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7. Proof & Congruence
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Math II

7.01 Parallel lines and transversals

Lesson
Practice
Lesson

A line that intersects two (or more) other lines is called a transversal.  Angles formed on the same side of a transversal are called consecutive angles.

In the applet below, $f$f is a transversal of $g$g and $h$h. Moving the slider changes the orientation of $g$g. Move the slider in the applet below and notice how the measures of the highlighted angles change:

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When $g$g and $h$h are parallel, the measure of the interior angles add to $180^\circ$180°.  When they aren't parallel, $g$g intersects $h$h forming a third angle, and adding the measures of the two interior angles and this third angle always makes $180^\circ$180°.

 

Consecutive angles

Let's use the applet below to explore the different pairs of consecutive angles formed by parallel lines.  Notice that angles between the parallel lines are called interior angles, and angles outside the parallel lines are exterior angles. 

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Formally, we can say that if two lines are parallel, then consecutive interior angles are supplementary.  We refer to this as the consecutive interior angles postulate. Using properties of linear pairs allows us to prove that consecutive exterior angles are also supplementary, and this is referred to as the consecutive exterior angles theorem.

Postulate or theorem?

The consecutive interior angles postulate is assumed to be true, so we call it a postulate. Other results, such as the consecutive exterior angle theorem, are derived from the postulates and other results, so we call it a theorem.

 

Alternate angles

If we look at a pair of angles on opposite sides of the transversal, we can refer to them as alternate angles.  Alternate angles between the parallel lines are alternate interior angles, while alternate angles outside the parallel lines are alternate exterior angles.

Use the applet to explore the different pairs of alternate angles.  What do you notice about their values?

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We can use properties of linear pairs and vertical angles to prove that if two lines are parallel, then pairs of alternate interior angles are congruent - this is called the alternate interior angles theorem. It's also true that pairs of alternate exterior angles are congruent - this is called the alternate exterior angles theorem.  

 

Corresponding angles

Corresponding angles are the angles that occupy the same relative position at each intersection.  We can slide the slider in the applet below to see all four pairs of corresponding angles.  What relationship do these angles have?

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Using similar results to those we discussed above, we can prove that if two lines are parallel, then the corresponding angles are congruent - this is called the corresponding angles theorem.

 

Angles formed by a transversal to two parallel lines

If two lines are parallel, then:

  1. Consecutive interior angles are supplementary (the consecutive interior angles postulate)
  2. Consecutive exterior angles are supplementary (the consecutive exterior angles theorem)
  3. Alternate interior angles are congruent (the alternate interior angles theorem)
  4. Alternate exterior angles are congruent (the alternate exterior angles theorem)
  5. Corresponding angles are congruent (the corresponding angles theorem)

All angle relationships summarized in a diagram

 

Practice questions

Question 1

Consider the diagram below.

  1. Which relationship describes the marked angles?

    Alternate exterior angles

    A

    Alternate interior angles

    B

    Consecutive interior angles

    C

    Consecutive exterior angles

    D

    Corresponding angles

    E

Question 2

Consider the diagram below.

Two vertical parallel lines are intersected by a slanted transversal line. An angle labeled $\left(x\right)$(x) and $84^\circ$84° is formed in-between the parallel lines. The two labeled angles are on the opposite sides of the transversal. The angle labeled $\left(x\right)$(x) is below the transversal line and has the left vertical line as one of its sides. The angle labeled $84^\circ$84° is above the transversal line and has the right vertical line as one of its sides.

  1. Solve for $x$x.

Question 3

Consider the diagram below.

Four line segments are drawn, forming angles. One segment is horizontal, another segment is vertical and the two other segments are slanted. The horizontal and the vertical segments intersect such that the bottom endpoint of the vertical line intersects with the left endpoint of the horizontal line, forming a right angle as indicated by a small blue square. The two slanted segments slants upwards from left to right and are parallel to each other as indicated by the single arrowhead. The bottom-left endpoint of the first slanted segment intersects with the horizontal and vertical segments, splitting the right angle into two angles. The angle formed between the first slanted segment and the vertical segment measure $26^\circ$26° as labeled. The angle formed between the first slanted segment and the horizontal segment is labeled "$p$pº." To its right is the second slanted segment whose bottom-left endpoint intersects with the horizontal segment. The angle formed between the second slanted segment and the horizontal segment is labeled "$q$qº." The horizontal line acts as a transversal line to the slanted parallel lines such that both the $p$pº and $q$qº angles are at the same position relative to their respective intersection points.

  1. Solve for $p$p.

  2. Solve for $q$q.

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