Question

3 - 5 study guide and intervention proving lines parallel
identify parallel lines if two lines in a plane are cut by a transversal and certain conditions are met, then the lines must be parallel.
if

• corresponding angles are congruent,
• alternate exterior angles are congruent,
• consecutive interior angles are supplementary,
• alternate interior angles are congruent, or
• two lines are perpendicular to the same line,

then
the lines are parallel.
example 1: if (mangle1 = mangle2), determine which lines, if any, are parallel. state the postulate or theorem that justifies your answer.
example 2: find (mangle abc) so that (mparallel n).
we can conclude that (mparallel n) if alternate interior angles are congruent.(mangle bad=mangle abc)(3x + 10=6x - 20)(10 = 3x-20)(30 = 3x)(10=x)(mangle abc=6x - 20=6(10)-20) or (40)
exercises
find (x) so that (lparallel m). identify the postulate or theorem you used.

1. ((5x - 5)^{circ}) ((6x - 20)^{circ})
2. ((4x + 20)^{circ}) (6x^{circ})
3. ((3x + 15)^{circ})
4. ((8x + 8)^{circ}) ((9x + 1)^{circ})
5. (2x^{circ}) ((3x - 20)^{circ})
6. ((5x + 20)^{circ}) (70^{circ})

chapter 3
30
glencoe geometry

Explanation:

Step1: Identify angle - relationship for problem 1

Since the angles \((5x - 5)^{\circ}\) and \((6x - 20)^{\circ}\) are alternate exterior angles, and for \(l\parallel m\), alternate exterior angles are congruent. So we set up the equation \(5x-5 = 6x - 20\).

Step2: Solve the equation for \(x\)

Subtract \(5x\) from both sides: \(-5=x - 20\). Then add 20 to both sides: \(x=15\). The postulate used is the Alternate - Exterior Angles Postulate.

Step3: Identify angle - relationship for problem 2

The angles \((4x + 20)^{\circ}\) and \(6x^{\circ}\) are corresponding angles. For \(l\parallel m\), corresponding angles are congruent. So we set up the equation \(4x + 20=6x\).

Step4: Solve the equation for \(x\)

Subtract \(4x\) from both sides: \(20 = 2x\). Divide both sides by 2: \(x = 10\). The postulate used is the Corresponding Angles Postulate.

Step5: Identify angle - relationship for problem 3

The angles \((3x + 15)^{\circ}\) are right - angles (since the lines are perpendicular to the transversal and for \(l\parallel m\), corresponding angles are congruent). If we assume the right - angle is \(90^{\circ}\), then we set up the equation \(3x+15 = 90\).

Step6: Solve the equation for \(x\)

Subtract 15 from both sides: \(3x=75\). Divide both sides by 3: \(x = 25\). The postulate used is the Corresponding Angles Postulate.

Step7: Identify angle - relationship for problem 4

The angles \((9x + 1)^{\circ}\) and \((8x + 8)^{\circ}\) are corresponding angles. For \(l\parallel m\), corresponding angles are congruent. So we set up the equation \(9x + 1=8x + 8\).

Step8: Solve the equation for \(x\)

Subtract \(8x\) from both sides: \(x+1 = 8\). Subtract 1 from both sides: \(x = 7\). The postulate used is the Corresponding Angles Postulate.

Step9: Identify angle - relationship for problem 5

The angles \(2x^{\circ}\) and \((3x - 20)^{\circ}\) are alternate interior angles. For \(l\parallel m\), alternate interior angles are congruent. So we set up the equation \(2x=3x - 20\).

Step10: Solve the equation for \(x\)

Subtract \(2x\) from both sides: \(0=x - 20\). Add 20 to both sides: \(x = 20\). The postulate used is the Alternate - Interior Angles Postulate.

Step11: Identify angle - relationship for problem 6

The angles \((5x + 20)^{\circ}\) and \(70^{\circ}\) are alternate exterior angles. For \(l\parallel m\), alternate exterior angles are congruent. So we set up the equation \(5x+20 = 70\).

Step12: Solve the equation for \(x\)

Subtract 20 from both sides: \(5x=50\). Divide both sides by 5: \(x = 10\). The postulate used is the Alternate - Exterior Angles Postulate.

Answer:

1. \(x = 15\), Alternate - Exterior Angles Postulate
2. \(x = 10\), Corresponding Angles Postulate
3. \(x = 25\), Corresponding Angles Postulate
4. \(x = 7\), Corresponding Angles Postulate
5. \(x = 20\), Alternate - Interior Angles Postulate
6. \(x = 10\), Alternate - Exterior Angles Postulate