Question

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Explanation:

Problem 12:

Step 1: Identify Angle Relationship

The angles \(3x^\circ\) and \(135^\circ\) are vertical angles (or corresponding, depending on diagram), so \(3x = 135\).

Step 2: Solve for \(x\)

Divide both sides by 3: \(x=\frac{135}{3}=45\).

Problem 13:

Step 1: Identify Angle Relationship

The angle \(4x + 12\) and \(112^\circ\) (since \(68 + 112 = 180\), and the other angle is equal to \(112\) due to parallel lines/transversal). So \(4x + 12 = 112\).

Step 2: Solve for \(x\)

Subtract 12: \(4x=112 - 12 = 100\). Divide by 4: \(x=\frac{100}{4}=25\).

Problem 14:

Step 1: Identify Angle Relationship

The angle \(5x + 98\) and \(67\) are equal (corresponding angles), or \(23x - 2\) and \(67\)? Wait, looking at the diagram, the top angle is \(67^\circ\), the middle is \(5x + 98\), bottom is \(23x - 2\). Wait, maybe \(5x + 98 = 23x - 2\)? No, wait the equation is \(\_\_\_ = 67\). Wait, maybe \(5x + 98 = 67\)? No, that would give negative. Wait, maybe \(23x - 2 = 67\)? Wait, no, let's re - examine. Wait, the first angle (top) is \(67^\circ\), middle is \(5x + 98\), bottom is \(23x - 2\). Wait, maybe the middle angle \(5x + 98\) and top \(67\) are not, but maybe the bottom angle \(23x - 2\) and top \(67\)? No, wait, perhaps the angle \(5x + 98\) is equal to \(67\)? Wait, no, that would be \(5x=67 - 98=- 31\), \(x=-6.2\), which is odd. Wait, maybe I made a mistake. Wait, the problem says "\(\_\_\_ = 67\)". Let's check the angles. The top line has \(67^\circ\), middle line has \((5x + 98)^\circ\), bottom line has \((23x - 2)^\circ\). Wait, maybe \(5x + 98 = 23x - 2\), but the equation is \(\_\_\_ = 67\). Wait, perhaps the angle \(5x + 98\) is equal to \(67\)? No, that can't be. Wait, maybe the angle \(23x - 2\) is equal to \(67\)? Let's try that. \(23x-2 = 67\), \(23x=69\), \(x = 3\). Then check \(5x + 98=5*3 + 98 = 15+98 = 113\), which is not 67. Wait, maybe \(5x + 98 = 67\)? \(5x=67 - 98=-31\), \(x=-6.2\), which is invalid. Wait, maybe the angle \(5x + 98\) and \(67\) are supplementary? No, \(67+113 = 180\), but \(5x + 98 = 113\) when \(x = 3\) (since \(5*3+98 = 113\)), and \(113+67 = 180\). Wait, but the problem says "\(\_\_\_ = 67\)". Oh! Wait, maybe the angle \(5x + 98\) and \(67\) are supplementary? No, the equation is \(\_\_\_ = 67\). Wait, maybe I misread the diagram. Let's assume that the angle \(5x + 98\) is equal to \(67\) (even though it gives negative, but maybe the diagram is different). Wait, \(5x+98 = 67\), \(5x=67 - 98=-31\), \(x=-6.2\) (invalid). Wait, maybe the angle \(23x - 2 = 67\), \(23x=69\), \(x = 3\). Then \(5x + 98=5*3+98 = 113\), and \(113 + 67=180\), so they are supplementary. But the problem says "\(\_\_\_ = 67\)". Maybe the angle \(5x + 98\) is equal to \(67\) is wrong, and the correct angle is \(23x - 2\). So \(23x-2 = 67\), \(23x=69\), \(x = 3\).

Problem 15:

Step 1: Identify Angle Relationship

The two angles are \((10x - 19)^\circ\) and \((3x + 9)^\circ\). First, we need to see what type of angles they are. If they are vertical angles, corresponding angles, etc. But to check congruence, set them equal: \(10x-19=3x + 9\).

Step 2: Solve for \(x\)

Subtract \(3x\) from both sides: \(7x-19 = 9\). Add 19 to both sides: \(7x=9 + 19=28\). Divide by 7: \(x = 4\). Then check the angles: \(10x-19=10*4 - 19=21\), \(3x + 9=3*4+9 = 21\). So they are congruent. The term for the pair: if they are vertical angles (if the lines are intersecting), or corresponding angles (if parallel lines). But from the diagram, they seem to be vertical angles (formed by two intersecting lines). So the term is vertical angles (or corresponding, but vertical is more likely). They are congruent, and \(x = 4\).

Answers:

Problem 12:

The angle is \(3x\), so \(3x = 135\), \(x=\boldsymbol{45}\).

Problem 13:

\(4x + 12 = 112\), \(x=\boldsymbol{25}\).

Problem 14:

Assuming the angle is \(23x - 2\) (or \(5x + 98\) was a miscalculation, but based on \(23x-2 = 67\)), \(23x - 2 = 67\), \(x=\boldsymbol{3}\).

Problem 15:

The term is vertical angles (or corresponding angles, depending on diagram). They are congruent. Solving \(10x-19 = 3x + 9\) gives \(x=\boldsymbol{4}\).

Answer:

s:

Problem 12:

The angle is \(3x\), so \(3x = 135\), \(x=\boldsymbol{45}\).

Problem 13:

\(4x + 12 = 112\), \(x=\boldsymbol{25}\).

Problem 14:

Assuming the angle is \(23x - 2\) (or \(5x + 98\) was a miscalculation, but based on \(23x-2 = 67\)), \(23x - 2 = 67\), \(x=\boldsymbol{3}\).

Problem 15:

The term is vertical angles (or corresponding angles, depending on diagram). They are congruent. Solving \(10x-19 = 3x + 9\) gives \(x=\boldsymbol{4}\).