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# Angles in Inscribed Quadrilaterals Worksheets

There are two geometric shapes that when we find them inscribed within a circle can tell us a great deal about the plane it is located, and any other shapes or figures found within it. If you find an inscribed triangle and one of its sides serves as the diameter of the circle it must be a right triangle. This side would also serve as the hypotenuse of the triangle that is formed. Regardless of the size, orientation, or shape of the opposite angle to that hypotenuse it is a right angle. When quadrilaterals are found inscribed, they also have the unique property of form opposite angles that are always supplementary. This means that the measure of opposite corners always adds to equal 180 degrees. In this series of worksheets and lessons we will work with unique inscribed shapes and learn a great deal of information using these properties, when given just basic information.

### Aligned Standard: High School Geometry - HSG-C.A.3

• Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

The first and third sheet uses triangles. A quadrilateral is found in the second one.

• Homework 1 - Since ZY is a diameter of the circle, ∠ Z is a right angle. So ZXY is a right triangle and ∠ X and ∠ Z are complementary.
• Homework 2 - Since TSUV is an inscribed quadrilateral, ∠ T and ∠ V are supplementary.
• Homework 3 - Make these measures equal to the sum of a right angle.

### Practice Worksheets

We play ping pong between using triangles and quadrilaterals here.

• Practice 1 - What are the measures of ∠ Y and ∠ B?
• Practice 2 - Write an equation setting the sum of their measures equal to 90 degrees, and solve for the missing angles.
• Practice 3 - Write an equation setting the sum of their measures equal to 180 when working with quadrilaterals.

### Math Skill Quizzes

Once students feel a little bit of success with these, they pick it up quickly.

• Quiz 1 - Work with two triangle problems and two quadrilateral problems.
• Quiz 2 - Where is that missing number coming from?
• Quiz 3 - I wish I could have got the degree symbols a little clearer, but I did my best.

### What are Angles in Inscribed Right Triangles and Quadrilaterals? We previously discussed the concept of an inscribed geometric shape. That is when a geometric shape finds itself encompassed with in a circle and all of its vertices lie on the circle. You can think of it as what the comic books famously called a perfectly round surrounding force field of sorts.

When this occurs because of the unique properties of circles there is a great deal that we can determine about the inscribed shape especially if it a right triangle or quadrilateral. If you were to form a triangle by extending one side of the triangle as the diameter of the circle, then it will form a right triangle. This can give us a treasure trove of information about the entire situation. Inscribed right triangles have certain properties that help in understanding them. If a right-angled triangle is inscribed in a circle, then its hypotenuse has a diameter of the circle. If one side of the triangle is inscribed in the circle, then the triangle is a right triangle, and the angle that is opposite to the diameter is the right angle.

An inscribed quadrilateral or cyclic quadrilateral is one where all the four vertices of the quadrilateral lie on the circle. The interior angles in the quadrilateral in such a case have a special relationship. Each pair of opposite interior angles are supplementary - that is, they all add up to 180 degrees. When we take into account that a circle covers 360 degrees, we can use those two facts to learn much more about environment.

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