Geometry Honors Midterm Review 2017 Answer Key

1.)

2x+80=4x+8 *now substitute 36 for x for the angles and solve

-2x -2x 4(36) + 8 = 152 2(36) + 80= 152 *remember alternate exterior angles are =

80= 2x +8 180-152= 28 will be the other angles that are supplementary (linear pairs)

-8 -8

72 = 2x

2 2

36 = x

2.) GJ is double PQ so 2(19) = 38

3.) 5/12 = 3/y *label the base of the small triangle y to find that side first use a proportion then cross

multiply to find the base. So 5y = 36 divide each side by 5 then y =7.2

Now use Pythagorean theorem to find the hypotenuse of the right triangle so a^2 + b^2 = c^2

remember a and b are your sides and c is the hypotenuse so 3^2 + 7.2^2 = c^2 then 9 + 51.84 = 60.84

now the last step is to take the square root of 60.84 = 7.8

4.)

. XZ = 36 Since the entire line segment is 36 using the segment addition postulate therefore

4x+2 + 2x - 2 = 36 combine like terms and solve for x now

6x = 36

6x = 36

6 6

X = 6

*now substitute 6 in for x and solve: 4(6) + 2 = 26 and 2(6) - 2 = 10 now check 26+10= 36

2x+80

4x+8

x y z 4x+2 2x-2

5.) *hint draw out the following two triangles separately and make your markings as you do

each step of the proof.. Triangle JXL and Triangle LYJ

Statements Reasons

1.)JK = LK Given

2.) <KJL = <KLJ Definition of Isosceles Triangle (base angles are congruent)

3.) <JYL and < LXJ are right angles Given

4.) <JYL = <LXJ since they both =90 Definition of Right Angles

5.) JL = JL Reflexive Property

6.) Triangle JXL = LYJ AAS Triangle Congruency

7.) JY = LX CPCTC (corresponding parts of congruent tri angles are

congruent)

*remember proofs can go in different order/wording but it should be similar to this, especially

the last 3 steps

6.) *see picture of graph paper at the end…

7.) Since angle bisectors mean that both angles are congruent -

Set them up to equal each other 6x-80 = 2x+130

+80 +80

6x = 2x + 210

-2x -2x

6x

4x = 210

4 4

x = 52.5

*Now go back and substitute in 52.5 for x for each angle and solve.

6(52.5) – 80 = 2(52.5) + 130 =

315-80=235 105 + 130 = 235

Both are supposed to be equal so both angles = 235 *NOT TO SCALE!! That means the entire

original angle would be 235 + 235 so 470 degrees *again sorry NOT to scale!!! The midterm

question is the same concept though..

6x-80

2x+130

b

8.) *see graph at the end 9.) *see graph at the end 10.) *see graph at the end

11.)

*Since 125 and a are linear pairs and therefore supplementary 180-125 = 55 so a=55 now

since a and b are alternate interior angles and congruent that means b = 55 the angle

supplementary to angle a and b would be 125 and so forth..

12.) Remember find the difference and sum of the two sides so 12-4 = 8 and 12+4 =16 so

therefore 8<x<16 would be possible values of the third side.

13.)

<BED is 180-142=38 since they are a linear pair and supplementary.

<BDE would be 180-90 – 38 = 52 *also remember <DBE + <BDE = <BEF *the two remote

interior angles of a triangle added together equal the exterior angle. So 90 + 52=142.

<BDE is congruent to <DBA since they are alternate interior angles so <DBA = 52 *this also

makes sense since <ABE and <CBE are alternate interior angles so 90 + 52=142

<CBE = 38 since it’s alternate interior angles to <BED *check the top 3 angles are linear so they

must add up to 180= (38 + 90 + 52)

<D = 128 (the exterior angle of triangle DBE is equal to the sum of the two remote interior

angles so <DBE + <BED which is 90 + 38 = 128

125

a

14.) GJ is double DE so 3x + 25 = 2(4x + 5) now solve for x. Distributive property first that gives

you 3x + 25 = 8x + 10

-3x -3x

25=5x + 10

-10 -10

15 = 5x

5 5

3 = x

*now go back and substitute in 3 for x in DE 4(3) + 5 = 12 + 5 = 17 now check to make sure

substitute in 3 for GJ to see if it’s double 3(3) + 25 = 9 + 25 = 34 so yes it’s correct!

15.) *see graph paper at the end

16.) *see graph paper at the end

17.) 3x + 2 is the longest side across from the largest angle based on info given *I will go over

this in class as well, we need to add an additional angle

18.) Angle addition property so the two angles = the larger outside angle therefore

3x -1 + x -1 = 150 now solve for x combine like terms so 4x-2=150 add 2 to each side (addition

property) and divide by 4 (division property) to get x =38. Substitute 38 for x solve for both

angles to prove 3(38) -1 = 113 & 38 -1 = 37 now add the two angles to prove they equal 150

combined so 113 + 37 = 150

19.) *see graph paper at the end

20.)

A M. B

Since M is the midpoint AM = MB 10x + 4 = 3x + 10 AB = 10x +4 + 3x + 10

-3x -3x 13x + 14

7x + 4 = 10 13(6/7) + 14

-4 -4 11.1 + 14

7x = 6 25.1

7 7

X = 6/7

10x + 4 3x +10

So x + 95= 132 * remember two remote interior angles = the opposite exterior angle

-95 -95

Therefore x = 37

22.) 6x + 4 + 30 = 10x CD: 6(8.5) +4 = 55

6x + 34 = 10x DE = 30 (already given)

-6x -6x CE = 10(8.5) = 85

34 = 4x

4 4

8.5 = x

23.) 6x +4 + 8x +2 + 2x-2 = 180 6(11) + 4 = 70

16x + 4 =180 8(11) + 2 = 90

-4 -4 2(11) – 2 = 20

16x = 176

16 16

X = 11

24.) 6x – 5 = 2x + 3 6(2) -5 and 2(2) + 3 so 7 would be for both angles that equal each

-2x -2x other and the total angle ABC would be 2(7) which equals 14

4x -5 = 3

+5 +5

4x = 8 x = 2

180-85=95

25.) A = D so set up x + 50 = 80 therefore x = 30 then 80 + 40 =120 so subtract 180-120 =60

since the angles are all congruent you have angles that are 80, 40, and 60

26.) IK = JH given

HI = JK given

HK = HK reflexive

Triangle KHI = Triangle KJH SSS Triangle Congruency

27.) *ADD AB = BC!!! So then 6x-4 = 2x + 8 4x = 12 x = 3 then 6(3) -4 = 14 and 2(3) + 8 = 14

therefore the triangle can be equilateral, isosceles, and acute

28.) see graph at the end

29.) AX = XC so AX = 3.4 and the triangles are congruent via SAS so BC = 8.2

30.) 4x + 10 =2x + 16 4(3) + 10 = 22 and 2(3) + 16 = 22 and 8(3) + 5 = 29

-2x -2x

2x + 10 = 16

-10 -10

2x = 6

X = 3

31.) WZ = 14.5 remember all segments from the same point on the angle bisector to the

opposite side of the angle is congruent.

32.) 6x-9 + 2x + 30 = 121 combine like terms so 8x + 21 = 121 then 8x = 100 so x = 12.5

6(12.5) – 9 = 66

2(12.5) + 30 = 55

66 + 55 + x = 180

So x = 59

33.) 74

21 21+74+ <Z = 180 So <Z is 85

Smallest to largest: side YZ, side XZ, side XY

x

y

z

34.) AB = ED given

BC = DC given

AC = EC given

<A = <E given

Triangle ABC = EDC *could be SSS or SAS

35.) C is the midpoint of AD and BE given

BC = CE and AC = CD definition of midpoint

<BCA = <DCE vertical angles are congruent

Triangle ABC = DCE SAS

36.) Linear Pair theorem, definition of supplementary angles, <1 + <2 + < 3 = 180, Substitution

property, subtraction property

37.) BE is smaller than ED

38.) BA = DA and <BCA = <ADC would have to be given for each situation *CA=CA would not

need to be given since that’s already there as reflexive property

39.) <CBF =<CDG given

<ABC = <ADC congruent supplements theorem

AC bisects <BAD given

<BAC = <CAD definition of angle bisector

AC = AC reflexive property

Triangle ABC = Triangle ADC AAS

AD = AB CPCTC (congruent parts of congruent triangles are congruent)

40.) 7s-22 = s + 2 *all sides equal since it’s an equilateral triangle *also all angles =60 degrees

+22 +22

7s = s +24

-s -s

6s = 24

S=4 7(4) -22 = 6 4+2 = 6 *all sides equal 6 since it’s equilateral

*ADDITIONAL QUESTION: An angle measure is 10 more degrees than 4 times its complement.

Find the measure of its complement.

4x+10 + x = 90 so 5x +10 = 90 5x = 80 x = 16 so one angle is 4(16) + 10 = 74 and it’s

complement would be what x equals so that would be 16 (also proven by 90-74).