Question

1. use the diagram to answer questions a - c (each is a separate problem). for each problem, justify your answer.

a. if m∠b = 74°, then
i. m∠f = _______ because __.
ii. m∠g = _______ because __.
b. if m∠e = 155°, then
i. m∠c = _______ because __.
ii. m∠d = _______ because __.
c. if m∠a = x° and m∠g=(x - 20)°, then
i. m∠x = _______ because __.
ii. m∠f = _______ because __.

1. find the values of angles a - c in the diagram to the right

a. m∠a = ________
b. m∠b = ________
c. m∠c = ________

Explanation:

Step1: Identify angle - relationships in part a - i

Since the two lines are parallel, $\angle b$ and $\angle f$ are corresponding angles. Corresponding angles are equal when two parallel lines are cut by a transversal.
$m\angle f=m\angle b = 74^{\circ}$

Step2: Identify angle - relationships in part a - ii

$\angle b$ and $\angle g$ are alternate - exterior angles. Alternate - exterior angles are equal when two parallel lines are cut by a transversal.
$m\angle g=m\angle b = 74^{\circ}$

Step3: Identify angle - relationships in part b - i

$\angle e$ and $\angle c$ are vertical angles. Vertical angles are equal.
$m\angle c=m\angle e = 155^{\circ}$

Step4: Identify angle - relationships in part b - ii

$\angle e$ and $\angle d$ are same - side interior angles. Same - side interior angles are supplementary (their sum is $180^{\circ}$) when two parallel lines are cut by a transversal.
$m\angle d=180^{\circ}-m\angle e=180 - 155=25^{\circ}$

Step5: Identify angle - relationships in part c - i

$\angle a$ and $\angle g$ are same - side interior angles. So $m\angle a + m\angle g=180^{\circ}$. Substitute $m\angle a=x^{\circ}$ and $m\angle g=(x - 20)^{\circ}$ into the equation: $x+(x - 20)=180$. Combine like terms: $2x-20 = 180$. Add 20 to both sides: $2x=200$. Divide by 2: $x = 100$. So $m\angle a=100^{\circ}$

Step6: Identify angle - relationships in part c - ii

$\angle a$ and $\angle f$ are alternate - exterior angles. So $m\angle f=m\angle a = 100^{\circ}$

Step7: Identify angle - relationships in question 6 - a

The angle adjacent to the $72^{\circ}$ angle on the straight - line has a measure of $180 - 72=108^{\circ}$. Using the angle - sum property of a triangle (the sum of the interior angles of a triangle is $180^{\circ}$), and considering the triangle formed by the parallel lines and the transversal, $m\angle a=180-(108 + 39)=33^{\circ}$

Step8: Identify angle - relationships in question 6 - b

$m\angle b = 39^{\circ}$ because of the property of alternate interior angles (the two parallel lines and the transversal)

Step9: Identify angle - relationships in question 6 - c

$m\angle c=108^{\circ}$ because of the property of corresponding angles (the two parallel lines and the transversal)

Answer:

a. i. $74^{\circ}$, corresponding angles are equal
ii. $74^{\circ}$, alternate - exterior angles are equal
b. i. $155^{\circ}$, vertical angles are equal
ii. $25^{\circ}$, same - side interior angles are supplementary
c. i. $100^{\circ}$, same - side interior angles are supplementary
ii. $100^{\circ}$, alternate - exterior angles are equal

1. a. $33^{\circ}$

b. $39^{\circ}$
c. $108^{\circ}$