Question

state the theorem or postulate that is related to each angle pair then find the measure of each angle. 12. ( mangle 3 = (23x + 11)^circ ), ( mangle 4 = (14x + 21)^circ ) ( x = square ) ( mangle 3 = square ) ( mangle 4 = square ) 13. ( mangle 1 = (7x + 15)^circ ), ( mangle 2 = (10x - 9)^circ ) ( x = square ) ( mangle 1 = square ) ( mangle 2 = square ) these are 2 different problems, the angle measures are not the same for #12 and #13. diagram: intersecting lines with angles labeled 1, 2, 3, 4, 5

Explanation:

Problem 12:

Step1: Identify Angle Relationship (Vertical Angles)

∠3 and ∠4 are adjacent and form a linear pair? Wait, no, looking at the diagram, ∠3 and ∠4—wait, maybe ∠3 and ∠4 are supplementary? Wait, no, the diagram shows two intersecting lines? Wait, no, the lines: maybe ∠3 and ∠4 are same - side interior or something? Wait, no, the problem says "state the theorem or postulate". Wait, maybe ∠3 and ∠4 are supplementary? Wait, no, let's check the angle measures. Wait, maybe ∠3 and ∠4 are supplementary (linear pair)? Wait, no, the given \( m\angle3=(23x + 11)^\circ\) and \( m\angle4=(14x + 21)^\circ\). Wait, maybe they are supplementary (sum to \(180^\circ\))? Wait, no, maybe vertical angles? Wait, no, the diagram: let's assume that ∠3 and ∠4 are supplementary (linear pair) because they are adjacent and form a straight line. So \(m\angle3 + m\angle4=180^\circ\).

Step2: Set Up the Equation

\((23x + 11)+(14x + 21)=180\)
Combine like terms: \(23x+14x + 11 + 21=180\)
\(37x+32 = 180\)

Step3: Solve for x

Subtract 32 from both sides: \(37x=180 - 32=148\)
Divide both sides by 37: \(x=\frac{148}{37} = 4\)

Step4: Find \(m\angle3\) and \(m\angle4\)

For \(m\angle3\): substitute \(x = 4\) into \(23x+11\)
\(m\angle3=23\times4 + 11=92 + 11=103^\circ\)
For \(m\angle4\): substitute \(x = 4\) into \(14x + 21\)
\(m\angle4=14\times4+21 = 56+21 = 77^\circ\)

Step1: Identify Angle Relationship (Vertical Angles or Corresponding Angles? Wait, the diagram: ∠1 and ∠2—maybe they are equal (corresponding angles or vertical angles)? Wait, the problem says "the angle measures are not the same for #12 and #13". Wait, \(m\angle1=(7x + 15)^\circ\) and \(m\angle2=(10x - 9)^\circ\). Maybe they are equal (vertical angles or alternate interior angles). So set \(7x + 15=10x - 9\)

Step2: Solve for x

Subtract \(7x\) from both sides: \(15 = 3x-9\)
Add 9 to both sides: \(15 + 9=3x\)
\(24 = 3x\)
Divide by 3: \(x = 8\)

Step3: Find \(m\angle1\) and \(m\angle2\)

For \(m\angle1\): substitute \(x = 8\) into \(7x + 15\)
\(m\angle1=7\times8+15=56 + 15=71^\circ\)
For \(m\angle2\): substitute \(x = 8\) into \(10x - 9\)
\(m\angle2=10\times8-9=80 - 9=71^\circ\) Wait, but the problem says "the angle measures are not the same for #12 and #13"—wait, maybe I made a mistake. Wait, maybe ∠1 and ∠2 are supplementary? Let's check. If \(7x + 15+10x - 9=180\)
\(17x + 6=180\)
\(17x=174\)
\(x=\frac{174}{17}\approx10.24\), but the problem says "the angle measures are not the same for #12 and #13"—maybe the first assumption (they are equal) is correct, and the note means that the angle measures in #12 and #13 are different, not within #13. So proceeding with \(x = 8\), \(m\angle1 = 71^\circ\), \(m\angle2=71^\circ\)

Answer:

(Problem 12):
\(x = 4\), \(m\angle3=103^\circ\), \(m\angle4 = 77^\circ\)

Problem 13: