Question

angles are congruent or supplement. give your reasoning.
example 3
if m∠6 = 142°, find each measure.
a. m∠1 =
b. m∠2 =
c. m∠3 =
d. m∠4 =
e. m∠5 =
f. m∠7 =
g. m∠8 =
example 4
if m∠5 = 82°, find each measure.
a. m∠1 =
e. m∠6 =
b. m∠2 =
f. m∠7 =
c. m∠3 =
g. m∠8 =
d. m∠4 =
h. m∠9 =
example 5
if m∠12 = 121° and m∠6 = 75°, find each measure.
a. m∠1 =
e. m∠5 =
b. m∠2 =
f. m∠7 =
c. m∠3 =
g. m∠8 =
d. m∠4 =
h. m∠9 =
example 6
if m∠7 = 38° and m∠10 = 102°, find each measure.
a. m∠1 =
f. m∠6 =
b. m∠2 =
g. m∠8 =
c. m∠3 =
h. m∠9 =
d. m∠4 =
i. m∠11 =
e. m∠5 =
j. m∠12 =
if m∠2 = 41°, m∠5 = 94°, and m∠10 = 109°, find each measure.
a. m∠1 =
b. m∠3 =

Explanation:

Step1: Identify vertical - angle relationship

Vertical angles are congruent. For example, in a set of intersecting lines, if we have two lines intersecting, the angles opposite each other are equal.

Step2: Identify corresponding - angle relationship

When we have parallel lines cut by a transversal, corresponding angles are congruent. For example, if two parallel lines are intersected by a third line, the angles in the same relative position are equal.

Step3: Identify alternate - interior/exterior - angle relationship

Alternate - interior angles and alternate - exterior angles formed by parallel lines and a transversal are congruent.

Step4: Identify supplementary - angle relationship

Adjacent angles that form a straight - line are supplementary (their sum is 180°).

Let's take Example 3:
Given \(m\angle6 = 142^{\circ}\)
a. \(\angle1\) and \(\angle6\) are alternate - exterior angles. So \(m\angle1=m\angle6 = 142^{\circ}\)
b. \(\angle2\) and \(\angle6\) are corresponding angles. So \(m\angle2=m\angle6 = 142^{\circ}\)
c. \(\angle3\) and \(\angle6\) are adjacent and supplementary. So \(m\angle3=180 - 142=38^{\circ}\)
d. \(\angle4\) and \(\angle3\) are vertical angles. So \(m\angle4 = m\angle3=38^{\circ}\)
e. \(\angle5\) and \(\angle3\) are alternate - exterior angles. So \(m\angle5=m\angle3 = 38^{\circ}\)
f. \(\angle7\) and \(\angle3\) are corresponding angles. So \(m\angle7=m\angle3 = 38^{\circ}\)
g. \(\angle8\) and \(\angle6\) are vertical angles. So \(m\angle8=m\angle6 = 142^{\circ}\)

Answer:

Example 3:
a. \(142^{\circ}\)
b. \(142^{\circ}\)
c. \(38^{\circ}\)
d. \(38^{\circ}\)
e. \(38^{\circ}\)
f. \(38^{\circ}\)
g. \(142^{\circ}\)