High School: Geometry

High School: Geometry

Congruence HSG-CO.A.1

1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Congruence is the overarching theme of the next dozen standards in geometry, so let's tackle that one before running down the rest of the line. Pun intended.

In its most basic sense, "congruency" means being in a state of agreement. No, not with your partner, children, or even your students. In the world of geometry, two figures are called congruent if one can be "carried onto" another.

What does that mean? In brief, it means that all corresponding pairs of sides and angles are equal. They're kind of like "identical twins" without the hassles of sibling rivalry or having to decide how long they'll be dressing alike.

More than just congruence, students should know the precise definitions of an angle, circle, perpendicular line, parallel line, and line segment. After all, they can't work with something if they don't know what it is. That's like trying to kick a field goal without knowing what those big yellow posts are at each end of the field.

Speaking of which—those huge yellow posts at either end? Yeah, the goal posts. Each clearly has two massive poles standing straight up in the air. They are parallel. The one at the bottom of each, which is parallel to the ground but connects them? It's perpendicular to the first two and to the one in the center that reaches into the grass below and holds up the whole thing.

There are several angles here, too. For example, one at each end of the bottom of that square "U" formed by the two parallel posts and their perpendicular bar.

Okay, so what about those undefined notions? Don't worry. It's a lot simpler than it sounds. It all boils down to the fact that we have to start somewhere.

In football, for example, the kicker has to trust that when he sends the ball to the other team, it's going to move in a certain direction based on where and with how much force his foot hits it. He may not realize it, but he trusts the laws of physics.

In geometry, when your students set out to prove or construct something, they have to accept a few concepts that seem perfectly understandable, but are not actually formally defined. They can think of them as the laws of geometry.

In this case, these "notions" are a point, a line, the distance along a line, and the distance around a circular arc. A point is simply a location, represented by a dot. A point has no size, no length, no nothing. Only a location.

A line is simply is straight, infinite length. It's named simply by any two points on the line. It can also be named by a single letter, normally shown in the lower case. The distance along a line is a decided or arbitrary length along the line.

The distance around a circular arc is like the distance along a line, but it's not straight. It's a length equidistant from a point rather than equidistant from a line.

You may also see these called "fundamental" or "primitive" notions. They're the same thing, but try to be consistent. Your students might get confused if you refer to the same thing in three different ways.

Using these undefined notions, we can define an angle as a two lines or line segments or rays that share a common endpoint and a circle as a set of points equidistant from a given point (better known as "the center").

What about all this business with lines? Well, since lines in the real world can't extend forever, we have line segments, which are pieces of the line that run between two endpoints, but don't extend beyond them.

Parallel lines are lines that never intersect. Ever. No matter how far you extend them. Perpendicular lines not only intersect, they meet each other head-on, at a 90° angle.

Students should be able to not only define but also recognize and draw a proper angle, circle, perpendicular line, parallel line, and line segment, and more. These are concepts that most students shouldn't have a problem with. If some of your students are struggling, give lots of definitions and examples so that they master these concepts before moving on.

Drills

  1. You are snowboarding down a slope. Which of the following is consistently perpendicular?

    Correct Answer:

    None of the above

    Answer Explanation:

    To be perpendicular, two lines must form a 90° angle. Your leg is continually adjusting for balance against the board and the slope of the mountain. They may be at 90° for brief moments, but not consistently. The same is true for the angle between your outstretched arms and your body. If any of these were perpendicular throughout your run, you'd never make it down.


  2. Which of the following would be considered a true line, in geometric terms?

    Correct Answer:

    The direction along which an astronomer is looking through a telescope for a new planet thought to be farther from Earth than Neptune

    Answer Explanation:

    Remember that a line is infinite in length. Though long, the lines of (A), (C), and (D) are, in fact, finite. However, the line of sight along which the scientist in (B) is stretching to find this new alleged farthest planet is unknown. Even if that planet were in a galaxy far, far away, the line that the astronomer uses to find it is infinite.


  3. Harry Potter and the Sorcerer's Stone. The night of the big Hollywood premiere (all the way back in 2001, if you can believe it). All the stars are there. So are about a hundred white-hot klieg lights. You know, those massive beams that stretch into the sky for grand openings, major events, and, yes, big celebs. As they move back and forth, they are bound to form which of the following?

    Correct Answer:

    All of the above

    Answer Explanation:

    As they move across the sky, the rays of light will continue to cross each other and form angles. Given the number of lights stretching upward and the various directions in which they move, some will end up parallel to one another at times. That makes (A) and (B) true. These long beams of light are restricted by a point on the each end. (Though the bulb itself is the more obvious end point, it does end somewhere way up high.) So, they are also line segments. That means (D) is the right answer. We're afraid you can't magic yourself out of that one.


  4. Which of the following is an incorrect way to name a line?

    Correct Answer:

    By three points on the line

    Answer Explanation:

    Since two points define a line, that's all we need to name it. Whether the points we choose are the two closest or the two farthest, we've still defined the same line. We can also use a lowercase letter to name a line if points become difficult to deal with. What we should never do is define a line by three points. That's just excessive.


  5. Let's say you opened your laptop and positioned the screen so it's exactly at 90°—a right angle—from your keyboard. Now, let's say you could take the screen and push it all the way down beyond 90°, until the back of the screen is flat against your desk. It looks as if the angle disappeared, but it hasn't. What is the angle called, and what is its measurement?

    Correct Answer:

    Straight angle at 180°

    Answer Explanation:

    Though it may not look like an angle, it is. Because it's also a straight line, its measurement is then double what a right angle would be. That means we have a 2(90°) = 180° angle. The other terms (linear, collinear, and horizontal) are all similar in definition to the word "straight," but they aren't the right words to modify types of angles.


  6. Points that lie on the same straight line are said to be what?

    Correct Answer:

    Collinear

    Answer Explanation:

    The term "horizontal" disregards the possibility of sloped or vertical lines, both of which are definitely a possibility. "Finite" can be used when talking about a line or line segment, but they do not refer to the points along a line, so answer (C) is incorrect. "Ordered" refers to, you guessed it, the "order" in which numbers, shapes, or other objects are placed but have no relation to whether points on a line are straight or not. "Collinear" refers to points that lie in a straight formation along a line because they are all "co-" (which means jointly or mutually) linear.


  7. There are two points on a circle, A and B. Is there a difference between and ?

    Correct Answer:

    No, the two arcs are the same unless stated otherwise

    Answer Explanation:

    This one's a bit tough. Unless stated otherwise, we always assume the arc indicated between two points is the shorter or minor arc. If you do want to indicate the larger or major arc for any reason, all you need to do is add an extra point in the middle of A and B, for instance, J. That will show that J is part of the arc. We could then name our arc .


  8. Assuming we start at point A and travel in an arc to B. Our trajectory is defined by what two measurements?

    Correct Answer:

    Length and angle measure

    Answer Explanation:

    The length measure is the distance along the curved line that is your arc; the angle measure is the angle that the arc makes compared to the center of its circle. Area has to do with the amount space within a two-dimensional boundary, not distance. Slope may seem right, but it's not specific enough for circles. The radius is the distance from the center of a circle to a point on the circle, and circumference goes all the way around the circle, not just an arc length. While sector has to do with circles, it isn't a measurement; it's a pie-shaped part of a circle. The best way to define our trajectory on an arc is with length and angle.


  9. Two distinct lines intersect. Which of the following is NOT definitively true?

    Correct Answer:

    The lines are perpendicular.

    Answer Explanation:

    Two distinct lines means that they aren't right on top of one another. Their intersection, then, is at one location: a point. Since parallel lines never intersect, we know that the two lines can't be parallel either. Since we now have lines that share a point, the spaces in between the lines are angles. Perpendicular lines intersect at 90° angles, but we don't know whether the angles formed are 90° or not. That means we can't say that the lines are perpendicular just yet.


  10. A line segment has endpoints labeled J and K and a collinear point M. Which of the following must be true?

    Correct Answer:

    M is on line JK

    Answer Explanation:

    The term "collinear" means that M is on the same line as J and K, so we automatically know that (C) is right. "Collinear" doesn't necessarily mean that M is in between J and K. Since a line segment is defined by its endpoints, M doesn't necessarily have to be on line segment JK because it doesn't have to be between J and K. The only answer that must be true is (C).


More standards from High School: Geometry - Congruence

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