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## Angles within Inscribed Right Triangles and Quadrilaterals

#### Circles - HSG-C.A.3

What are Angles in Inscribed Right Triangles and Quadrilaterals? Inscribed Right Triangles have certain properties that help in understanding them: 1. If a right-angled triangle is inscribed in a circle, then its hypotenuse has a diameter of the circle. 2. 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.

### Printable Worksheets And Lessons  #### 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 sybols a little clearer, but I did my best.