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Getting ready for congruence

Finding missing triangle angle measures and identifying parallel lines from angle measures on transversals help prepare us to learn about congruence.
Let’s refresh some concepts that will come in handy as you start the congruence unit of the high school geometry course. You’ll see a summary of each concept, along with a sample item, links for more practice, and some info about why you will need the concept for the unit ahead.
This article only includes concepts from earlier courses. There are also concepts within this high school geometry course that are important to understanding congruence. If you have not yet mastered the Intro to Euclidean geometry lesson, the Rigid transformations overview lesson, or the Properties & definitions of transformations lesson, it may be helpful for you to review those before going farther into the unit ahead.

Using angle relationships

What is this, and why do we need it?

When lines intersect, especially when a transversal intersects a pair of parallel lines, the intersections form angles with special relationships. We will use these angle relationships to explain how to construct parallel or perpendicular lines, which in turn help us to bisect angles and line segments. We will also use parallel sides to reveal more properties of parallelograms.

Practice

Problem 1
Exactly two of the lines in the following figure are parallel.
The figure is not to scale.
Four lines, A, B, C, and D, appear parallel. There is a transversal line that cuts through all four of the lines. The lower left angle created by the transversal intersecting line A is ninety-one degrees. The upper right angle created by the transversal intersecting line B is ninety-two degrees. The upper left angle created by the transversal intersecting line C is ninety-one degrees. The lower right angle created by the transversal intersecting line B is eighty-nine degrees
Which two lines are parallel?
Choose 2 answers:
Choose 2 answers:

For more practice, go to Angle relationships with parallel lines.

Where will we use this?

Here are a few of the exercises where reviewing angle relationships might be helpful:

Finding missing angle measures in triangles

What is this, and why do we need it?

The three angle measures in any triangle add up to 180, degree. We'll use congruence along with other concepts, like the fact that the interior angle measures of a triangle sum to 180, degree, to find missing measurements.

Practice

Problem 2
Find the value of x in the triangle shown below.
A right triangle with a sixty-three degree angle and an angle labeled x degrees.
x, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
degree

For more practice, go to Find angles in triangles.

Where will we use this?

Here are the first few exercises where reviewing angle measures in triangles might be helpful.

Want to join the conversation?

  • male robot hal style avatar for user RN
    For the first practice question,how do we know that line a and line d have alternate interior angles and corresponding angles,and why does it not work for line b and line c?
    (14 votes)
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    • mr pink green style avatar for user David Severin
      Any pair of lines have these angles, so the question is are they congruent (if you use alternate interior or alternate exterior or corresponding)?
      or are they supplementary (if you use same side interior or same side exterior)?
      For any one line and the transversal, you can use the fact that adjacent angle are supplementary and vertical angles are congruent.
      Thus, for b and c, alternate interior angles would be 88 degrees (180-92) and 91 degrees which are not congruent, thus not parallel. For corresponding angles, you would get the same results of 88 and 91 degrees which are not congruent.
      (0 votes)
  • male robot hal style avatar for user Sean0618li
    what exactly does congruence mean?
    (3 votes)
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  • marcimus red style avatar for user leonard
    I don't get this. could i get help?
    (3 votes)
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  • aqualine ultimate style avatar for user Jaxson w
    my brain hurts
    (3 votes)
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  • blobby green style avatar for user Daniel
    How do you know which one is parallel and which one is not?
    (0 votes)
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  • blobby green style avatar for user real
    ratatat pum pum ratatatatatatata pum pum boom (im shooting you)
    (1 vote)
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  • male robot hal style avatar for user Sonit
    i am a little confused here. what is congruence exactly??
    (1 vote)
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    • mr pink green style avatar for user David Severin
      Congruence compares two line segments, shapes, or 2-d figures. It two things are exactly the same shape and same size, then they are congruent. If two line segments are congruent, that would mean their lengths would be the same. If two figures are congruent (like congruent triangles), then their angles are the same and their side lengths are the same.
      (1 vote)
  • aqualine ultimate style avatar for user James Boersma
    oranges are blue strawberrys are orange. so by this theory 7+7=90
    (0 votes)
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  • blobby green style avatar for user 358844
    What provides congruence ?
    (0 votes)
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  • blobby green style avatar for user Stella Ness
    How do you know those are parallel and not that there could be more then 2 answers there?
    (0 votes)
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