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Trapezoid Arch and Panel Design Problem - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

When identifying missing angle measurements in trapezoid design and trapezoidal arc problems, knowledge of trapezoid and angle properties is necessary. In a trapezoid design problem, one part of a shape is given while the rest of the shape needs to be imagined. With knowledge of missing angles, it is possible to find the vertex and the other angles by knowing about trapezoid propertiesand corresponding angles.

There are 2 common problems when you're learning about triangles and trapezoids. The first one is the triangle and trapezoid design where you're given one part of a shape so what you have to imagine is that there're going to be adjacent figures that are congruent. So in your mind you have to imagine that there's going to be another kind of design over on this side. There's going to be another congruent design over on this side and that's going to continue for however many designs you have. So the first key step when you're trying to find missing angles is to ask yourself well how many do you have in the total design. The reason why that's important, is because you're looking to find this vertex angle of the isosceles trapezoid. The way you're going to calculate x is you're going to say well if all of these designs are congruent you know that they're going to sum to 360 degrees because it's going to be a circle. So if I want to find just one of these congruent angles I'm going to have to divide by the number of designs that we have. So if I told you that this was a part of a 12 piece design you would find this isosceles vertex angle by dividing 360 by 12.
Once you know that angle then you can find the 2 base and then you know that you have vertical pair of angles excuse me not vertical you have a linear pair of angles, so these 2 angles have to be supplementary and you have corresponding congruent angles, so you know that you can just find the rest of your angles too and so we have 2 markings there and one marking here. So the key to solving this problem is finding that vertex angle by dividing 360 by how many you have.
The second key type of problem that you'll see is the trapezoidal arch problem. In this one you're going to have congruent isosceles trapezoids that make up an arch and you're probably going to be asked to find the angles inside that trapezoid. So what you have to imagine is that this is part of a polygon that extends down below where you can't really see.
The other thing you have to remember is that if there were 1, 2, 3, 4 pieces up here there's going to be 1, 2, 3, 4 pieces down here, so you're going to have n equals 8. You're going to have an 8 sided polygon. So the first thing you're going to have to do to find x is you're going to have to find this interior angle. The reason why is if we zoom in over here if I draw in we'll have angle x and angle x and I know that these have to be congruent because we have congruent isosceles triangles. You're going to find this angle using your polygon angle sum formula which says 180 times n minus 2 divided by n. So in this case you would say that n is 8 and you'd plug it in you'd get some number for your interior angle. Once you know that number then you can say that x plus x plus this number must sum to 360 degrees.
Once you know x then you can find your y by knowing that consecutive angles must be supplementary. So those are the keys to keep in mind when you're taking about a trapezoid arch problem and a triangle with trapezoid design problem.

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