Question

1. if m∠2 = 98°, m∠3 = 23° and m∠8 = 70°, find each measure. a. m∠1 = b. m∠4 = c. m∠5 = d. m∠6 = e. m∠7 = f. m∠9 = g. m∠10 = 4. if m∠3 = 54°, find each measure. a. m∠1 = b. m∠2 = c. m∠4 = d. m∠5 = e. m∠6 = f. m∠7 = g. m∠8 = h. m∠9 = i. m∠10 = j. m∠11 = k. m∠12 = l. m∠13 = m. m∠14 =

Explanation:

Step1: Identify vertical - angle relationships

Vertical angles are equal. For example, if we have two intersecting lines, the angles opposite each other are vertical angles.

Step2: Identify supplementary - angle relationships

Supplementary angles add up to 180°. For example, if two angles form a linear pair, they are supplementary.

Step3: Identify complementary - angle relationships

Complementary angles add up to 90°. If there is a right - angle formed by two angles, they are complementary.

Problem 3:

Given \(m\angle2 = 98^{\circ}\), \(m\angle3=23^{\circ}\), \(m\angle8 = 70^{\circ}\)

• a. \(\angle1\) and \(\angle2\) are supplementary. So \(m\angle1=180 - m\angle2=180 - 98=82^{\circ}\)
• b. Using the angle - sum property of a triangle (the sum of interior angles of a triangle is 180°), if we consider the triangle with angles \(\angle1\), \(\angle3\) and \(\angle4\), then \(m\angle4=180-(m\angle1 + m\angle3)=180-(82 + 23)=75^{\circ}\)
• c. \(\angle5\) and \(\angle4\) are supplementary. So \(m\angle5 = 180 - m\angle4=180 - 75 = 105^{\circ}\)
• d. \(\angle6\) and \(\angle8\) are vertical angles. So \(m\angle6=m\angle8 = 70^{\circ}\)
• e. \(\angle7\) and \(\angle6\) are supplementary. So \(m\angle7=180 - m\angle6=180 - 70 = 110^{\circ}\)
• f. \(\angle9\) and \(\angle7\) are vertical angles. So \(m\angle9=m\angle7 = 110^{\circ}\)
• g. \(\angle10\) and \(\angle9\) are supplementary. So \(m\angle10=180 - m\angle9=180 - 110 = 70^{\circ}\)

Problem 4:

Given \(m\angle3 = 54^{\circ}\)

• a. \(\angle1\) and \(\angle2\) are complementary (since there is a right - angle). If we assume the right - angle is formed by \(\angle1\) and \(\angle2\), and we know nothing else about their individual relationships for now. But if we consider the linear - pair and vertical - angle relationships:
• \(\angle1\) and \(\angle3\) are vertical angles, so \(m\angle1=m\angle3 = 54^{\circ}\)
• Then \(m\angle2=90 - m\angle1=90 - 54 = 36^{\circ}\)
• b. \(m\angle2 = 36^{\circ}\) (from above)
• c. \(\angle4\) and \(\angle3\) are supplementary. So \(m\angle4=180 - m\angle3=180 - 54 = 126^{\circ}\)
• d. \(\angle5\) and \(\angle4\) are vertical angles. So \(m\angle5=m\angle4 = 126^{\circ}\)
• e. \(\angle6\) and \(\angle5\) are supplementary. So \(m\angle6=180 - m\angle5=180 - 126 = 54^{\circ}\)
• f. \(\angle7\) and \(\angle6\) are vertical angles. So \(m\angle7=m\angle6 = 54^{\circ}\)
• g. \(\angle8\) and \(\angle7\) are supplementary. So \(m\angle8=180 - m\angle7=180 - 54 = 126^{\circ}\)
• h. \(\angle9\) and \(\angle8\) are vertical angles. So \(m\angle9=m\angle8 = 126^{\circ}\)
• i. \(\angle10\) and \(\angle9\) are supplementary. So \(m\angle10=180 - m\angle9=180 - 126 = 54^{\circ}\)
• j. \(\angle11\) and \(\angle10\) are vertical angles. So \(m\angle11=m\angle10 = 54^{\circ}\)
• k. \(\angle12\) and \(\angle11\) are supplementary. So \(m\angle12=180 - m\angle11=180 - 54 = 126^{\circ}\)
• l. \(\angle13\) and \(\angle12\) are vertical angles. So \(m\angle13=m\angle12 = 126^{\circ}\)
• m. \(\angle14\) and \(\angle13\) are supplementary. So \(m\angle14=180 - m\angle13=180 - 126 = 54^{\circ}\)

Answer:

Problem 3:

a. \(m\angle1 = 82^{\circ}\)
b. \(m\angle4 = 75^{\circ}\)
c. \(m\angle5 = 105^{\circ}\)
d. \(m\angle6 = 70^{\circ}\)
e. \(m\angle7 = 110^{\circ}\)
f. \(m\angle9 = 110^{\circ}\)
g. \(m\angle10 = 70^{\circ}\)

Problem 4:

a. \(m\angle1 = 54^{\circ}\)
b. \(m\angle2 = 36^{\circ}\)
c. \(m\angle4 = 126^{\circ}\)
d. \(m\angle5 = 126^{\circ}\)
e. \(m\angle6 = 54^{\circ}\)
f. \(m\angle7 = 54^{\circ}\)
g. \(m\angle8 = 126^{\circ}\)
h. \(m\angle9 = 126^{\circ}\)
i. \(m\angle10 = 54^{\circ}\)
j. \(m\angle11 = 54^{\circ}\)
k. \(m\angle12 = 126^{\circ}\)
l. \(m\angle13 = 126^{\circ}\)
m. \(m\angle14 = 54^{\circ}\)