU ML C ZULL OD WU NL 2 Write whether the triangles below are congruent by ASA AAS or neither BC ZC LB LZ DB XZ ZD 2X X A O B C D 5 What additional information would be enough to 6 What additional information would be enough to prove that ADBC AXZC by AAS prove that ARVU AWUV by AAS R Z XZ GF 2X LG YZ EF XY GF determine whether W DA ZW LR OB C OD LRVU LWUV VW UR RV WU Optional Complete problems 2 5 7 12 from your textbook on separate lined paper finichod check the odd problems for solutions at the back of the textbook

opposite adjucent spolenice 2 Your Turn 1 Statements 1 LN NP 2 LM 20 3 LLNM LPNO Choose from the phrases in the reason bank above to complete each proof Given LN NP and 2M 20 Prove A LMN A PON 4 ALMNA PON Reason Bank Vertical Angles are Congruent Statements 1 LT LR 2 LSQT LSQR Given 2T LR and ZSQT ZSQR Prove A QTSA QRS 3 QS QS 4 AQTS AQRS Reflexive Property of Congruence Reasons signat T 2 Given Reasons So no AAS Triangle Congruence Theorem M SORP N S O R EXPLAIN

Use the figure to find the value of x 3x 1 E 2x

urements Communicate effectively based on purpose task and audience using appropriate academic vocabulary and medium Angle vence ENGAGE Transformations and Angles Complete the following activity with your teacher in order to continue your learning on lines angles and triangles For triangle ABC the points D E and F are the midpoints of each side of the triangle Use a straightedge to roughly sketch the perpendicular bisector for sides AB BC and AC Name the point where the lines intersect 0 B D ob B E A anothe F For the following triangle ABC use a straightedge to sketch the angle bisectors for each of the angles Locate the location where the lines intersect Hint Simply draw a perpendicular line from these points C Hint Bisect LA LB and C

Write a two column proof 1 Given GJ GM GK bisects ZJGM Prove AGJK AGMK J M K Statement Rea

4 Given ZQ S ZTRS RTQ Prove AQRT ASTR Q T R S Statement Reason

4 Given ZQ ZS ZTRS ZRTQ Prove AQRT ASTR R Statement Reaso

B Prove ABDH AFDH D DH bisects ZBDF 12 H F Statement Reason

lumn proof Given GJ GM GK bisects ZJGM Prove AGJK AGMK J G M K Statement Reason

8 Given ZI 22 DH bisects ZBDF Prove ABDH AFDH B D 12 H F Statement Reason

Write a two column proof 1 Given EF FH ZE H Prove HG DE D G E F H Statement Reason

4 Given ZQ ZS ZTRS ZRTQ Prove AQRT ASTR R Statement R