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Lines *p* and *q* are parallel. Find the measures of all of the numbered angles. Explain how you found each measure.

No matter which triangle you start with, you can extend the three sides and add a line parallel to one side.

In the following problems, do not use your protractor or anything else to measure the angles. Instead, look at the above picture and use what you know about lines and angles.

a. |
In the picture above, what is m∠1 + m∠2 + m∠3? Explain how you got your answer. |

b. |
In the picture above, ∠1 is the equal in measure to one of the angles in the triangle. Which one? |

c. |
In the picture above, ∠2 is the equal in measure to one of the angles in the triangle. Which one? |

d. |
In the picture above, ∠3 is the equal in measure to one of the angles of the triangle. Which one? |

e. |
Use your answers to questions (a)-(d) to explain why m∠4 + m∠5 + m∠6 is 180°. Explain why this would be true for any triangle, and not just the one pictured. |

A central angle is an angle with its vertex at the center of a circle:

a. |
If the central angle cuts off a quarter-circle, what is the measure of the central angle? |

b. |
If the central angle cuts off a semicircle, what is the measure of the central angle? |

c. |
If the central angle cuts off one-third of a circle, what is the measure of the central angle? |

d. |
Find a general rule for central angles based on how much of the circle they cut off. |

In the below figure, a central angle and an inscribed angle cut off (intercept) the same arc of a circle:

a. |
Make a conjecture: Which of the two angles is larger? |

b. |
How much larger is it? |

c. |
How did you make your decision? |

Problem H1 adapted from Connected Geometry, developed by Educational Development Center, Inc. p. 72. © 2000 Glencoe/McGraw-Hill. Used with permission. www.glencoe.com/sec/math

Problem H2 developed by Educational Development Center, Inc. © 2000 Glencoe/McGraw-Hill. Used with permission. www.glencoe.com/sec/math

**Cuoco, Al; Goldenberg, E. Paul; and Mark, June (December, 1996). Geometric Approaches to Things. In the paper “Habits of Mind: An Organizing Principle for Mathematics Curriculum.” The Journal of Mathematical Behavior, 5, (4), pp. 375-402. **

Reproduced with permission from the publisher. Copyright © 2002 by Elsevier Science, Inc. All rights reserved.

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Geometric Approaches to Things

Reasoning may vary. One way to find the measures of the angles is as follows: m∠5 is 45°, since it is a vertical angle to the given 45° angle. Since angles inside a triangle add up to 180°, m∠7 must be 70°. This means that m∠6 is 110° because m∠6 and m∠7 add up to 180°. Similarly, m∠8 must be 115°. m∠11 is 110°, since it is vertical to∠6. m∠12 is 70°, m∠10 is 115°, and m∠9 is 65°. Since∠12 and∠3 are corresponding, m∠3 is 70°. Similarly, since∠9 and∠4 are corresponding, m∠4 must be 65°. Finally, using vertical angles, m∠1 is 65°, and m∠2 is 70°.

a. |
m1, m2, and m3 add up to 180° because they lie on a straight line. |

b. |
1 is the same as 5 because they are corresponding angles. |

c. |
2 is the same as 6 because they are vertical angles. |

d. |
3 is the same as 4 since they are corresponding angles. |

e. |
Since m1, m2, and m3 add up to 180°, and because they respectively equal 5, 6, and 4, it follows that m4, m5, and m6 add up to 180°. Since the reasoning that led us to the conclusion did not use any specific angle measure in the given picture, it holds in general. Note that this is a proof that shows there are 180° in every triangle! |

**Problem H3**

a. |
The measure is 90° (or 360° / 4). |

b. |
The measure is 180° (or 360° / 2). |

c. |
The measure is 120° (or 360° / 3). |

d. |
If a central angle cuts off an arc of one n-th of the full circle, its measure is 360° / n. |

a. |
The central angle is larger. |

b. |
It is twice as large. |

c. |
Answers will vary. One way to answer the question is to use the software to sketch the figure, then measure the two angles. If you tested this for several cases, it would lead to a conjecture that can be made here: Angles inscribed in circles have measures that are half the measure of the arc they intercept (cut off), which is equivalent to the measure of the central angle. Try it. |