Question

7x-12° 3x+28° type of angle pair these angles are equation x= angle measurements= 3x° 6x° type of angle pair these angles are equation x= angle measurements= challenge question: find x and y. (7y - 20)° (5x - 38)° (3x - 4)°

Explanation:

First Angle Pair (Top Diagram)

Step1: Identify angle pair type

The two angles \(7x - 12^\circ\) and \(3x + 28^\circ\) are alternate exterior angles (since the lines are parallel, alternate exterior angles are equal).

Step2: Set up equation

Since alternate exterior angles are equal, we set \(7x - 12 = 3x + 28\).

Step3: Solve for \(x\)

Subtract \(3x\) from both sides: \(7x - 3x - 12 = 28\) → \(4x - 12 = 28\).
Add 12 to both sides: \(4x = 28 + 12\) → \(4x = 40\).
Divide by 4: \(x = \frac{40}{4} = 10\).

Step4: Find angle measurements

Substitute \(x = 10\) into \(7x - 12\): \(7(10) - 12 = 70 - 12 = 58^\circ\).
Substitute \(x = 10\) into \(3x + 28\): \(3(10) + 28 = 30 + 28 = 58^\circ\).

Second Angle Pair (Middle Diagram)

Step1: Identify angle pair type

The angles \(3x^\circ\) and \(6x^\circ\) are supplementary (they form a linear pair, so their sum is \(180^\circ\)).

Step2: Set up equation

\(3x + 6x = 180\).

Step3: Solve for \(x\)

Combine like terms: \(9x = 180\).
Divide by 9: \(x = \frac{180}{9} = 20\).

Step4: Find angle measurements

\(3x = 3(20) = 60^\circ\), \(6x = 6(20) = 120^\circ\).

Challenge Question (Bottom Diagram)

Step1: Analyze the right angle and linear pair

The angle with the right angle symbol and \((3x - 4)^\circ\) form a linear pair with the angles on line \(l\). Also, the sum of angles in a triangle is \(180^\circ\), but here we have a right angle (\(90^\circ\)) and the other two angles \((5x - 38)^\circ\) and \((7y - 20)^\circ\) should satisfy:
First, the angle \((3x - 4)^\circ\) and the right angle: Wait, actually, the two lines \(l\) and \(m\) are parallel? Wait, no, the triangle has a right angle, and the angles \((5x - 38)^\circ\) and \((3x - 4)^\circ\) and the right angle? Wait, no, let's re - examine. The angle with the right angle symbol and \((3x - 4)^\circ\) are adjacent and form a linear pair? Wait, no, the diagram shows a right angle (square) and \((3x - 4)^\circ\) at the bottom, and the two angles \((7y - 20)^\circ\) and \((5x - 38)^\circ\) at the top with line \(l\) and \(m\) parallel? Wait, actually, since \(l\) and \(m\) are parallel, the alternate interior angles or we can use the fact that the sum of angles around the right angle: Wait, the right angle is \(90^\circ\), so \((3x - 4)+90 + (5x - 38)=180\)? No, wait, the three angles in the triangle - like figure: the right angle, \((3x - 4)^\circ\), and the angle between \(l\) and the triangle? Wait, no, let's correct. The two angles \((5x - 38)^\circ\) and \((3x - 4)^\circ\) and the right angle: Wait, actually, the angle with the right angle symbol is \(90^\circ\), and \((3x - 4)^\circ\) is adjacent to it, so \(90+(3x - 4)+(5x - 38)=180\)? No, that's not right. Wait, the lines \(l\) and \(m\) are parallel, so the angle \((5x - 38)^\circ\) and the angle inside the triangle (with the right angle) are related. Wait, actually, the sum of the angles \((5x - 38)^\circ\), \((3x - 4)^\circ\), and \(90^\circ\) should be \(180^\circ\) (since they are on a straight line? No, \(l\) and \(m\) are parallel, so the transversal forms angles. Wait, let's start over.

The angle \((3x - 4)^\circ\) and the right angle ( \(90^\circ\)) and \((5x - 38)^\circ\): Wait, the three angles \((7y - 20)^\circ\), \((5x - 38)^\circ\), and the right angle (plus \((3x - 4)^\circ\))? No, the diagram has a right angle (square) at the bottom vertex, so the triangle has a right angle. So the sum of the two non - right angles \((7y - 20)^\circ\) and \((5x - 38)^\circ\) should be \(90^\circ\) (since the sum of angles in a triangle is \(180^\circ\), and one angle is \(90^\circ\)). Also, the angle \((3x - 4)^\circ\) and the right angle are adjacent, so \(90+(3x - 4)=180-(5x - 38)\)? No, that's complicated. Wait, actually, the angle \((3x - 4)^\circ\) and \((5x - 38)^\circ\) and the right angle: Wait, the lines \(l\) and \(m\) are parallel, so the alternate interior angles: \((5x - 38)^\circ\) and the angle inside the triangle (the one with the right angle) are equal? No, let's use the fact that the sum of angles on a straight line is \(180^\circ\). The angle at the bottom: \(90^\circ+(3x - 4)^\circ+(5x - 38)^\circ = 180^\circ\)? Wait, no, the three angles \(90^\circ\), \((3x - 4)^\circ\), and \((5x - 38)^\circ\) are on a straight line (since \(l\) and \(m\) are parallel, the transversal creates a linear pair with the triangle's angles). So:

Step1: Set up equation for \(x\)

\(90+(3x - 4)+(5x - 38)=180\)
Combine like terms: \(90 - 4 - 38+3x + 5x=180\)
\(48 + 8x=180\)
Subtract 48 from both sides: \(8x=180 - 48=132\)
Wait, that gives \(x = 16.5\), which is not an integer. Maybe my initial assumption is wrong. Wait, maybe the angle \((3x - 4)^\circ\) and the right angle are complementary? No, the right angle i…

Answer:

s:

First Angle Pair:

• Type of angle pair: Alternate Exterior Angles
• These angles are: Equal
• Equation: \(7x - 12 = 3x+28\)
• \(x=\boldsymbol{10}\)
• Angle Measurements: \(\boldsymbol{58^\circ, 58^\circ}\)

Second Angle Pair:

• Type of angle pair: Linear Pair (Supplementary)
• These angles are: Supplementary
• Equation: \(3x + 6x=180\)
• \(x=\boldsymbol{20}\)
• Angle Measurements: \(\boldsymbol{60^\circ, 120^\circ}\)

Challenge Question (assuming the above calculation):

• \(x=\boldsymbol{16.5}\)
• \(y=\boldsymbol{\approx9.36}\) (or \(y=\frac{131}{14}\) if we keep it as a fraction, since \(7y = 65.5=\frac{131}{2}\), so \(y=\frac{131}{14}\approx9.36\))