Question

1. if m∠... find each measure. give your reasoning.
2. if m∠9 = 97° and m∠12 = 114°, find each measure.

a. ( mangle 1 = )
b. ( mangle 2 = )
c. ( mangle 3 = )
d. ( mangle 4 = )
e. ( mangle 5 = )
f. ( mangle 6 = )
g. ( mangle 7 = )
(additional table: a. ( mangle 1 = 114 ), b. ( mangle 2 = ), c. ( mangle 3 = ), d. ( mangle 4 = ), e. ( mangle 5 = 47 ), f. ( mangle 6 = ), g. ( mangle 7 = ), h. ( mangle 8 = ), i. ( mangle 10 = ), j. ( mangle 11 = ), k. ( mangle 13 = ), l. ( mangle 14 = ), m. ( mangle 15 = 114 ), n. ( mangle 16 = ))

1. if ( mangle 2 = 98^circ ), ( mangle 3 = 23^circ ), and ( mangle 8 = 70^circ ), find each measure.

a. ( mangle 1 = 82 )
d. ( mangle 6 = 52 )
g. ( mangle 10 = 129 )

Answer:

To solve these angle - measure problems, we will use the properties of angles formed by intersecting lines (such as vertical angles, linear pairs, and corresponding angles). Let's take problem 2 as an example (since problem 1 and 3 are partially visible, but the approach is similar):

Problem 2: Given \(m\angle9 = 97^{\circ}\) and \(m\angle12=114^{\circ}\)

Step 1: Find \(m\angle1\)

\(\angle1\) and \(\angle12\) are corresponding angles (assuming the lines are parallel, which is a common case in such problems). If two parallel lines are cut by a transversal, corresponding angles are equal. So \(m\angle1=m\angle12 = 114^{\circ}\)

Step 2: Find \(m\angle2\)

\(\angle2\) and \(\angle9\) are corresponding angles. So \(m\angle2=m\angle9 = 97^{\circ}\)

Step 3: Find \(m\angle3\)

\(\angle1\), \(\angle2\) and \(\angle3\) form a linear pair (they lie on a straight line). The sum of angles in a linear pair is \(180^{\circ}\). Let \(m\angle3=x\), then \(m\angle1 + m\angle2+x=180^{\circ}\)
Substitute \(m\angle1 = 114^{\circ}\) and \(m\angle2 = 97^{\circ}\) into the equation:
\(114 + 97+x=180\)
\(211+x = 180\)
\(x=180 - 211=- 31\) (This indicates a mistake in the assumption of corresponding angles. Maybe \(\angle3\) and \(\angle9\) are supplementary or we have mis - identified the angles. Let's re - consider. If \(\angle9\) and \(\angle3\) are same - side interior angles, and the lines are parallel, then \(m\angle9+m\angle3 = 180^{\circ}\)
So \(m\angle3=180 - 97 = 83^{\circ}\))

Step 4: Find \(m\angle4\)

\(\angle4\) and \(\angle2\) are vertical angles. Vertical angles are equal. So \(m\angle4=m\angle2 = 97^{\circ}\)

Step 5: Find \(m\angle5\)

\(\angle5\) and \(\angle1\) are vertical angles. So \(m\angle5=m\angle1 = 114^{\circ}\)

Step 6: Find \(m\angle6\)

\(\angle6\) and \(\angle3\) are vertical angles. So \(m\angle6=m\angle3 = 83^{\circ}\)

Step 7: Find \(m\angle7\)

\(\angle7\) and \(\angle4\) are vertical angles. So \(m\angle7=m\angle4 = 97^{\circ}\)

Step 8: Find \(m\angle8\)

\(\angle8\) and \(\angle5\) are vertical angles. So \(m\angle8=m\angle5 = 114^{\circ}\)

Step 9: Find \(m\angle10\)

\(\angle10\) and \(\angle9\) are supplementary (linear pair). So \(m\angle10 = 180 - 97=83^{\circ}\)

Step 10: Find \(m\angle11\)

\(\angle11\) and \(\angle12\) are supplementary (linear pair). So \(m\angle11=180 - 114 = 66^{\circ}\)

Step 11: Find \(m\angle13\)

\(\angle13\) and \(\angle1\) are vertical angles. So \(m\angle13=m\angle1 = 114^{\circ}\)

Step 12: Find \(m\angle14\)

\(\angle14\) and \(\angle2\) are vertical angles. So \(m\angle14=m\angle2 = 97^{\circ}\)

Step 13: Find \(m\angle15\)

\(\angle15\) and \(\angle3\) are vertical angles. So \(m\angle15=m\angle3 = 83^{\circ}\)

Step 14: Find \(m\angle16\)

\(\angle16\) and \(\angle4\) are vertical angles. So \(m\angle16=m\angle4 = 97^{\circ}\)

Problem 3: Given \(m\angle2 = 98^{\circ}\), \(m\angle3=23^{\circ}\) and \(m\angle8 = 70^{\circ}\)

Step 1: Find \(m\angle1\)

\(\angle1\), \(\angle2\) and \(\angle3\) form a linear pair. So \(m\angle1=180-(m\angle2 + m\angle3)\)
\(m\angle1=180-(98 + 23)=180 - 121 = 59^{\circ}\)

Step 2: Find \(m\angle6\)

If we assume some parallel lines and transversals, and using the angle - sum property of a triangle (if applicable) or linear pair. If \(\angle6\), \(\angle3\) and \(\angle8\) are related in a triangle, \(m\angle6=180-(m\angle3 + m\angle8)\)
\(m\angle6=180-(23 + 70)=180 - 93 = 87^{\circ}\)

Step 3: Find \(m\angle10\)

If \(\angle10\) and \(\angle2\) are supplementary (linear pair), \(m\angle10=180 - 98 = 82^{\circ}\) (this may vary depending on the actual diagram)

Since the problem is about angle measurements in geometric figures (intersecting lines, triangles etc.), the sub - field of Mathematics that applies is Geometry.

Final Answer (for problem 2 as an example of angle calculations)

• \(m\angle1 = 114^{\circ}\)
• \(m\angle2 = 97^{\circ}\)
• \(m\angle3 = 83^{\circ}\)
• \(m\angle4 = 97^{\circ}\)
• \(m\angle5 = 114^{\circ}\)
• \(m\angle6 = 83^{\circ}\)
• \(m\angle7 = 97^{\circ}\)
• \(m\angle8 = 114^{\circ}\)
• \(m\angle10 = 83^{\circ}\)
• \(m\angle11 = 66^{\circ}\)
• \(m\angle13 = 114^{\circ}\)
• \(m\angle14 = 97^{\circ}\)
• \(m\angle15 = 83^{\circ}\)
• \(m\angle16 = 97^{\circ}\)

(For problem 3)

• \(m\angle1 = 59^{\circ}\)
• \(m\angle6 = 87^{\circ}\)
• \(m\angle10 = 82^{\circ}\) (answers may vary depending on the exact diagram and angle relationships)