Question

1. complete the proof by justifying each statement.

given: ∠gij and ∠kih are vertical angles, m∠gij = 16x - 20 and m∠kih = 13x + 7
prove: x = 9
statements\tanswer\treasons
∠gij and ∠kih are vertical ∠s\th\t
∠gij ≅ ∠kih\t\t
m∠gij = m∠kih\t\t
m∠gij = 16x - 20, m∠kih = 13x + 7\t\t
16x - 20 = 13x + 7\t\t
3x - 20 = 7\t\t
3x = 27\t\t
x = 9\t\t
a. addition property
b. def of angle or segment bisect
c. def of congruent angles or seg
d. def. of complementary ∠s
e. def. of supplementary ∠s
f. distributive property
g. division property
h. given
i. linear pair postulate
j. multiplication property
k. reflexive property of = or ≅
l. segment addition postulate
m. simplify or combine like terms
n. substitution property
o. subtraction property
p. symmetric property of = or ≅
q. transitive property of = or ≅
r. vertical angle theorem

1. complete the proof below.

given: ∠1 and ∠4 form a linear pair, m∠3 + m∠1 = 180
prove: ∠3 ≅ ∠4
statements\tanswer\treasons

1. ∠1 and ∠4 form a linear pair\t\t
2. ∠1 and ∠4 are supplementary\t\t
3. m∠1 + m∠4 = 180\t\t
4. m∠3 + m∠1 = 180\t\t

m∠1 + m∠4 = m∠3 + m∠1\t\t
m∠4 = m∠3\t\t
∠4 ≅ ∠3\t\t
a. addition property
b. def of angle or segm
c. def of congruent ar
d. def. of complemer
e. def. of suppleme
f. distributive pro
g. division prope
h. given
i. linear pair
j. multiplicati
k. reflexive
l. segmer
m. simplif
n. subs

Explanation:

Step1: Apply vertical - angle theorem

By the vertical - angle theorem, vertical angles are congruent. Since $\angle GIJ$ and $\angle KIH$ are vertical angles, $\angle GIJ\cong\angle KIH$, so $m\angle GIJ = m\angle KIH$. Given $m\angle GIJ=16x - 20$ and $m\angle KIH = 13x+7$, we have the equation $16x - 20=13x + 7$.

Step2: Subtract $13x$ from both sides

Using the subtraction property of equality, $(16x-13x)-20=(13x - 13x)+7$, which simplifies to $3x-20 = 7$.

Step3: Add 20 to both sides

Using the addition property of equality, $3x+( - 20+20)=7 + 20$, so $3x=27$.

Step4: Divide both sides by 3

Using the division property of equality, $\frac{3x}{3}=\frac{27}{3}$, and $x = 9$.

For the second proof:

Step1: Use the definition of a linear pair

Since $\angle1$ and $\angle4$ form a linear pair, by the definition of a linear pair, $\angle1$ and $\angle4$ are supplementary, so $m\angle1 + m\angle4=180$ (Reason: Definition of supplementary angles, E).

Step2: Given information

We are given that $m\angle3 + m\angle1=180$ (Reason: Given, H).

Step3: Set the two equations equal

Since $m\angle1 + m\angle4=180$ and $m\angle3 + m\angle1=180$, we can set $m\angle1 + m\angle4=m\angle3 + m\angle1$ (Reason: Substitution Property, N).

Step4: Subtract $m\angle1$ from both sides

Using the subtraction property of equality, $(m\angle1 - m\angle1)+m\angle4=(m\angle3 + m\angle1 - m\angle1)$, so $m\angle4=m\angle3$.

Step5: Use the definition of congruent angles

Since $m\angle4=m\angle3$, by the definition of congruent angles, $\angle4\cong\angle3$ (Reason: Definition of congruent angles, C).

Answer:

For the first proof:

Statements	Answers	Reasons
$\angle GIJ\cong\angle KIH$	R	Vertical Angle Theorem
$m\angle GIJ = m\angle KIH$	C	Def of Congruent angles or segments
$m\angle GIJ=16x - 20,m\angle KIH = 13x + 7$	H	Given
$16x - 20=13x + 7$	N	Substitution Property
$3x-20 = 7$	O	Subtraction Property
$3x=27$	A	Addition Property
$x = 9$	G	Division Property

For the second proof:

Statements	Answers	Reasons
$\angle1$ and $\angle4$ are supplementary	E	Def. of supplementary $\angle$s
$m\angle1 + m\angle4=180$	E	Def. of supplementary $\angle$s
$m\angle3 + m\angle1=180$	H	Given
$m\angle1 + m\angle4=m\angle3 + m\angle1$	N	Substitution Property
$m\angle4=m\angle3$	O	Subtraction Property
$\angle4\cong\angle3$	C	Def of Congruent angles or segments