2D Geometry Questions and Answers

29 Given Prove Plan R Parallelogram RSTV also XY VT 21 LS First show that RSYX is a parallelogram S

In Exercises 19 to 22 classify each statement as true or false In Exercises 19 and 20 recall that the symbol C means is a subset of 19 Where Q quadrilaterals and P polygons QCP 20 Where Q OCP quadrilaterals and P parallelograms

3 MNPQ is a and m M 110 Find a QP b NP parallelogram Suppose that MQ M P N c mZQ d m P 5 MN

11 Given that mA 2x 3 and m B 3x 23 find the measure of each angle of ABCD

In Exercises 27 to 30 use the definition of a parallelogram to complete each proof 27 Given Prove R V 1 RS VT 2 3 4 RS VT RV 1 VT and ST VT RSTV is a parallelogram Statements S PROOF Reasons 1 2 Given 3 If two lines are to the same line they are to each other 4 If both pairs of opposite sides of a quadrilateral are II the quadrilateral is a

9 Given that AB 3x 2 BC CD 5x A D 4x 1 and 2 find the length of each side of ABCD B Exercises 9 16 C

For Exercises 5 to 8 MNPQ is a parallelogram with diagonals QN and MP 7 8 M Exercises 5 8 5 a If QN 12 8 find QR a If QR 7 3 find RN If QR R If MR and MP N b If MR 5 3 find MP b If MP 10 6 find RP 2x 3 and RN x 7 find QR RN and QN 5 a 7 and MP 12a 34 find MR RP

In Exercises 17 and 18 consider RSTV with VX 1 RS and VY I ST 17 a Which line segment is the altitude of RSTV with respect to base ST b Which number is the height of RSTV with respect to base ST 18 a Which line segment is the altitude of RSTV with respect to base RS R 12 20 b Which number is the Exercises 17 18 height of RSTV with respect to base RS 16 15

26 Quadrilateral RSTV has RS TV and RS TV Using intuition what type of quadrilateral is RSTV R S

5 Fill in each blank with always sometimes or never a The corresponding angles of two parallel lines have measures that are equal b A closed figure is c The diameter of a circle is bigger than a radius of that circumference d Isosceles triangles are right triangles 6 a Determine each angle measure in the triangle b Then state whether the triangle is acute right or obtuse AND c also state whether the triangle is scalene isosceles or equilateral x 30 2x 120 C simple 4x 15

7 DF DE EF E 60 D G 50 F Exercises 7 8 8 If DG is the bisector of FDF then DG I

10 A 35 Exercises 9 10 L E D 45 10 B

In Exercises 15 to 18 describe the triangle AXYZ not shown as scalene isosceles or equilateral Also is the triangle acute right or obtuse 15 m X 43 and m Y 47 16 m X 60 and LY LZ 17 m X mZY 40 18 m X 70 and m Y 40 19 Two of the sides of an isosceles triangle have lengths of 10 cm and 4 cm Which length must be the length of the base

14 Is it possible to draw a triangle whose sides measur a 7 7 and 14 9 b 6 7 and 14 c 6 7 and 8 9

In Exercises 15 to 18 construct angles having the given measures 15 90 and then 45

20 Describe how you would construct an angle measuring 75 21 Construct the complement of the acute angle Q shown Q

19 Describe how you would construct an angle measuring 22 5 3

7 In Exercises relationship sets intersect 11 to 14 determine whether the sets have a subset Are the two sets disjoint or equivalent Do the 11 L equilateral triangles E equiangular triangles 12 S triangles with two sides A triangles with two Ls 13 R right triangles O obtuse triangles

23 A surveyor knows that a lot has the shape of an isosceles triangle If the vertex angle measures 70 and each equal 23 side is 160 feet long what measure does each of the base

22 Is it possible for a triangle to be a an acute isosceles triangle b an obtuse isosceles triangle c an equiangular isosceles triangle

In Exercises 31 to 34 suppose that BC is the base of isosceles AABC not shown 3 3 31 Find the perimeter of AABC if AB 8 and BC 10 32 Find AB if the perimeter of AABC is 36 4 and BC 14 6 33 Find x if the perimeter of AABC is 40 AB x and BC x 4

38 Because of construction along the road from A to B Alinna drives 5 miles from A to C and then 12 miles from C to B How much farther did Alinna travel by using the alternative route from A to B A B

18 Given Prove M is the midpoint of NQ NP RQ with transversals PR and NQ NP OR

B D 1 2 Exercises 5 6 6 Given Prove 7 Given Prove 8 Given Prove 9 Given Prove 10 Given Prove 21 and 22 are right s AB bisects CAD AABC AABD P is the midpoint of both MR and NQ AMNP AROP MN OR MN QR AMNP AROP LR and LV are right s 21 42 ARST AVST 21 42 23 24 ARST AVST Exercises 7 8 S M 1 2 R R 3 4

In Exercises 1 to 4 state the reason SSS SAS ASA AAS or HL why the triangles are congruent 1 Given Prove 2 Given Prove 3 Given Prove 4 Given Prove 21 42 LCAB LDAB ACABADAB LCAB LDAB AC AD ACABADAB LM and LR are right angles MN QR MP RP AMNP AQRP P is the midpoint of MR and LN 40 AMNP AQRP C B D 1 2 Exercises 1 2 M Exercises 3 4 P R A N

Find the equation of the graph given below Notice that the cosine function is used in the answer template representing a cosine function that is shifted and or reflected Use the variable x in your equation rather than the multiplication x symbol Provide your answer below 2 0 5 3 6 TT 2 1 0 0 5 TT

Write a system of equations associated with the augmented matrix Do not try to solve 11 5 5 O 9x y z 7 x 5y 3 9x y 7 x 3y 3 5x 5y 9 9x 7 x y 3 5x 5y 9

y 4 csc x 7 1 Drag the movable red point to shift the function the black point to set the vertical asymptotes and the blue point at the correct set of coordinates You may click on a point to verify its coordinates Note Make sure to move the points in the direction of the phase shift represented in the function Provide your answer below 2TT 31 2 8 71 g 2 6 5 4 3 2 0 0 10 2 2 1 TT 2 0 3m 2 211

For the function f x csc x set the moving point to a possible values of f x Provide your answer below 1 1 T 2TT 4 2 0 0 23 0 1 1 1 1 2TT 1 1

Find the LCM of y2 1 and y 7y 6

Make a table of values using multiples of 4 for x If an answer is undefined enter UNDEFINED y sec x X 0 Ala 3x K 5x 4 3x 2 7 4 2x 5 Use the entries in the table to sketch the graph of the function for x between 0 and 2x

Provide your answer below Drag the black dot to shift your graph in the desired direction Use the blue draggable dot to change the period Drag the orange dot to change the amplitude and or reflect with respect to the x axis The horizontal distance between the vertical dotted green lines corresponds to one period Note Reference the sinusoidal function in the form A sin Bx C D 6 5 4 3 2 t y sin 2x 1 2 1 27 0 6 9 1

A If the parabola opens up down left or right B The location of the focus C The equation of the directrix 1 y 1 48 x 2 y 1 8 x 3 x 1 40 y 4 x 1 8 y 5 y 1 28 x 6 x 1 36 y 7 x 1 12 y 8 y 1 36 x

1 Center 5 1 passing through 8 2 2 Center 0 0 passing through 4 3 3 Center 2 3 passing through 1 3 4 Center 1 2 passing through 1 0 5 Center 5 9 passing through 2 9 6 Center 4 3 passing through 2 2 7 Center 7 2 passing through 1 6 8 Center 3 3 passing through 3 2 9 Center 3 2 passing through 5 2

Deandre wants to paint a rectangular region of a wall with one layer of paint He has 12 ft of tape to tape around the border of the region he paints He is considering three regions with lengths of 1 ft 3 ft and 4 ft Answer the questions below to find which of these regions would require the most paint a Fill in the table to find the width and the area for each region Region 11 Region 2 Region 31 Length 1 ft 3 ft 4 ft Width 0ft b Which of these regions would require the most paint Region 1 O Region 2 Region 3 Area 0

N In Exercises 21 to 24 the triangles named can be proved congruent Considering the congruent pairs marked name the additional pair of parts that must be congruent in order to use the method named 3 1 21 SAS 22 ASA W 23 SSS M 24 AAS E D E AABD ACBE X AWVY AZVX N AMNO AOPM G Y QN P C Z H

5 N A Exercises 25 26 26 Given Prove DC AB and AD BC AABC ACDA Statements 1 DC AB 2 LDCA LBAC 3 4 B 5 AC AC 6 PROOF Reasons 1 2 3 Given 4 If two lines are cut by a transversal alternate interior Ls are 5 6 ASA

In Exercises 13 to 18 use only the given information to state the reason why AABC ADBC Redraw the figure and use marks like those used in Exercises 9 to 12 3 A 3 4 12 B Exercises 13 18 D

In Exercises 9 to 12 congruent parts are indicated by like dashes sides or arcs angles State which method SSS SAS ASA or AAS would be used to prove the two triangles congruent 3 1 9 10 11 12 A C M AA R B N E S D

6 In a right triangle the sides that form the right angle are the legs the longest side opposite the right angle is the hypotenuse Some textbooks say that when two right triangles have congruent pairs of legs the right triangles are congruent by the reason LL In our work LL is just a spe cial case of one of the postulates in this section Which postu late is that

6 Find x Degrees 7 Explain how you got your answer A What properties did you use B What formula did you use

and h x z 19 Then what MUST be true 20 Explain how you arrived at your answer A What Algebra Property did you use to come to your solution B What was the logic process you went through to determine the ans a

Center 2 4 4 Radius is 4 2 6 What is the equation of this circle y Input the correct sign or along with the appropriate value WITHOUT space

If 4XYZ 130 find a and Ab b Z 16 The relationship between a and b is that they are 17 Set up an equation to model this information Use a for the measure of a and b for the measure of b Write your answer WITHOUT spaces

B C A is a right angle and C 30 13 Set up an equation to model this information use B for the m Use the button to input the equation Write your answer WITHOUT spaces 14 Solve the equation for B

Using the equation to determine the following for the next two questions 3 y 9 81 10 Find the Center of the circle 11 Find the Radius of the circle units

15 Name the property AB CD then AB EF CD EF O Reflexive Property O Symmetric Property O Transitive Property O Additive Property

12 What is the relationship between angle a and angle Oasb Oa b O a b O a b b

Using the figure below to determine the following for the next three questions 1 Name a Secant line O AF GH EB DC 2 Name a Central Angle

23 Consider any triangle and one exterior angle at each vertex What is the sum of the measures of the three exterior angles of the triangle