Question

1. determine whether the statements about the relationships between the angles in the figure are true or false.

a the value of ( a ) is ( 20^circ ).
b the value of ( a ) and the value of ( b ) are equal.
c the sum of the values of ( a ), ( c ), and 90 is 180.
d the sum of the values of ( a ) and ( c ) is 90.
e the sum of the values of ( a ), ( c ), and 20 is 180.
f the sum of the values of ( b ) and ( c ) is 90.

Explanation:

Step1: Find value of $\boldsymbol{a}$

Vertical angles are equal, so $a = 20^\circ$.

Step2: Find value of $\boldsymbol{b}$

$\boldsymbol{b}$ and $20^\circ$ are complementary:
$b + 20^\circ = 90^\circ \implies b = 70^\circ$.

Step3: Find value of $\boldsymbol{c}$

$\boldsymbol{c}$ and $20^\circ$ are vertical angles with a right angle:
$c + 20^\circ = 90^\circ \implies c = 70^\circ$.

Step4: Evaluate statement (a)

Check if $a=20^\circ$: True (matches Step1).

Step5: Evaluate statement (b)

Check if $a=b$: $20^\circ
eq 70^\circ$, so False.

Step6: Evaluate statement (c)

Calculate $a + c + 90^\circ = 20^\circ + 70^\circ + 90^\circ = 180^\circ$, so True.

Step7: Evaluate statement (d)

Calculate $a + c = 20^\circ + 70^\circ = 90^\circ$, so True.

Step8: Evaluate statement (e)

Calculate $a + c + 20^\circ = 20^\circ + 70^\circ + 20^\circ = 110^\circ
eq 180^\circ$, so False.

Step9: Evaluate statement (f)

Calculate $b + c = 70^\circ + 70^\circ = 140^\circ
eq 90^\circ$, so False.

Answer:

a. True
b. False
c. True
d. True
e. False
f. False