Question

example 6 if m∠7 = 38° and m∠10 = 102°, find each measure. a. m∠1 = b. m∠2 = c. m∠3 = d. m∠4 = e. m∠5 = f. m∠6 = g. m∠8 = h. m∠9 = i. m∠11 = j. m∠12 = k. m∠13 = l. m∠14 = m. m∠15 = n. m∠16 =

Explanation:

Step1: Identify angle - relationships

Assume the figure has parallel - lines and transversals. If we consider the properties of angles formed by parallel lines and transversals such as vertical angles (equal), corresponding angles (equal), alternate - interior angles (equal), and same - side interior angles (supplementary).
Let's assume some basic angle - relationship rules. For example, if two parallel lines are cut by a transversal, vertical angles are congruent.

Step2: Find \(m\angle1\)

Since \(\angle7\) and \(\angle1\) are vertical angles, \(m\angle1 = m\angle7=38^{\circ}\)

Step3: Find \(m\angle2\)

\(\angle1\) and \(\angle2\) are supplementary (linear pair). So \(m\angle2 = 180^{\circ}-m\angle1=180 - 38=142^{\circ}\)

Step4: Find \(m\angle3\)

\(\angle3\) and \(\angle1\) are corresponding angles (assuming parallel lines), so \(m\angle3 = m\angle1 = 38^{\circ}\)

Step5: Find \(m\angle4\)

\(\angle4\) and \(\angle2\) are corresponding angles, so \(m\angle4 = m\angle2 = 142^{\circ}\)

Step6: Find \(m\angle5\)

\(\angle5\) and \(\angle7\) are alternate - interior angles, so \(m\angle5 = m\angle7 = 38^{\circ}\)

Step7: Find \(m\angle6\)

\(\angle6\) and \(\angle8\) are vertical angles. \(\angle8\) and \(\angle7\) are supplementary (linear pair), so \(m\angle8=180 - 38 = 142^{\circ}\), then \(m\angle6 = 142^{\circ}\)

Step8: Find \(m\angle8\)

\(m\angle8 = 142^{\circ}\) (as calculated above)

Step9: Find \(m\angle9\)

\(\angle9\) and \(\angle10\) are supplementary (linear pair), so \(m\angle9=180 - 102 = 78^{\circ}\)

Step10: Find \(m\angle11\)

\(\angle11\) and \(\angle9\) are vertical angles, so \(m\angle11 = m\angle9 = 78^{\circ}\)

Step11: Find \(m\angle12\)

\(\angle12\) and \(\angle10\) are vertical angles, so \(m\angle12 = m\angle10 = 102^{\circ}\)

Step12: Find \(m\angle13\)

Assuming appropriate parallel - line and transversal relationships, if \(\angle13\) and \(\angle9\) are corresponding angles, \(m\angle13 = m\angle9 = 78^{\circ}\)

Step13: Find \(m\angle14\)

\(\angle14\) and \(\angle10\) are corresponding angles, so \(m\angle14 = m\angle10 = 102^{\circ}\)

Step14: Find \(m\angle15\)

\(\angle15\) and \(\angle11\) are corresponding angles, so \(m\angle15 = m\angle11 = 78^{\circ}\)

Step15: Find \(m\angle16\)

\(\angle16\) and \(\angle12\) are corresponding angles, so \(m\angle16 = m\angle12 = 102^{\circ}\)

Answer:

a. \(m\angle1 = 38^{\circ}\)
b. \(m\angle2 = 142^{\circ}\)
c. \(m\angle3 = 38^{\circ}\)
d. \(m\angle4 = 142^{\circ}\)
e. \(m\angle5 = 38^{\circ}\)
f. \(m\angle6 = 142^{\circ}\)
g. \(m\angle8 = 142^{\circ}\)
h. \(m\angle9 = 78^{\circ}\)
i. \(m\angle11 = 78^{\circ}\)
j. \(m\angle12 = 102^{\circ}\)
k. \(m\angle13 = 78^{\circ}\)
l. \(m\angle14 = 102^{\circ}\)
m. \(m\angle15 = 78^{\circ}\)
n. \(m\angle16 = 102^{\circ}\)