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Geometry
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Lines and Angles

These problems require knowledge of basic definitions of lines and
angles. They may include ideas about parallel and perpendicular lines,
angle relationships, or intersections. They may also include a review
of algebraic topics. A very few problems may require more advanced knowledge of
geometric topics, but most should be appropriate for the beginning
geometry student.
 Related Resources

Interactive resources from our Math Tools project:
Geometry: Points and lines
Geometry: Angles
Geometry: Perpendicular & Parallel Lines
The closest match in our Ask Dr. Math archives:
High School: Euclidean/Plane Geometry
NCTM Standards:
Geometry Standard for Grades 912
Access to these problems requires a Membership.

Analyzing Angles
 Annie Fetter

Geometry, difficulty level 2. Given some information about angles in a picture, figure out some of
the angle measures.
... more>>

Another Ambiguous Angle?
 Annie Fetter

Geometry, difficulty level 3. Given isosceles triangle ABC, with BE perpendicular to AD. If angle ACB is x, what is angle CBE?
... more>>

The Apprehendable Angle
 Annie Fetter

Geometry, difficulty level 2. Triangle ABC is isosceles with a vertex angle (at the top) of 50 degrees. If D is on AC, E is on BC, and AD = BE, what is the measure of angle ADE?
... more>>

Bouncing Billiards
 Annie Fetter

Geometry, difficulty level 1. Help Maria decide the angle that will sink her cue shot after four
bounces.
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Circle Intersections
 Annie Fetter

Geometry, difficulty level 2. What is the maximum number of times that six circles of the same size could intersect?
... more>>

Circles and Tangents
 Annie Fetter

Geometry, difficulty level 3. AOD is a diameter of circle O. B is any point on the circle. At B, a tangent is drawn to the circle. From the center, O, a line is drawn parallel to AB, meeting the tangent at P. Prove that PD is tangent to
the circle.
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Finding a Supplement
 Terry Trotter & Annie Fetter

Geometry, difficulty level 1. Two lines, AB and CD, intersect at point E. Angles AED and CEB have measures that can be expressed as 2x^2 + 3x + 15 and 3x^2 + x  20. Find all of the possible measures of angle BED.
... more>>

Find This Angle
 Annie Fetter

Geometry, difficulty level 3. Given triangle ABC, where E is on AB and DB is perpendicular to AE. AC = BC, angle ACB is twice the measure of angle CBD, and angle DBE = 90 degrees. What is angle CBD?
... more>>

Folding a Piece of Paper in Thirds
 Annie Fetter

Geometry, difficulty level 3. Show that the given folds mark a piece of paper exactly in thirds.
... more>>

How Many Angles?
 Annie Fetter

Geometry, difficulty level 2. Given a picture of two parallel lines and two transversals, how many
different angle measures are represented by the 14 labeled angles?
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Lost at Sea?
 Annie Fetter

Geometry, difficulty level 2. Help Darius and Skip figure out whether they're in the United States
or Canada using the bearings Skip takes.
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Making a Map
 Annie Fetter

Geometry, difficulty level 3. Given a list of relationships between five different objects, explain which ones you could uniquely place on a map.
... more>>

Measuring the Circumference of the Earth
 Annie Fetter

Geometry, difficulty level 3. At our office on the autumnal equinox, the angle formed by the sun's rays and a stick is 40 degrees. If our office is 2749 miles from the equator, what's the circumference of the Earth?
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One Last Bit of Fun
 Annie Fetter

Geometry, difficulty level 3. In parallelogram ABCD, the bisector of angle ABC intersects AD at P. If PD=5, BP=6, and CP=6, what is the value of AB?
... more>>

Partitioning the Plane
 Annie Fetter

Geometry, difficulty level 3. Using six lines, what's the maximum number of regions into which you
can divide a plane?
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Pentagon Puzzle
 Annie Fetter

Geometry, difficulty level 1. Explore the interior angles of pentagons.
... more>>

Picking a Point
 Steve Risberg

Geometry, difficulty level 2. A point is randomly selected from within the rectangle having vertices
at (0,0), (2,0), (2,3) and (0,3). What's the probability that the
xcoordinate of the point will be less than the ycoordinate?
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Plane and Simple?
 Annie Fetter

Geometry, difficulty level 3. Explore splitting the plane into regions using lines.
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Points, Lines, and Planes
 Annie Fetter

Geometry, difficulty level 3. Determine the number of lines defined by 5 or 6 points, and the number of planes defined by 4 or 5 points. Expand this for N points.
... more>>

Precision Paper Folding
 Annie Fetter

Geometry, difficulty level 3. After folding down two corners of a rectangular piece of paper, explain what the resulting angle measure will always be.
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A Rectangle Dissection
 Annie Fetter

Geometry, difficulty level 3. Cut the given rectangle into three pieces as indicated by the dashed
lines. Rearrange the pieces to form a square, and explain why the
resulting shape actually is a square.
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Sighting in a Rifle
 Annie Fetter

Geometry, difficulty level 3. Analyze the given target of four shots, and tell me how far to move the scope on my rifle so that the group of shots will be centered around the bull'seye.
... more>>

Splitting a Plane
 Annie Fetter

Geometry, difficulty level 3. What's the greatest number of regions into which you can split a plane using six lines?
... more>>

Splitting Space
 Annie Fetter

Geometry, difficulty level 4. What's the greatest number of regions into which you can divide space using four planes?
... more>>

Two Times Two Angles
 Annie Fetter

Geometry, difficulty level 3. The measures of two angles are in the ratio 5:3. The measure of the larger angle is 30 greater than half the difference of the angles. Find the measure of each angle.
... more>>

Up Periscope!
 Annie Fetter

Geometry, difficulty level 2. Given a diagram of a broken submarine periscope, explain which angles need to be equal in order to for the image to reach the eyepiece.
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Walking Fast
 Annie Fetter

Geometry, difficulty level 2. A woman with a 28" inseam takes 5 steps for every 4 steps I take. I have a 34" inseam and cover 20 yards in 20 steps. Am I covering more ground because my legs are longer, or because my legs form a larger angle with each step?
... more>>

What's My Angle?
 Annie Fetter

Geometry, difficulty level 2. Given line AE, point C between A and E, ray CB perpendicular to line AE, and point D on the same side of AE as point B. If CD equals DE and angle CDE is x degrees, what is the angle measure of angle BCD?
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